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1. 0 5 190490 190490 1 0 190480 190480 4 5 2 0 150470 150470 2 5 190460 190460 590 3 5 g legend mS m Figure 31 Meigem long range filter range 40m sill 6 5 mS m with search window radius 40 m fixed varying minimum distance 09 X m 637500 637550 637600 637650 637700 637750 5330300 5330300 2 E 5330200 5330200 2 ki g IB 1 0 0 5 0 0 5330100 5330100 0 5 zi oth 330000 5330000 637500 637550 637600 637650 637700 637750 o p X m a 0 6 m b legend MSU Figure 32 Carnuntum Magnetic susceptibility short range filter range 1m sill 0 005 MSU using a minimum distance of 0 2 m search window radius 0 6m 54 X m 637500 637550 637600 637650 637700 637750 5330300 5330300 2 4 amp 5330200 5330200 gt IB 1 0 0 5 0 0 5330100 5330100 0 5 zi oth 330000 5330000 637500 637550 637600 637650 637700 637750 2 n X m 1 5m d legend MSU Figure 32 Continued Carnuntum Magnetic susceptibility short range filter range 1m sill 0 005 MSU using a minimum distance of 0 2 m search window radius 1 5m 313 short SW 0 6 dist 0 2 short SW 1 dist 0 2 30 20 10 0 10 30 20 10 0 10 rho 0 964 wo C Lo al E 5 ri i 4 ct 4 ct v v 5 a 5 ES zal ln in a e B B e e n ptt ER et j o 8 5 3 lo 9 ts 30 20 10 0 10 30 20 10 0 10 short SW 0 6 dist 0 2 short SW 1 dist 0 2 a 0 6 m vs 2 5 m b 1 m vs 2 5 m shor
2. display very strong variations in the center left corner the region with the water draining channels displays moderately strong variations and the rest of the image shows only small variations which are possibly forced by the method used in stead of being produced by a genuine presence of a short range component If we were to calculate a separate variogram for the different regions we would probably find very large differences The sills of the short range components would obviously be very different but it is very probable that also the ranges themselves would differ because the variability found in the electric conductivity are made by very different processes It is a coincidence that the ranges of the variogram components these processes lead to are more or less the same Nonetheless the variogram itself might be better modeled by two or more short range components 3 3 6 Long range To filter the long range component a search window with radius 100 m which is equal to the range and a minimum distance of 20 m is chosen The resulting filtered image clearly shows that both short and long range components are still present See fig 34a This is to be expected as the chosen minimum distance is much larger than the range of the short range component 4 m Except for the scale of the values of the electric conductivity there is not a large difference between the Ordinary Kriged image and Factorial Kriged long range ima
3. Next to Masking Option select Mask all Duplicates but First and press Run File Selection Sampling Another selection method is to make a random subset of points a Sampling selection The samples are not entirely randomly chosen First we have to choose a data file and if wanted a previous selection Also we have to create a new selection variable Then we have to define a grid and is mostly analogous to the creation of a grid object see B so this will not be elaborated upon Now in each grid cell only one point is chosen Which point is chosen depends on whether you select Random Point point is chosen randomly in the grid cell or Center Point the point closest to the center is chosen If Random Point is chosen there is a choice to set a Seed for the Random Number Generator which will choose different configurations for different numbers Pressing Run will create and save the new selection This procedure is useful not only to create subsets but in particular for the generation of variograms if the dataset is particularly dense Using all data available will ensure that there is a superabundance of available pairs possibly in the range of hundreds of thousands or more for every single lag distance even the very small ones Since a stable experimental variogram needs only approx 100 pairs for each lag distance one can see that this superabundance creates overly long computati
4. and press Add The new model can be changed manually after pressing Edit see B One also has to create a Neighborhood before proceeding Just submit a name next to New File Name and press Add One can also use a previously defined neighborhood The neighborhood contains information on the search window number of samples used minimum distance Now press Edit We arrive at the Neighborhood Definition window The first tab called Sectors allows you to define a Search Ellipsoid The directions of the major axes U and V are found after pressing rotation One can also adapt the directions of the axes If desirable one can also define the minimum amount of samples that are in the search window before an estimation is done You can also define the amount of sectors to be used and the maximum number of samples per sector to be used This last one basically limits the amount of samples to be used for estimation For example if the number of sectors is 1 the amount of samples to be used will always be a maximum of the Optimum Number of Samples per Sector The tab Advanced allows one to tweak the sample selection procedure among which selecting a Minimum Distance Between two Selected Samples see 2 4 1 Now press Run to save the neighborhood and return to the Standard Co Kriging win dow Back on the Standard Co Kriging window one can define whether to do Ordinary Kriging or Factorial Kriging by in t
5. surements namely the ice wedges from the glacial period and the naturally eroded water draining channels Fig 17 The measurements albeit different in nature are equally densely packed 3 3 2 Experimental variogram and model The attained experimental variogram is shown in figure 16 Again there are two com ponents Contrary to the magnetic susceptibility measurements we find a small range component with a range that is much larger than the average minimum distance between datapoints A model is fit with two components The small range component has a range of 4 m and a sill of 0 8 mS m The long range component has a range of 90 m and a sill of 15 3 ol ms m Distance m Figure 16 Carnuntum electric conductivity experimental variogram dots and model line mS m 3 3 3 Ordinary Kriging The Ordinary Kriged estimation of the electric conductivity is shown in figure 17 We find the features that were discussed earlier 3 3 4 Factorial Kriging 3 3 5 Short range To filter the short range component a search window with radius 5 m which is larger than the range of 4 m and a minimum distance of 1 m is chosen The ratio of 1 5 is well below 1 3 as suggested earlier see Ch 3 1 5 The filtered image is shown in figure 18 It clearly shows the geological features of interest here are also some very faint residuals of the driving lanes This is to be expected the driving lanes were not t
6. the range of the component The experimental variogram that results from the filtered image could also mean that of 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 10 20 30 40 50 10 20 30 40 50 Distance m Distance m a short range component mS b long range component mS m Figure 20 Meigem electric conductivity Left The variogram of the short range compo nent image Right The variogram of the long range component image in reality the different components exhibit this spatial variability and that the model we created is in fact incorrect This thought however has not been further explored Very often the original dataset also exhibits a trend which is the spatial variability over a lag distance greater than the range of the largest component This trend is filtered in Factorial Kriging as the local mean Ch 2 2 4 One could reason that subtracting all filtered images corrected by a factor of the components with a limited range from the image obtained from Ordinary Kriging would yield the trend This was attempted with the Meigem dataset The correctness of the trend was evaluated both visually and by its variogram fig 21 When using the factors obtained by dividing the sill of the component in the model by the sill of the filtered image of the component the supposed trend still contains traces of the components The correct factors were found manually which w
7. an anisotropic decaying sine wave as a model for the short range component X m 42 637500 637550 637600 637650 637700 637750 1 0 0 8 0 6 5330300 5330300 0 4 0 2 n 0 0 5330200 5330200 gt 0 2 0 4 5330100 5330100 0 6 0 8 1 0 5330000 5330000 637500 637550 637600 637650 637700 637750 J Figure 24 Carnuntum magnetic susceptibility MSU the remainder of the short range variability after the driving lanes are filtered out X m 43 4 Discussion The goal of this thesis is to provide a method for applying Factorial Kriging on a dataset with multiple spatial variational components In the following we will discuss whether we have achieved our goal First of all when applying Factorial Kriging it is important to use a correct model If a component has a range that is too short the filtered images will partly represent this wrongly modeled component and partly accommodate the real data It has been shown multiple times in this dissertation that the filtered image does not necessarily have exactly the same variogram as the model In Chapter 3 3 5 Carnuntum electric conductivity the filtered image of the short range component is not homogeneous with large amplitude differences in different locations However the model supposes a short range component that does not change over the domain In Chapter 3 4 2 we discussed how the model and the variogram of the filtered images are different even if the component is
8. but due to the large amount of points it is often very impractical Usually the experimental points are combined in a limited amount of lag classes First a class width or lag separation has to be defined This is analogous with creating a zone of two concentric circles around each sample location in real space If another sample resides inside the aforementioned zone it creates a pair that is used for the variogram calculation This is done for all valid pairs Afterwards the values are combined and averaged The average lag distance is h and the average variogram value is y h It is common that the largest lag distance of any class does not exceed half the largest dimension of the surface covering the data This measurement avoids that some points are not represented in the variogram for large lag distances If the experimental variogram is anisotropic or suspected to be one can also define directional classes on top of lag classes y h 3 The directional classes indicate certain directions in the 2D space Similar to lag separation an angular tolerance is chosen samples that reside inside the angular tolerance from the considered directional class form valid pairs that are used for the directional variogram Often a maximum bandwidth is also defined which determines the maximum Cartesian distance from a direction This avoids the risk that a point is chosen that is far from the intended direction for large lag values
9. 3 4 2 Comparison of filtered images and real components 3 4 3 Separating local structures from global patterns 4 Discussion A figures B ISATIS manual AA 46 59 1 Abstract When dealing with dense spatially distributed numerical datasets one frequently ob serves spatial variation acting on different scales simulateously When trying to obtain these components individually the common solution lies in applying a classical filtering method based on the spatial spectrum of the data or on regularization However these methods do not take advantage of the rich spatial variational information contained inside the data In this thesis we try to gain insight and determine a workable method based on the Factorial Kriging equations FK is a promising filtering method based on a statisti cal interpolation method called Kriging that is based on the spatial variability of the data Densely packed datasets that provide full area coverage provide an abundant and highly detailed source of information on the studied location However with extra detail comes extra complexity The first major finding of this thesis is that unlike the Ordinary kriging method FK estimations can not rely on highly localised information alone but requires information from all over the range of the component In order to avoid exponentially growing computational requirements a way to reduce complexity has to be found This solution lies
10. 42 376 15000 10000 5000 3 141592654 12 56637061 28 27433388 50 26548246 78 53981634 113 0973355 153 93804 201 0619298 Figure 19 Computation times vs search window area line with a fitted trend dots on It should be noted that a Kriged estimation itself does not retain the variogram as it is an interpolation method not a simulation method Thus it is expected that there will be some difference between the variograms of the filtered images and their corresponding components The comparison was done for the images of the short and long range component of the Meigem dataset fig 20 We see that the image holds quite well compared to the model for lag distances smaller than the range However for lag distances larger than the range the variability drops which means that if two points lie further apart than a certain threshold value their values are more similar where we would expect their values to be uncorrelated because the distance is larger than the range of the component A first thought is that to attain these images we tend to use a search window that is more or less equal to the range of the component so variability at larger lag distances is not kept into account However the drop in correspondence of the image variogram with the model for lag values larger than the range is not explained by this as the filtered image tends to converge to a stable image as the search window radius grows to and larger than
11. The variogram model Typically the experimental variogram y h is a function which increases from low values near the origin low h to larger values as h increases Often this function stabilizes around a maximum for large h values This maximum is called the sill This sill represents the total variance The lag h at which the sill is reached is h scatterplot Semivanogram o 9 o 9 qu Las Y of 4 oo e 9x 6 i j 9 6 N o Q A oz x T hd 96 2 x th 2h 3h 4h Distance Figure 1 The construction of an experimental variogram called the range which is the maximal extent at which there is a spatial relation For lag values lower than the range there exists a dependence between the observations which increases as observations lie closer to each other At lag values larger than the range the expected difference between observations is maximal and independent from the distance Theoretically the variogram is zero at h 0 However in practice there is always a min imal distance between the two closest points This can cause the variogram to have a positive value even when extrapolated to the origin The value at at the origin is called the nugget effect The nugget represents a random noise term for the measured variable The nugget can be caused by measurement errors but also by sources of variability caused by phenomena that operate at a smaller scale than the sm
12. Xo Z xo m xo M Xa 6 m In this form we do not have to suppose that the mean is stationary For the Ordinary Kriging equation it is supposed that m x is locally stationary which means that we suppose that in the neighborhood of x where measurements contribute to the Kriging estimation 6 the mean is supposed to be constant Combining this assumption with the general equation 6 and supposing that the sum of the weights is equal to 1 X Ded 7 Yields the Ordinary Kriging estimator n Xo 2 Ma Z Xe 8 Equations 7 and 8 together should obey the two criteria of an unbiased and optimal interpolation 3 and 4 The first criterion is easilly checked the second criterion requires the introduction of the variogram After some mathematical manipulations the ordinary point Kriging system in terms of the variogram is attained YI Hp y Xa Xo 0 1 n Xo n Xo p ue 9 Here w is a Lagrange multiplier used to add the condition 7 to the equation Ordinary Kriging is also able to provide a measure of the precision of the interpolation the Ordinary 2 Kriging variance sog n Xo Sox xo Y A yGxa xo Y 10 a 1 It allows to estimate the relative precision of the Kriged estimates The Ordinary Kriging system can be written in matrix form AJA B 11 A AT B 12 This yields the weights A4 which are used to find
13. fairly homo geneous The reason for this difference of behavior is probably that Factorial Kriging is still an estimation method as is Ordinary Kriging which gives optimal estimations but not necessarily an image that displays the component realistically The most important element of using Factorial Kriging is that the search window covers the entire range of the component that we want to filter This not only means adapt ing the search window dimensions but also making sure that the data are preferably homogeneously distributed over the entirety of the range This means using more data or using a larger minimum distance between samples to cover a larger area Next to this when using ISATIS you will want to use a minimum distance that is smaller or equal to the range of all components with a shorter range see Ch 3 1 6 Otherwise the filtered image will retain features of these short range components When using a different soft ware than ISATIS you should first determine how the software chooses the sample of the original datapoints inside the search window If it always chooses the nearest neighbor in tandem with imposing a minimum distance the warning still applies Otherwise no problems are foreseen A possible solution to the problem with the minimum distance apart from reducing it to the range of the shortest range component is to work with subsets of the data randomly selected with an appropriate amount of data t
14. mind when I found something interesting Eef Meerschman my tutor showed great interest into my topic and was always prepared to listen to me and discuss I hope that she enjoys early motherhood and 1 really appre ciate the time we worked together Ellen Van De Vijver did do a great job serving as a substitute tutor for Eef when she was home recovering from childbirth but even before that 1 enjoyed our conversations about geostatistics and faits divers Thanks go to Timothy Saey and Valentijn Van Parys for their work attaining and cleaning up the data I would base my thesis on and especially to Timothy for taking the time to give me background information which makes this thesis much more enjoyable to read or so I hope Now for the hidden person behind the man who did this a special thanks goes towards my girlfriend Nele for being ever so patient with me and making me believe in myself even as I frequently doubted myself I hope I can provide the same support next year when she is working on her thesis Contents 1 Abstract 2 Introduction DA NGO Aa A Be See ede Be th i a YG eh ee 2 2 oS ae Bk ER OE SS ee eee ee eS DIV s 4 2 es 4 amp bcd OOS d doe Ee be 2 9 We A e Dodo OS E ER Does Factorial 2 i 5 4 4 amp ccd bo ds cs or 4 de woe KO eR Boe Ae dons I EO ADD dk d Pu 2 3 1 Data collection methods da 44 k
15. of the short range component With a minimum distance of 5 m there are no signs of this anomalous structure but at a minimum distance of 6 m the structure starts to appear There is a strong indication that for minimum distances larger or comparable to the range of components with a smaller range than the one considered the filtered image will include at least a fraction of these smaller range components avoid this we could choose to use a minimum distance that is significantly larger than the range of the smallest range component To put more force to this proposition consider the ex perimental variograms constructed from the filtered image of the long range component one constructed with a minimum distance of 5 m the other one 12 m fig 8 The one for a minimum distance of 12 m clearly shows on top of the long range component an 21 extra component with a much smaller range where the one for a minimum distance of 5 m shows only a large range component A clear image of a correctly filtered long range component is found in figure 7 b The fig ure shows the influence of the field track running in the north southeast direction There seems to be other sources of variability which have not been identified However since the range of the long range component approaches the 1 2 dimension of the domain it might be difficult to filter out a genuine local source of variability from a trend X m X im 0865 48700 40710 48720 08
16. search window radius of 100 m varying minimum distance with corresponding variogram ov 637500 637550 637600 637650 637700 637750 5330300 5330300 5330200 5330200 5330100 5330100 a 637550 637600 637650 637700 637750 S 637500 X mi Figure 35 Carnuntum magnetic susceptibility MSU the remainder of the variability after the driving lanes are filtered out 98 0 5 0 4 3 0 1 0 0 QU A 0 1 0 2 0 3 0 4 0 5 N A short range B ISATIS manual This manual is meant to be used with Isatis version 2012 1 WinN T64 Different versions could present slightly different or new options New project When starting your first project in ISATIS or simply have the desire to start a new project it should first be created First go to the data file manager File gt Data File Manager Once arrived in the data file manager create a new study by selecting Study gt Create Each project gets its own space on a hard drive Choose an appropriate name Disable Automatic Location on Disk and choose the location of your own choice Choose None for Study for Default Parameters Then click Create If you want to change projects you can do this at any time by selecting in the Data File Manager Study Set And select the Study Name of your desire Warning all unsaved progress of your original project will be discarded Reading in the data ISATIS provide
17. search windows smaller than the range of the considered component we see structures appear that have a smaller range than this component This is not the result of any specific limits on shape that are set up but are purely resulting from the restriction of the area where information is used to make the filter This also means that the screening effect see Chapter 2 2 as used in Ordinary Kriging as a way of restricting the amount of datapoints to be used does not apply in the same way for Factorial Kriging Otherwise a search window with radius 7 m contains hundreds of datapoints which would be more than enough for the outer datapoints to be shielded by the ones closer to the location to be estimated Subsequently we evaluate a series of Factorial Kriged images where the search window is kept fixed at 8 m which is larger than the range and we vary the minimum distance between points that are used for Kriging The pictures fig 28 and the scatterplots fig 29 show that making the minimum distance larger has only a small influence on 20 the image quality Making the minimum distance large enough to become a significant fraction of the search window dimensions leads to the image becoming slightly more grainy The fact that the minimum distance can be pretty large before significant loss of quality ensues is a good sign This means that it might be possible to only use a small subsection of the available data to estimate the filter
18. select X gravity center Unit select length m y Field Type select Y gravity center Unit select length m z Field Type select numeric 32 bits Unit select length m t Field Type select numeric 82 bits Unit select float s ELEC Field Type select numeric 32 bits Unit select float Save the header file Save As Arriving back in the ASCII File Import window under Isatis File select Create a New File if not already selected and make a NEW Points File We arrive at the File and Variable Selector window and if necessary in case of a brand new project or desirable create a New Directory and make a New File Press 0 We arrive back in the ASCII File Import window Now press Import on the bottom of the window The new dataset will be made 60 Exploratory data analysis and the Variogram Statistics gt Exploratory Data Anal YSIS The first thing you will want to do with a new dataset is see how the data look like First we should select suitable variables in the dataset we just made First press Data File at the top of the window A new window called File and Variable Selector opens In the top left part the directory and available selections of your datapoints constructed by an area or subset more on that later are shown If you want to use only a limited selection of your data to be analyzed highlight the selection in the upper left p
19. surface and thus are not present in the variogram see Chapter 2 2 2 When extrapolating the experimen tal variogram to a 0 m lag the nugget effect seems to disappear entirely A model was fitted with 2 components a short range component with a range of 7 m and a sill of 4 5 1 mS m and a long range component with a range of 40 m and a sill of 6 5 mS m 1 The short range component can be identified as the component corresponding with the ice wedges No explanation has been found for the long range component 88690 88700 88710 88720 88740 88740 88750 190540 J 190540 190530 H 190530 190520 190520 190510 190510 190500 190500 y X 190490 190490 190480 190480 190470 190470 190460 190460 10 20 30 30 Distance im a Experimental Variogram dots and Model line b Ordinary Kriging Image mS m7 mS m7 Figure 6 Meigem left Variogram right Ordinary Kriging Image with outlier indicated by the arrow 3 1 3 Ordinary Kriging The Ordinary Kriging method was used to estimate the electrical conductivity on a regular grid The used variogram model was the one described earlier Each point used up to 5 nearest neighboring points in each of the 8 used sectors of space This means that optimally 40 neighboring points are used which is almost always the case The sectors 19 were used to counteract the anisotropy of the sampling density see Ch 2 3 3 The result of the Ord
20. that is nearest to the location to be estimated in the subset X 4 x B x K j TE x X X x X 3 X ae 3 gt Ka z X x de RA A X Ea NA gt x Z Z X x FOU 3 A SEE co ES x X A 2 x X X X 14 A 2 a Y 8 Y x x 8 ue xX E T x x x x EE OC x x 2 X V A x x x x x x x X A A x x d V v x t gt v x X x x A X BE 4 A x st X 068 x P4 X x ol i X x g x 4 z 3 X x x Li x x X X X x 2 x 2 b gt Y as A A x 9 A o x X 4A0 y 1 80 Kx 9 x X x X A x X 3 E X x x E g x x gt X X A x 0 30 E 2 2 i x 2 0230 x S i po X x x C 3 X a gt A X x x x oO A X x ES x d uA i kk 4 Es x 3 2 x X gt Xx x x y x X c y x x x v V a radius of search window b minimum distance Figure 4 The selection of samples to be used for estimation at the rectangle in the center Maximum number of points ISATIS allows the user to define a maximum number of points to be used for the estimation of the variable at a location If more than this maximum amount of points lie within the restrictions search window minimum distance points per segment ISATIS tends to only choose the points nearest to the location to be estimated As we do not want this number to influence the results of this th
21. the estimator Z xo and its variance Sox Xo properties of the Kriging system The Kriging system does not use any information on the measurements directly The measurements are used to construct the variogram y x which are then used in the Kriging equations Aside from this the Kriging estimates only depends on the spatial distribution of the observations The Kriging equations also show us that next to the variogram between the observation points and the point to be estimated the B matrix the variogram between the observation points the A matrix also plays a large role Kriging is an exact interpolator meaning that the estimation Z x at an observation point will be equal to the observed variable Z x and its associated Kriging variance sor X will be equal to zero There are no restrictions to the weights apart from eq 7 It is entirely possible that a weight is negative or larger than 1 This means that Kriging is able to extrapolate above or below extreme values but this is not without its risks it is possible that unphysical estimations are obtained screening effect Kriging incorporates information on the sampling configuration If multiple observation points lie closely together the weights they receive for interpolation will decrease Also the observation point closest to the point of estimation will receive the highest weight and subsequently the points lying behind the closest point will receive a lo
22. the long range component and a minimum distance smaller than the range of the short range component As the requirements on the search window are most important a compromise has to be made regarding the minimum distance We 28 637500 637550 637600 637650 637700 537750 1 0 0 8 0 6 5330300 5330300 0 4 0 2 5330200 5330200 y 0 0 0 2 0 4 5330100 5330100 0 6 0 8 i 0 5330000 5330000 637500 637550 637600 637650 637700 637750 X m N A Figure 13 Filtered image of Carnuntum Magnetic susceptibility short range filter MSU search window radius of 2 5 m minimum distance of 0 2 m 29 took a minimum distance of 8 m and a search window radius of 80 m This implies that we use a fairly large amount of datapoints 54 807 8 x 7 10 x 4 400 to estimate the filtered image at each pixel Again we see the occurrence of the short range component in the image of the long range component fig 14 similar to what happened in Chapter 3 1 6 This is also confirmed by the variogram of the filtered long range component fig 15 It seems to contain a nugget but this is basically just the short range component which has a range too short to be properly displayed The difference between the ranges of the short range and long range components are too large to be able to check the hypothesis developed there namely that reducing the minimum distance to a value smaller than the range of the smallest co
23. u the mean of the variable The squared differences are S E Z x Z x h Z x Z x h dst o 16 with 0 1 1 if k k and 0 otherwise If the process is non stationary the last compo nent Z x with basic variogram g h is unbounded with gradient b Each component Z x can be estimated similarly to Ordinary Kriging 9 The estimation again is a linear combination of the values of its neighbors However to obtain an unbiased estimation the sum of the weights should be equal to 0 n X 375517 Miv Xa 8 49 Xo b g X4 Xo a1 n Xo n Xo Y Ag 0 17 f 1 with x being the Lagrange multiplier for each component The system has to be solved for each component and is of the same complexity as the Ordinary Kriging system Each component will generally yield different A values and using these the filtered image for each component can be constructed Note that we do not need to know the solution of any component except for the considered component to make a filtered image This means that all components can be reconstructed independently If there is a long range trend this does not necessarily obstruct the reconstructions as often we can assume the mean to be locally stationary in the search neighborhood of xo We can rewrite 15 as 5 Z x Y Z x u x 18 k 1 The local mean u x can be considered a long range component To be complete the local mean also has to be estimat
24. window is only relevant for shorter range components see Jaquet 1989 Goovaerts and Webster 1994 2 3 The datasets 2 3 1 Data collection methods A very useful way of taking measurements in a non destructive way is by making use of the properties of the electromagnetic spectrum For frequencies that are not too high electromagnetic waves merely interact with soil by scattering without energy loss Soils with different electromagnetic properties such as permittivity and permeability will scat ter electromagnetic fields differently and will thus result in image contrast when emitted and measured The department of Soil Management of the Faculty of Bio Engineering at UGent has high tech electromagnetic induction EMI measuring devices at its disposal These de vices are able to perform simultaneous measurements of electric conductivity and magnetic 12 susceptibility at different depths The measurements of the electric conductivity and magnetic susceptibility were made using the DUALEM 215 This sensor consists of a 2 41 m long tube and has one trans mitter and four receiver coils at a different spacing see Fig 3 but also with different orientations to perform measurements at different depths The sensor is mounted on a non metallic sled and pulled by an all terrain vehicle across the surface of the terrain of interest In order to try to cover the terrain as fully as possible in order to attain a complete image o
25. 0540 190520 190510 Y m H 190470 190460 Figure 30 Meigem long range filter range 40m sill 6 5 mS m X m 88690 88700 88710 88720 88730 88740 88750 190520 190510 190500 Uu A Y m 190450 190480 190470 150460 X m BB690 88700 88710 88720 88730 88740 88750 A Y m 190540 190530 190520 190510 150500 150450 150480 190540 190540 190520 190510 190500 w AX Y m 150450 150480 150470 190460 X m 88690 88700 88710 88720 88730 88740 88750 190540 190530 190520 190510 190500 190450 190480 190470 190460 b 20 m X m 88690 88700 88710 88720 88730 88740 88750 X m BB690 88700 88710 88720 88730 88740 88750 190540 190540 190520 190510 190500 190490 190480 190470 190460 190530 190520 190510 190500 190490 190480 190470 190460 ur A Y m 190540 190530 190520 190510 X m 190530 190520 190510 190500 190450 190480 190470 190460 190540 190530 190520 190510 190500 150450 150480 150470 150460 u A ur A Y m Y m 8690 88700 88710 88720 88720 88740 88750 E 5 m fixed varying search window radius 02 190540 190530 190520 190510 190480 190470 190460 X m 88690 88700 88710 88720 88730 88740 88750 190530 150520 1905
26. 10 190500 uw A 190450 190480 190470 150460 25 m X m 88690 88700 88710 88720 88730 88740 88750 190540 190540 190520 190510 190480 190470 190460 190530 190520 190510 190500 190490 190480 190470 150460 150540 190540 190520 190510 190500 ur A 150450 150480 150470 190460 uw A 1 Hn 0 5 0 5 0 5 0 0 5 0 25 5 9 a5 wa Doc mS 1 with minimum distance X m X m 88690 88700 88710 88720 88720 88740 88750 88690 88700 88710 88720 88720 88740 88750 150540 J 150540 190540 190540 190530 190530 190530 190530 190520 190520 190520 190520 190510 190510 190510 190510 190500 190500 y 190500 190500 y gt gt 190490 190490 190490 190490 190480 190480 190480 190480 190470 190470 190470 190470 190460 190460 190460 190460 X m X x a 4 m b 5 190540 J 190540 190540 J 190540 190530 190530 190530 190530 190520 190520 190520 190520 190510 190510 190510 190510 190500 190500 y 190500 190500 gt gt gt 190490 190490 190490 190490 190480 190480 190480 190480 150470 150470 150470 190470 190460 190460 190460 190460 X m X m 22690 88700 88710 88720 88730 88740 88750 88690 B8700 88710 88720 88730 88740 88750 3 0 190540 190540 2 5 190540 190540 2 0 1 5 150520 150520 1 0 150510 190510 0 0 190500 y 190500 y gt
27. 190500 5 gt T y 0 5 150450 150450 190450 150450 1 0 1 5 190480 190480 190480 190480 2 0 190470 190470 190470 190470 2 5 3 0 190460 190460 150460 190460 3 5 N A e legend mS m7 Figure 25 Meigem short range filter range 7m sill 4 5 mS m with minimum distance 0 m no min dist varying search window radius given Continued in figure 26 46 X m X m 88690 88700 88710 88720 88730 88740 88750 88690 88700 88710 88720 88720 88740 88750 150540 150540 150540 150530 150530 150530 190520 190520 190520 190510 190510 190510 150500 190500 5 5 190500 5 gt qu 150450 190450 190450 190480 190480 190480 150470 150470 150470 150460 150460 150460 X m a 5 X m X 3 0 150540 150540 235 190530 190530 2 0 15 190520 190520 1 0 190510 190510 0 5 z E s 0 0 5 190600 5 5 190500 5 gt 0 5 190490 190490 1 0 1 5 190480 190480 2 0 190470 190470 2 5 3 0 190460 190460 3 5 N A e legend E mS Figure 26 Continuation of figure 25 Meigem short range filter range 7m sill 4 5 mS m 1 with minimum distance 0 m no min dist varying search window radius given AT Small Range search window Hm Local mean search window zm amall Range search window Lm ur arras amall Range search window fm ug KOPUTH arras 10 5 0 5 10 5 i Small Rang
28. 60 3 0 190540 190540 2 5 190520 190520 20 1 5 190520 190520 1 0 190510 190510 0 5 4 2 0 0 190500 g 7 190500 F gt re 0 5 190490 150450 1 0 190480 190480 1 5 2 0 190470 190470 9 6 190460 180460 ES 3 5 5 a g legend 1 mS Figure 28 Meigem short range filter range 7m sill 4 5 mS m with search window radius 8 m fixed varying minimum distance 50 Small Range search window 8m 10 5 0 5 4 z 5 3 za y c E a u E I ROS s Hu a 3 E 5 c n a 5 z u E z e u v 3 o 5 E 3 d e E Ss 3 3 a 5 E u E a a LI rt Small Range search window Small Range search window im a 0 vs 0 5 m b 0m vs 1 Small Range search window fm Small Range search window 8m 10 5 0 5 351p wz ug we abura 351p wg ug ms abura Small Range sw Om 2m dist Small Range zw Um 3m dist Small Range search window 8m Small Range search window 8m c 0m vs 2 m d 0m vs 3m Small Range search window 8m Small Range sw Um dm dist 351p wp ug we abura wur Figure 29 Scatterplots of Meigem short range filter range 7m sill 4 5 mS m with search window radius 8 m fixed varying minimum distance Comparing minimum distance of 0 m with longer minimum distance ol 190540 190530 190520 190510 190500 Y m 150450 190480 190470 190460 190540 19
29. 720 48740 88750 4865 48700 40710 48720 08720 48740 88750 150540 d 150540 150540 150540 150530 150530 150530 150530 1505 20 1505 20 1505 20 1505 20 150510 150510 150510 150510 5 150500 150500 L 5 150500 150500 set nd 15045 15045 15045 150450 150480 150480 150480 150480 150470 150470 150470 H 150470 150450 e 150450 150450 150450 88700 48710 48720 48730 2E7140 88750 48700 48710 487 20 48730 8750 X m X m a short range b long range Figure 7 Meigem short range component with search window 8m minimum distance 0m left and long range component with search window 40m minimum distance 5m right A possible explanation for the occurrence of the short range component in the image of the large range component is in the inner workings of ISATIS We are able to choose a minimum distance between points to be krigged which is necessary for Factorial Kriging in dense datasets to limit the size of the Kriging system and at the same time allow that the datapoints cover the whole area This last aspect is necessary to correctly filter a com ponent But when choosing a minimum distance ISATIS always includes the datapoint that lies nearest to the location that we want to estimate The next datapoint included in the Factorial Kriging system meets the minimum distance requirement In this case it lies at least 5 m from the first datapoint However this means that in a very dense dataset the first datapo
30. E o x T o o O D i T o 190450 8 b 88675 88725 88775 Easting coordinate m Figure 5 a Aerial photograph showing polygonal crop marks and a former field track with a rectangle delineating the test area and b a closeup of the test area 3 1 1 Data exploration Due to the possible long range spatial trends it is often difficult to identify outliers value that is a local outlier might not so special when compared to values at that are far away which would make it hard to identify when displayed using classical data exploration methods such as a histogram However variogram clouds often make it possible to identify local outliers In the Meigem dataset a few adjacent points are clearly different than others In the upper left corner of figure 6b there is a small scale disruption with a large amplitude This disruption is a clear departure from the overall spatial variational trend However upon closer inspection the disruption is not a discontinuity but a gradual but quick deviation from the local value We decided to leave the outliers also because it might provide some insight in how the Factorial Kriging procedure works 18 3 1 2 Experimental variogram and model The experimental variogram has a smooth variation fig 6a It was constructed using 20 classes with lag distance of 2 m with a maximum lag distance of 40 m Larger lag distances than 40 m exceed half the largest dimension of the measured
31. aken into account when modeling the variogram but this does not mean that they are not present But the fact is that the disturbance caused by the driving lanes are small enough compared to the amplitude of the image to be only a minor nuisance It is expected that the driving lanes can be eliminated almost entirely by correctly modeling the variogram component associated with the driving lanes and choosing a minimum distance smaller than the driv ing lane distances 0 7 m For the reasoning behind this claim we refer to Chapter 3 4 3 32 X qu 537500 637550 637600 637650 637700 637750 5330300 f 5330300 5330200 5330200 5330100 4 5330100 5330000 A 5330000 637500 637550 637600 637650 637700 637750 X m iw X Image jua Figure 17 Carnuntum electric conductivity mS m t Ordinary Kriged image The features of interest are 1 Ice wedges 2 water draining channels 3 aqueduct 33 Image X m 637750 637650 637700 637500 637550 637600 5330300 5330300 330200 330200 5330100 5330100 5330000 5330000 637500 637550 637600 637650 637700 637750 Figure 18 Carnuntum electric conductivity filtered image of short range mS m t search window radius 5 m minimum distance 1 m X m 34 It should also be noted that the amplitude of variation in the filtered image is far from homogeneous In the upper left corner shows the region that contained ice wedges and
32. alitative differences that are hard to distin guish from the variogram because they have a similar range it is sometimes possible to filter them by modeling using a priori knowledge In Chapter 3 4 3 we have been able to separate structures produced by the ruins of a building from artifacts produced by driv ing lanes from the measurements by modeling the driving lanes as one component and applying Factorial Kriging on the remainder of the model while using a search window corresponding with the component range When dealing with features that are to small to observe in the variogram it might be a better practice to approximately model this component guess range isotropy shape of the model and give it a very small sill We have not tried this so it remains to be seen wether this approach works 45 Appendices A figures X m X m 88690 88700 88710 88720 88730 88740 88750 88690 88700 88710 88720 88730 88740 88750 d 7 d 190540 190540 tas 190540 190540 190530 190530 190530 190530 190520 190520 150520 150520 190510 190510 190510 190510 3 lt 190500 190500 g 150500 190500 E 5 1590450 150450 150450 190450 190480 190480 190480 190480 150470 150470 150470 190470 190460 190460 190460 190460 X m X m 3 0 190540 d 150540 190540 d 150540 225 190530 190540 190530 190540 2 0 155 150520 150520 150520 150520 1 0 190510 190510 190510 190510 0 5 ME Ex lt E E D D 190500 190500 y 190500
33. allest sampling distance These basic components of a variogram model are shown in fig 2 Sill Po MUT O 2 2 9 Range 5 E 3 o Z4 Nugget 0 0 5 10 15 20 Laa h Figure 2 Shown Basic models of variograms 1 Spherical 2 Exponential 3 Gaussian The fitting of a theoretical model to the experimental variogram in practice is often done using a select few mathematical functions The mathematical limitations on the possible functions are that the resulting matrix is invertible and positive definite see section 2 2 3 The following contains a list of the most commonly used models a is the range see also fig 2 Nugget effect yo h Co h gt 0 Spherical model h Sh nc e 5 3 4 2a 2 a m h Ci h gt a no c i e 9 a 0 i e 9 e Exponential model Gaussian model Power Ws s que In the exponential and Gaussian models a practical range is used defined by the h value for which the function reaches 95 of the sill Cj If possible the Gaussian model is to be avoided as it causes instability during the inversion of the Kriging matrices see section 2 2 3 It represents a large degree of homogeneity of the variability over short distances The power model is unbound It represents increasing variability with increasing distance and may indicate a spatial trend non stationary mean However it is possible that the spatial dimension of
34. allis UNIVERSITEIT GENT Faculty of Sciences Factorial Kriging of Soil Sensor Images using ISATIS Sam Vanloocke Master dissertation submitted to obtain the degree of Master of Statistical Data Analysis Promotor Prof Dr Ir Marc Van Meirvenne Tutor Ir Eef Meerschman Faculty of Bio Engineering Department of Soil Management Academic year 2011 2012 allis UNIVERSITEIT GENT Faculty of Sciences Factorial Kriging of Soil Sensor Images using ISATIS Sam Vanloocke Master dissertation submitted to obtain the degree of Master of Statistical Data Analysis Promotor Prof Dr Ir Marc Van Meirvenne Tutor Ir Eef Meerschman Faculty of Bio Engineering Department of Soil Management Academic year 2011 2012 The author and the promotor give permission to consult this master dissertation and to copy it or parts of it for personal use Each other use falls under the restrictions of the copyright in particular concerning the obligation to mention explicitly the source when using results of this master dissertation Foreword First of all I would like to thank my promotor Prof Marc Van Meirvenne He was the one who introduced me to and made me enthousiastic about the scientific field of spatial statistics He provided me with an interesting research topic where I could contribute due to its novelty He also helped me with providing much insight into the topic while still keeping an open
35. art of the window The upper right part of the window shows the available variables Pressing the variables puts them in a stack of variables shown in the bottom of the window Go to the map you created and select at least the variable ELEC then press OK Back in the Exploratory Data Analysis window the location of your chosen variables are shown on top along with the chosen selection and below that the chosen variables are shown to be selected Right next to this you can press Statistics to review some basic statistics on the variable In the bottom 8 buttons are shown to be pressed Hover over them for a moment to see what they do Evidently you need to have at least 2 variables selected to make a scatterplot The the analysis tools can be adapted to personal choices by following Application Calculation Parameters in the window of the graphic This graphic window is obtained by first pressing their icon with a variable selected waiting for the calculation to end or if you are impatient aborting the calculation by going to the main window mentioning the version number of ISATIS and pressing the button showing red stop sign which becomes a yellow smiley face if no calculations are done Changing the graphical parameters is extremely useful for the creation of the experi mental variogram Arriving in the Variogram Calculation Parameters window we see some options that look familiar We can load previous parame
36. between an observation Z x at a location x and a second observation Z x h at x h depends only on h This means that the variance is a function only of h Var Z x h Z x E Z x h Z x 27 h 1 y h is the semi variance of variogram function Here the vector h represents the lag or spatial distance between two observations An extra assumption has to be made regarding the mean E Z x h Z x 0 2 Or the mean m is supposed to be constant Together these two assumptions 2 and 1 form the intrinsic hypothesis The experimental variogram In the simple case that the variable that is considered is correlated equally in all directions isotropical correlations the variogram function is one dimensional From 1 we find that the variogram at a certain lag distance h can be estimated by averaging the mean squared difference of all pairs of observation at a distance h from each other This can only be done in the case that the samples are taken from a regular grid In an irregular grid we will find for each lag distance that is present in the sample only one corresponding pair In that case we can resort to indirect methods using a variogram cloud or using lag classes 2 against the lag distance h It is often used in explorative data analysis to identify pairs The variogram cloud is simply the plot of the experimental values of of extreme differences One can use a variogram cloud to fit a model directly
37. dist 0 m Computation times compared with search window radius and area Radius Surface area computation time m mi s The surface area is proportional to the amount of datapoints contained within Expos ing the relationship between the surface area and the computation time should expose the complexity of the matrix inversion operation Of multiple trends for the computational complexity of the matrix inversion process the one corresponding with the Coppersmith Winograd algorithm with a complexity order of O n 8 did fit best fig 19 This means that the computation time increase with a power of 2 376 of increasing amount of data points used for the estimation This also means that doubling the search window radius or halving the minimum distance will multiply the computation time with 2 3 27 3 4 2 Comparison of filtered images and real components The results of the Factorial Kriging method are encouraging One could ask oneself how ever whether the resulting images correspond with the real life patterns we expect them to represent This can be checked by calculating the variogram of the images and comparing them with the variogram of the component they represent If they do not correspond the images have a different variational configuration than the model that the filter was based 36 CPU time vs search window area 45000 40000 35000 30000 25000 measure CPU time s 20000 B Fit
38. draining channels in EC There are traces from an old Roman aqueduct running across the domain both EC and MS There are graves from the period that the gladiator school was in use south of the aqueduct in MS In this chapter the measurements of the the magnetic susceptibility is used and it is composed of only one measurement MS2HCP opposed to the electric conductivity measurements see Ch 3 3 1 Bath Campus Garden Cellblocks for Gladiators Cemetery Figure 10 Closeup of the Roman gladiator school at the site of Carnuntum 3 2 1 Data exploration This dataset is very dense There is a distinction between the driving direction and the direction perpendicular to the driving direction The driving lanes lie approximately parallel to each other at a distance of approximately 0 7 m On the driving lanes the 20 measurement points are very densely packed with a distance between nearest neighbor of approximately 0 2 m and at some locations even more dense 3 2 2 Experimental variogram and model As the dataset contains so many measurements it takes a very long time to calculate the experimental variogram As explained in chapter 2 2 the amount of pairs used to calculate a value of the experimental variogram should be at least 100 However due to the density of the dataset the amount of pairs for each lag class is much larger than 100 even if a large amount of lag classes are used Therefore it is sensible to use a subs
39. e Kriging system leads to the overrepresentation of information at a very small lag distance in the filter We use more information than expected in the immediate surroundings of the point to be estimated and less from further locations This will lead to the artifacts previously seen This phenomenon is also clearly shown in the variogram of the long range component fig 8b Unlike what we would expect the variogram also contains part of the short range component T he variogram of a well filtered image should not contain a component different from the one to be filtered 29 Figure 9 Basic problem while using a minimum distance area 1 almost always contains one used observation area 2 all observations almost never uses any observation search window 3 contains 24 3 2 Testcase 1 To test the method a different dataset is used The site of Carnuntum is set in Austria at the location of the ruins of a Roman gladiator school 7 The dataset contains many different measurements among which magnetic susceptibility of the soil MS fig 12 which is discussed in this chapter and the electric conductivity EC fig 17 which is discussed in the next chapter Apart from the gladiator school appearing in MS see fig 10 some other interesting features are present in the datasets These include ice wedges in EC from the last ice age comparable to the ones found in the Meigem dataset and a series of eroded water
40. e button again allows you to edit existing polygons move add and delete their vertices Scrolling the middle mouse button will zoom in and out To save the polygons select Application then Save and Run Now that you have created the polygon file this has to be applied to create a selection in File Selection From Polygons Select the polygon file that was just created create a new output file based on the object that you want to apply the selection on Select Sample selected if inside 1 polygon which mean that the selection variable becomes one if it is inside at least one polygon that you selected zero if not Then select the polygons which you want it to apply to Then press Run This selection is particularly useful to refine a grid if the area that covers the sample is particularly irregular This means that for OK or FK estimations will not be made outside areas of interest reducing the computation time by an order of the fraction of removed area ISATIS has many more selection tools many of them worthwhile for a Geostatistician For more information we refer you to the ISATIS Manual and Case studies book Images Display New Page First you have to submit a Page Name One can save complex representations of the dataset as is clearly shown in the Contents tab below We will limit us to simple images First go to the View Label tab and deselect Automatic from One Item Thi
41. e did by checking the variogram and controlling the trend visually These factors are somewhat different than the calculated ones Also we were still unable to filter out the short range component from certain regions of the image This might imply that the amplitude of the filtered image of the short range component is incorrect in certain areas of the domain Or that the correct factor is in fact not a constant across the entire domain but a function of location 38 X m X m 88690 88700 88710 88720 88730 88740 88750 88690 88700 88710 88720 88730 88740 88750 46 0 150540 4 150540 150540 j 190540 45 0 190540 H 190540 190530 H 190520 44 0 190520 190520 150520 150520 43 0 150510 190510 190510 190510 42 0 lt E 190500 190500 lt E 190500 190500 lt 3 Hao 150450 1590450 1590450 190490 40 0 190480 190480 190480 190480 35 0 35 0 190470 150470 1590470 150470 38 0 38 0 190460 190460 190460 190460 27 0 27 0 Jwa 1 b correct trend mS m EC 0 10 20 30 40 50 EC 1 25 1 25 1 00 1 00 0 75 0 75 0 50 0 50 0 25 0 25 v 10 20 30 30 50 ED sis 10 20 30 30 50 ED Distance m Distance m c mS m d mS m Figure 21 Meigem electric conductivity naive trend OK SR LR Left manually optimized trend OK 2 28R 2 3LR Right 39 3 4 3 Separating local structures from global patterns When modeling the Carnuntum magnetic susceptibility dataset Ch 3 2 somet
42. e search window Hm Local mean search window zm a 1m vs 8 m b 2m vs8 m Loral mean search window dm Local mean search window dm 10 5 p E amall Range search window fm ug KOPUTH arras Burg amall Range search window fm ug HMopurh arras abupy 10 5 g Local mean search window dm d 4mvs8m Figure 27 Scatterplots of Meigem short range filter with minimum distance 0 m varying search window radius Comparing search window radius of 8 m with search window up to 4 m 45 Local mean search window im Local mean search window mall Range search window fm amall Range search window fm ug KOPUTH arras ug arras pT Pure Local mean search window 5m Local mean search window e 5mvs 8m f 6 mvs 8m Local mean search window 7m 10 5 D 555 H En mall Range search window fm ug HMopiurh arras pT Rue Local mean search window Ta g 7mvs8m Figure 27 Continued scatterplots of Meigem short range filter with minimum distance 0 m varying search window radius Comparing search window radius of 8 m with search window 5 m 7 m 40 X m X m 190540 190540 190530 190530 190520 150520 190510 190510 150500 150500 gt 8 150450 150450 150480 150480 150470 150470 150460 190460 a 0m X m 150540 150530 150520 190510 g 7 190500 g gt gt 150450 150480 150470 1904
43. ed the statistic 13 will be an unbiased estimator The experimental variogram as a function of lag distance is an estimator of the spatial variability of the measured phenomenon As the dataset is very large the the amount of available couples for each lag class will be very large This makes it possible to refine the experimental variogram by using a smaller lag distance and more lag classes If the lag distance is small enough the experimental variogram will be a good approximation of the actual spatial variance at this distance if the assumptions of 2nd order stationarity apply If the the spatial variability is a smooth function of lag distance the obtained experimental variograms will also behave very smoothly This will be almost always the case when working with natural data as natural phenomena seldomly are discontinuous The high density of the data also means that the nugget of the experimental variogram will be a good approximation of the real nugget effect except if phenomena working on a comparable or even smaller scale than the minimal lag distance are important for the measured variable The nugget effect is a measurement of the noise of a variable Since natural phenomena are rarely noisy a very dense dataset will often show a very small nugget effect The high density of the datasets will lead to problems regarding the inversion of the Kriging system 12 Since it is advised to use samples that complet
44. ed as a linear combination of the neighbors The sum 11 of the weights is 1 to obtain unbiasedness This can be done by Kriging n X 3 519 Agy Ka Xg v Ko 0 0 1 n Xo n Xo S 3 21 19 f 1 To estimate a spatial component the neighborhood should cover at least 1ts range How ever in dense dataset there are so many data inside this neighborhood that only a small portion of them are used in practice due to the screening effect The biggest obstacle associated with solving the Factorial Kriging equations is the inversion of the matrix containing the interactions between the neighbors and its complexity rises quickly with increasing neighbors Also including a high amount of data tends to make the inversion of this matrix less stable Furthermore distant points are generally shielded by the nearest observations so that the actual range of the neighborhood is smaller than the specified neighborhood This means that the range of the estimated component will be smaller than the range determined by the earlier analyses construction of variogram Some so lutions to this problem have been proposed When working with data on a regular grid Galli et al 1984 proposed to use a smaller selection of the data only including every second or fourth point This ensured that the data will be selected from all over the search neighborhood Others have proposed to add the long range component and local mean together so that the search
45. ed by the range of the artifacts and the structures of interest If the range of the artifacts to subtract is very different from the range of the structures of interest no trick has to be applied and the component in the variogram corresponding with the structures of interest should be modeled separately instead If the artifacts are irregular and show no structure unlike to the straight and equidistant parallel driving lines we deal with it might be impossible to separate arti facts from useful structures in a similar matter because the artifacts can not be modeled correctly However it might be possible to try to model or estimate the structures of interest using properties that distinguish them from the artifacts If the artifacts show regularity but are different in different subareas such as change of driving direction one could image the subareas separately Each subarea receives its own variogram model The aspects that sets this method apart from other filtering methods is that Facto rial Kriging uses almost exclusively statistical information on spatial correlation only including a limited amount of a priori information Al 637500 637550 637600 637650 637700 637750 2 0 1 5 1 0 5330300 5330300 0 5 0 0 5330200 5330200 m 0 5 1 0 5330100 5330100 1 5 2 0 5330000 5330000 637500 637550 637600 637650 637700 637750 pa Figure 23 Carnuntum magnetic susceptibility MSU filtering the driving lanes by using
46. ed component at one location This could reduce computation times drastically There seems to be no clear and distinct value at which the minimum distance still yields a good image but when increased the image quality deteriorates quickly This enables the user to adapt the minimum distance according to his needs It appears that using a minimum distance that is a large fraction such as 1 3 of the range of the considered component still results in a good image A clear image of the filtered short range component is found in figure It shows the polygonal network caused by the ice wedges without much influence of other sources 3 1 6 Long range First the minimum distance is kept constant at 5 m and the search window radius is in creased in steps fig 30 Again we see that the structures we expect to see only show up when the search window radius becomes equal or larger than the range of the component 40 m Subsequently the search window radius is kept constant at 40 m the range of the long range component and the minimum distance between measurements used is increased fig 31 We see that not only the image becomes more grainy as the minimum distance increases but also that a different structure appears besides the long range component This structure has a much smaller scale It is strikingly similar to the short range com ponent Furthermore it seems to appear when the growing minimum distance approaches the range
47. ely cover the range of the variogram to calculate the solution of this matrix equation the matrix tends to become very big The inversion of a matrix tends to be very computer intensive with its growing size so a large part of this thesis will be dedicated at attempts to mitigate this effect 2 3 3 Anisotropy of density The density of the data is composed of two components as can be seen in figure 4 First there is the distance between two consecutive measurements inside the same driving lane This is determined by the time between two measurements constant and the vehicle velocity which can fluctuate depending on the vehicle and and driver The average distance inside a driving lane tends to be of the order of 10cm The distance between nearby measurements on separate driving lanes is usually around 70cm Of course these distances can be regulated to fit the requirements of the soil survey The difference between the information density inside the driving lanes and between consecutive driving lanes leads to an anisotropy of information density where generally the information density is highest parallel to the driving lanes and least dense perpendicular to the driving lanes This might lead to complications for Factorial Kriging of a short 14 range component if its range is of the order of the intra driving lane distance In that case the amount of information used in the direction parallel to the driving la
48. ent Now press Back in the tem contents for Raster window you can Display Current Item Back in the Contents window press the Legend button to add the legend to the image which will be located at the right side Now press Display again to show the image with legend If desired the legend can be relocated by dragging and dropping it both times press left button You can save the image as a file by going to Management Print in the My PageName window In the Print window check Print to File press Output Format and select a format that you can work with ex PNG can be read by most Ordinary image display software Press Output File Name and select the location where you want to save the image and submit a file name This concludes a basic tutorial which shows how to get to most of the results which were obtained for this thesis It should also put new ISATIS users on the way to become proficient in the use of this powerful software package Many more aspects of ISATIS were not touched upon but we are sure that with some patience so much more can be done 05 References El 2 M Van Meirvenne Geostatistics 1000256 Geostatistics UGent 2011 unpublished P Goovaerts Geostatistical tools for characterizing the spatial variability of microbi ological and physico chemical soil properties Biol Fertil Soils Springer Verlag 1998 27 p 315 334 R Webster M A Oliver Geostat
49. ery different scales The emergence of more and more detailed datasets that manifest variability on different scales has necessated the investigation of nested variation Facto rial Kriging is a method based on Kriging estimation that utilizes this nested variation in order to produce filtered images that represent the physical property caused by only one component at a time 2 1 Goals The main goal of this thesis is to provide a usable method for Factorial Kriging for high density datasets Of primary importance is to find out what is the optimal configuration to calculate the Factorial Kriged filtered images Apart from the correctness of the images we also want to find out what are the main restrictions for this method and other aspects to be aware of As a researcher often has to cope with time restrictions this the sis should provide the user with ways to reduce the computation time but minimize their drawbacks and some insight on how changing the configuration changes the computation time The research will be conducted principally with the ISATIS software package a powerful commercial geostatistical software package and one of the only software packages that provide a Factorial Kriging routine One of the end results of this thesis is a user manual for Factorial Kriging for high density datasets when using ISATIS so the reader can immediately test the method 2 2 Theory 2 2 1 Spatial statistics Geostatistics is a branch of s
50. esis it is always set very high 10000 unless stated otherwise Images make an image of an estimated component ISATIS requires a regular grid with quantitative values At each location of this regular grid the the variable to be 16 imaged should be estimated One can avoid estimation at certain points or regions by making use of selection variables 2 5 Research procedure To achieve the goal of finding an optimal configuration for Factorial Kriging we will execute a parameter sweep varying both the search window dimensions and the minimum distance First we must define a regular grid which allows us to plot an image of the estimated variable A satisfactory configuration will be achieved when the estimated image stabilizes thus if changing the parameter does not change the image qualitatively This can also be evaluated by making a scatterplot of subsequent estimations and seeing if the points are located on the first bisection line which indicates a perfect one on one match 3 Methods and Results 3 1 Initial method development The first dataset was collected on a rectangular field in Meigem fig 5 The dataset maps the electric conductivity up to a certain depth fig 6b The measurements contains a network of polygonal crop marks caused by ice wedges during the glacial period and a former field track running in a north southeast direction The data were attained using an EMI sensor 6 190550
51. et of the data while making sure that this dataset contains enough pairs in lag classes that are of our interest The subset was made by dividing the 2D space in small rectangles and randomly choosing one datapoint in each rectangle In our case the rectangles had a dimension of 1 m by 1 m The resulting subset 59455 datapoints was used for the calculation of the experimental variogram with 100 lag classes of 1 m and another experimental variogram with 50 lag classes of 0 2 m fig 11 The variogram behaves very smoothly Two components were fitted one spherical component with a very small range of 1 m and a sill of 0 005 MSU magnetic susceptibility units which we will call the short range component and one spherical component with a longer range of 80 m and a sill of 0 12 MSU The reason why we chose to include the short range component is because we expect the structures of interest Roman gladiator school building aqueduct graves to exhibit variability exclusively on a very small scale too Distance m D 10 20 30 40 50 7D BD 100 AAG AAG MPAGO TIPA MPAGO TIPA Variogram Warioqram TYH TYH Figure 11 Experimenta Variogram dots and model line of the Carnuntum magnetic susceptibility dataset MSU right zoom for small lag values 26 3 2 3 Ordinary Kriging The Ordinary Kriged images show variation at a large range and local distortions that can be identified as the co
52. f the terrain the vehicle is driving in close parallel lanes back and forth with a distance that is as constant as possible Obviously it is not always possible to drive straight forward because of obstructions like trees However this shouldn t necessarily lead to problems when analyzing the data It just means that the information density at the location of the three will be less dense A GPS was used to georeference the measurements with an accuracy of approximately 0 10 m The GPS and sensor are connected to a field computer to gather their information Measurements are made at constant time intervals with a frequency of 8 10Hz 5 A E y RUE y ARA z n is rc Mart M T Me A oer ix x dz D oO owl 7 vds 12 ue g a NS Me AS EANA ate Mir kon eq t XE d LEM Ve k a PE A de A o 4 4 n a LA QC P at ia P e A 5 4 i o ART AS E E m 3 m 1 1d 2 21 Figure 3 Transmitter and receiver dipole orientations and coil spacings of DUALEM 215 2 3 2 Dense data The fact that we are working with a dataset that fully covers the terrain of interest means that the data are very densely packed This abundance of information leads to a few complications and theoretical consequences As a consequence of the central limit theorem a statistic converges asymptotically to a certain value as the sample size increases If the used statistic is unbias
53. ge Reducing the minimum distance to 3 m See fig 34b shows a clear reduction in the amplitude of the short range component However we have not been able to filter out this short range component entirely Further reducing the minimum distance might improve the image but the process becomes very time consuming We suspect that the short range component might be better represented by more than one localized components The structures in the upper left corner might be better represented with a component with a shorter range than 3m When comparing the variogram of the long range component with minimum distance 20 m and 3 m it is clear that the short range component becomes less outspoken when the minimum distance becomes small Comparing both filtered images the small range components looks more blurred when the minimum distance 15 small 30 3 4 New insights and theoretical considerations 3 4 1 Computation times As mentioned before the computation times for matrix inversion are one of the largest obstacles to do Factorial Kriging successfully During the first Factorial Kriging estima tions on the Meigem dataset when the short range component was estimated with a fixed minimum distance of 0 m and increasing search window radius see Ch 3 1 5 the computation times were recorded The computation times the radii and surface area s of the search window are written down in Table 1 lable 1 Meigem EC short range component min
54. he window that is opened by pressing Special Model Options To perform OK just uncheck any boxes To perform Factorial Kriging check 64 the box next to Factorial Kriging Extra options will appear In the box on the left one can highlight the component for which a Factorial Kriging estimation should be made When the selection is made just press Apply to save and return to the Standard Co Kriging window You can test the setup by pressing the Test button This will at first give an overview of the locations of the input variable You can close in by scrolling When clicking on the variable once you see the output grid appearing no selection applied When clicking again on a location of your choice ISATIS takes the time to calculate the OK or FK estimation at that location and the weights of the used neighboring samples The used samples are highlighted and their corresponding weights are shown The used search win dow and if applicable sectors are also shown This view is very useful to determine if the Kriging procedure is set up correctly When working with very dense datasets and no minimum distance is instituted its is possible datapoints are located so closely to eachother that ISATIS will run into problems when inverting the Kriging matrix a so called inversion problem Normally ISATIS will stop the Kriging procedure when running into such problems but ISATIS can be forced to still proceed by checking the b
55. hing pe culiar appeared T he configuration that was used for the short range component was also applied to the long range component search window of 1 5 minimum distance of 0 2 m The resulting image still only shows short range variable structures as 15 expected only info over a short lag distance is used but the image showed the structures much more clearly The idea rose that the unwanted artifacts driving lanes can be separated from the desired structures by modeling the variogram of these artifacts as closely as pos sible and extracting the structures as a residual Since the driving lanes are parallel and approximately equidistant and seem to show an alternating perturbation possibly due to alternating driving directions it makes sense to model the perturbation as an attenuating wave sinusoidal in the direction perpendicular to the driving direction and as a constant in the driving direction We have to make sure that the period of the attenuated wave is approximately twice the distance between driving lanes There is no clear indication how fast the function should attenuate so a guess has to be made The way to model this in ISATIS is a little tricky and means some trial and error while trying to measure period and let this component fit the experimental variogram as close as possible It might be possible to deduce the model from the experimental variogram using a lot of small lag steps but we decided to estima
56. in using only a small subset of the data within the range of the compo nent We found that we can reduce the comutation times greatly by forcing a minimum distance between datapoints to be used for the estimation a procedure that is unique for the software package ISATIS that was used in this thesis This minimum distance be chosen a decent fraction of the range of the component while still retaining a decent quality of the filtered image Unfortunately a limiting requirement seems to be that the minimum distance should be smaller than the range of all other components with a smaller range We found some evidence that this limit is due to the inner workings of the implementation of the minimum distance in ISATIS but no defenite proof has been found Finally the Factorial Kriging procedure was used to filter localised structures from global patterns even as the two have similar ranges and are thus hard to separate in the variogram The pattern was modeled and the structures were filtered as the residual of the spatial variability within the considered range 2 Introduction It is quite common that spatial variation of a property in the environment occurs on different scales simultaneously The physical processes that bring about this variation are often caused by different sources For example soil conditions and their variability arise from plant roots agricultural activity and geological activity all are processes that work on v
57. inary Kriging estimation can be seen in figure 6b 3 1 4 Factorial Kriging The model contains two components working at a short range of 7 m and one working on a longer range of 40 m The information that is relevant for the long range component covers a much larger area around the location to be estimated than the small range component As the time to invert the matrix A rises very quickly with the amount of points considered we first execute the parameter sweep for the short range component and try to implement the learnt lessons for the long range component 3 1 5 Short range First the search window is varied keeping the minimum distance between samples used for the estimation at zero This means that all datapoints inside the search window will be included for the calculation of the filtered component image We proceeded by Krig ing the first variogram component for circular search windows with different diameter lengths in steps of 1 m from 1 m up to 8 m In figures 25 and 26 it can be seen that the image of the short range component stabilizes when the search window reaches the range of the short range component This is further illustrated in figure 27 which shows the scatterplots of consecutive images As the search window increases the dots are located closer to the bisection line This means that images tend to a stable configuration with increasing search window If we look at the filtered images of the small range component for
58. int to be chosen will almost always be located very close to the location that we try to estimate Subsequently we will find a circular region with a radius thats a little bit smaller than the minimum distance that almost never contains a datapoint that 22 0 6 0 5 4 3 0 0 2 mel 0 0 0 0 10 20 30 40 50 Distance m Distance m a mindist 5 m b mindist 12 m Figure 8 Variogram of the filtered images of the long range component for different minimum distance mS m is included in the Kriging system Beyond this region the distribution of included points will remain irregular but stabilizes We refer to figure 9 for a graphical representation of the problem If we would construct a density profile of the chosen datapoints relative to the point to be estimated we will find a density that is very high for very small distances up to approximately the average distance between datapoints then a region where the density drops to 0 up until approximately the minimum distance and finally rises to sta bilize at a fixed value Optimally a homogeneous density profile is needed as the chance for each datapoint to be used for estimation within the search window should be equal This is an unbiased approximation of the ideal instance that the chance for a datapoint to be included is 10096 so each datapoint is included The fact that ISATIS always includes the nearest datapoint in th
59. istics for Environmental Scientists Wiley Chich ester 2nd Edition 2007 P Goovaerts R Webster Scale dependent correlation between topsoil copper and cobalt concentrations in Scotland European Journal of Soil Science 1994 45 p 19 95 T Saey Integrating multiple signals of an electromagnetic induction sensor to map contrasting soil layers and locate buried features Ph D thesis UGent Belgium 2011 E Meerschman M Van Meirvenne P De Smedt T Saey M M Islam F Meeuws E Van De Vijver G Ghysels Imaging a polygonal network of ice wedge casts with an electromagnetic induction sensor Soil Science Society of America Journal 2010 9 p 2095 2100 Ludwig Boltzmann Institute for Archaeological Prospection and Vir tual Archaeology Carnuntum Roman urban landscape website http archpro lbg ac at casestudies austria 2012 69
60. lating the ellipse for which the major axes are the orientations of the two aforementioned variograms and their lengths are their respective ranges and subsequently calculating the distance between the origin and the point at which the ellipse cuts the direction of the variogram to be found If the sill of the directional variograms are different there is zonal anisotropy This can be modeled by adding a second anisotropic structure with a very large range in one direction so that this component disappears in the considered direction variogram properties The variogram provides a measure of the statistical distance between two points The smaller the variogram at a certain lag h the more information both points have in common 2 2 3 Ordinary Kriging As mentioned at the start of this chapter the Kriging interpolation method is an estima tion method that makes use of the spatial correlation of the data at hand Deriving the Kriging estimation starts at the criteria of unbiased and optimal interpolation E Z xo Z xo 0 3 s xg E Z xo Z xo minimum 4 With Z xo being the estimation at xy and Z xo being its real value Since the Kriging estimation method is a interpolation method the interpolated value can be written as a weighted linear combination of measurements in the neighborhood around n Xo 2 5 In its most general form the general Kriging equation is n
61. mponent would avoid the occurrence of the anomalies X im Image 627500 627590 6276 04 53756545 627700 6327754 4 0 ha e 2230300 2430300 1 0 2240 00 2440 00 ur x Y mj 1 0 5320100 5330100 ag 4 D 3 0 dd d 2340000 EST500 6375090 637650 637700 6477530 X m Java Figure 14 Filtered image of Carnuntum Magnetic susceptibility long range filter MSU search window radius of 80 m minimum distance of 8 m 30 0 10 20 30 40 50 60 70 80 90 100 0 00 L 0 10 20 30 40 50 60 70 80 90 100 Distance m Figure 15 Variogram of Carnuntum Magnetic susceptibility long range filter MSU search window radius of 80 m minimum distance of 8 m 39 9 Testcase 2 This dataset contains the electrical conductivity measurements recorded at the site of Car nuntum Contrary to the magnetic susceptibility see Ch 3 2 the electrical conductiv ity consists of measurements of 4 different electric conductivity measurements combined These 4 electric conductivity measurements have different penetration depths ECIPRP up to 0 5 m EC2PRP up to 1 0 m ECIHCP up to 1 5 m and EC2HCP up to 3 2 m These 4 measurements allow the construction of the electric conductivity from 0 m to 0 5 m 3 3 1 Data exploration The measurements of the electric conductivity on the Carnuntum site contain more in formation on geological features that did not show up in the magnetic susceptibility mea
62. nes might be much higher compared with the direction perpendicular to the driving lanes This can lead to artifacts The direction in which the vehicle moves alternates between consecutive driving lanes This might lead to perturbations in the data which have to be dealt with during the post processing stage of the data acquisition However it is possible that small artifacts are still present during the stage of data analysis 2 4 The software ISATIS ISATIS is a Geostatistical software that can be used in all steps of statistical data analysis ranging from data exploration data analysis to the creation of complex images and data representations and simulation based on user created models ISATIS is one of the only commercially available software packages that can perform Factorial Kriging and handle non regularly spaced datasets Furthermore the fact that the user can force ISATIS to impose a minimum distance between samples see further in this chapter without the use of a predetermined subset of the data makes this software a good choice to tackle the problems set by the research design ISATIS does have its flaws however such as its black box approach that impedes the user to have much direct control on the process and ISATIS forcing the datapoint nearest to the location to be estimated to be included in the Kriging system when using a minimum distance Most time intensive calculations are parallelized so working with a clus
63. ntours of a building in the center of the plot Even harder to discern but still somewhat visible is the ancient Roman aqueduct remains that runs from the bottom left corner to the upper right corner 637500 637550 637600 637650 637700 637750 5330300 5330300 5330200 w A 5330200 5330100 5330100 5330000 5330000 637500 637550 637600 637650 637700 637750 y Figure 12 Ordinary Kriged image of the Carnuntum magnetic susceptibility dataset MSU The features of interest are 1 Gladiator school see figure 10 2 Aqueduct 3 Graveyard X m 2f 3 2 4 Factorial Kriging 3 2 5 Short range Of interest are the structural remains of the Roman gladiator school However it is reasonable that the short range component range 1 m mostly represents deviations from the background due to the act of driving in close parallel lanes We also want to show that the image does not change when the search window radius becomes larger than the range of the short range component The search window is varied from 0 6 m to 2 5 m in steps The minimum distance between datapoints is 0 2 m Figures 32 show the the filtered images for a search window radius of 0 6 m and 1 5 m figure 13 shows the filtered image for a search window radius of 2 5m The images stabilize when the search window radius becomes larger than 1 m The correlations between subsequent plots become equal to one and the points on the corresponding scatterplots move to the bi
64. o allow a sufficiently quick evaluation of the Factorial Kriged images while still retaining enough information to correctly filter the images In this case the minimum distance should be set to 0 in ISATIS compensate 44 for loss in information due to working with subsets of the data many different subsets can be used and an average filtered image calculated This would also enable the calculation of a variance if desirable It is also reasonable to believe that if an existing feature in the dataset is not included in the model it will not simply disappear in the filtered images of the compenents that are modeled but will be included in some form especially if range of the component that is not modeled fits inside the used search window We have seen this occur when evaluating the Meigem dataset and the Carnuntum electric conductivity dataset where the filtered images show signs of the driving lanes which were not modeled While applying the proposed restrictions search window covers range of component minimum distance larger than range of smallest range component it is possible to use sparse samples by increasing the minimum distance This will reduce computation times significantly up to a factor of 5 when halving the sample size see Ch 3 4 1 without a considerable loss of image quality see Ch 3 1 5 where a minimum distance up to 1 3 of the component range has been tried When dealing with spatial features with qu
65. on times Especially since the creation of a variogram is a trial and error process before a final variogram can be made The fact that one sample is randomly chosen in each grid cell ensures that there will be enough pairs of samples that lie significantly closer to one another than the cell dimen sions as long as there are a large amount of grid cells This means that lag distances smaller than the cell dimension can also be represented in the experimental variogram File Selection From Polygons At last it is also possible to define selections over whether or not a datapoint resides in side a predetermined space It is possible to manually draw a polygon using the Polygon Editor File Polygon Editor Inside the Polygon Editorwindow select Application then New Polygon File Create a new polygon file and next to This Polygon File will contain select 2D polygons Now press create Now select Application then Auzil tary Data Here you can select any variable that will be used as a background for the creation of a polygon Back to the Polygon Editorwindow start creating a polygon by pressing the right mouse button on the location you desire Click on a location to create 66 an extra vertex Press the right mouse button to close the vertex first and last vertex created are connected Press right mouse button again to end the create polygon mode Pressing the right mous
66. op of the overall maximum and minimum neighbors a maximum of points to be used per segment should be specified This ensures that information in all directions from xp is represented more or less equally There are other interpolation method based on Kriging such as simple Kriging without or with a trend surface with varying local means lognormal Kriging block Kriging coKriging indicator Kriging 2 2 4 Factorial Kriging A statistical process Z X can be treated as a combination of different independent nested processes If this is true the variogram of Z X y h itself is a combination of different varlograms y h 1 b 14 h 13 If the processes are uncorrelated the total variogram can be written as a linear combination of S basic variograms g h S y h Y wg h 14 k 1 Here g h is the k th basic variogram used to construct y h not necessarily in any particular order and b is the coefficient that represents the relative contribution of g h 10 to the total variance The components need not necessarily represent any physical process but if they are modeled after such a process mainly by manipulation of the range of the components it will improve the interpretability of the resulting images If second order stationarity is in order for Z X it can be represented as the sum of its components Z x 3 2109 4 15 with the expectation of Z x equal to 0 and
67. ox next to Stop at First Inversion Problem under the Special Model Options which also contains the Factorial Kriging options However this will lead unresolved points in the image A way to avoid these problems is to use the Look for Duplicates tool or to work with a subset of the dataset see below Selections A selection is a variable with the same spatial distribution as the original but its values are composed of ones and zeros When choosing variables in any File and Variable selector window mostly you can choose a selection to apply to the chosen variable The selection can only be applied to variables that belong to the same object like points grid Ordinarily ISATIS will know what to do with the selection It is possible to apply a logical operation on a selection NOT or on two selections OR AND XOR to make more complex selections File Selection Logical Operations Tools Look for Duplicates A fist selection method for your dataset that is advisable if some datapoints lie very close to eachother is to look for duplicates and mask them In the Look for Duplicates window first select your data file In the File and Variable selector window select your variable and create a new variable that will contain information on whether or not a datapoint is a duplicate This new variable must be selected for the New Selection Name Next submit 65 a small value next to Minimum Distance
68. r ME Ge ON a EE a Cala AE 2 310 ANOOP O dS aai a al X aeu A A 5 2L DheGornware DALIS kos Eck amp oo ES ara a BRGY SAR 2 4 1 Short manual and ISATIS specific methods 2 0 Juesearch procedure a ew eke XO Bow AGES Do owe 3 Methods and Results 3 1 Initial method development Gran x A 3 1 2 Experimental variogram and model Ordinary s sere sas bI B XREBIRRORVE ARBRES sp uod xe me ME NE m EU RU es 5 A ty wee ode II IOHBPdlPG xs cima SAS folie ee oe we oe eo e E A E o AA exploration dk ve xo te oec Poe a ads he e da 3 2 2 Experimental variogram and model B20 Ordinary FOIOS cera bo tecum ek She ek Gh SO Se d Diu Factorial KENE de we ds RU eae epe ee deua Sos ai 922 0 OM uev toe ue ero tae a ese SE ccc ool Data exploration e s soea Eu IY e AUS ee C 3 3 2 Experimental variogram and model 10 12 12 13 14 15 15 17 3 4 So Orinar ESO 4 oca 2208264083 4462 CE e eee S ood pineal Ghee Sn EA Moe A ur d ahd ae ue NR ee ee d edd 30 0 MONO TADO 3 RIADA Aa ad SEU 5 New insights and theoretical considerations od Computacion TIMES 6 sonda a ceo xL RERO RA ee we Ew Rees
69. rid can be adapted by sliding the slidebar or manually submitting the value Back in the Create Grid File window window press Run on the bottom left to save the erid file 63 Kriging and Factorial Kriging Estimation gt Co Kriging In the Standard Co Kriging window the first thing we do is selecting the dataset to be used Press the Input File button Here we select a possible selection subset on the top left and a variable on the top right We do not need to select anything for the other two possible input variables Variance of Measurement error Kriging Weights Press OK Back on the Standard Co Kriging window press Output File Now select the de sired grid with eventual selection on the right For example we can use a cropped grid to reduce redundant calculations Cropping is done by making a polygon and applying a selection operation see B After selecting the right grid file a New Variable has to be made Submit an appropriate name an press OK Back on the Standard Co Kriging window press Model In this window a model can be selected ISATIS will only accept variogram models that correspond with the the correct input variable submitted earlier If the model was constructed using exactly the same data as the input variable both variable and selection apply then the model will be accepted Otherwise you can also make a new model submit a name next to New File Name
70. s an easy way to read in datasets of a myriad of predefined data types such as Gslib Excel AutoCAD File Import Neither the Meigem nor the Carnuntum datasets are stored as predefined types The dataset can be read in as an ASCI type We are going to handle the importing of the Carnuntum electric conductivity named elektrisch csv in our case in the following First make sure that first line does not contain the names of the variables However be sure to remember the names z y z t ELEK File import ASCII On the ASCII File Import screen do the following Search the file elektrisch csv Uncheck the box Header is Contained in the ASCII Data File Make a new header file Click Build New Header 59 We arrive in a new window called Editing a new header First click New Header File choose an appropriate location and name for the header file In tab data organization Type of File select Points Dimension select 2D points In tab Options Check CSV input Comma Separated Value for Values Separator check Other and type for Decimal Symbol type This will depend on how the data are ordered in the original dataset so be sure to check Leave the rest as it is In tab Base Fields Create 5 fields New field and name them x y z t ELEC Select the 5 variables subsequently and edit the format of the variables x Field Type
71. s removes the text containing the name of the variable that will be imaged from the image itself In the Contents tab on the left select Raster and push the arrow pointing to the right The ltem contents for Raster window appears First select the variable that you want to represent here it is a variable from a grid file You can already take a look at your image by pressing the Display Current Item on the bottom of the window You can submit a Legend Title Now choose a color scale There are some preset color scales such as greyscales rainbow We used a custom made color scale that was based on the Rainbow Reversed scale make your own scale press the Color Scale button In the Selector window select New Color Scale Give the scale a name Then press the Edit button It shows the Color Scale Definition window You can Load Color Scale but we will define our own color scale Next to Bounds Definition select User Defined Classes press Palette 67 Name and select Rainbow Reversed READONLY You can change the Number of Classes but we leave them at default 32 It is advisable to first Calculate from file which sets the bounds of the scale at the minimum and maximum value and adjust after wards Pressing Bounds brings up the Contiguous Bounds window As a starter just change the maximum and minimum bounds then press OK For Undefined Values we chose Transpar
72. satisfied press OK in the bottom right Back in the Variogram Fitting window press the Model button in the lower left corner and submit a file name or replace an existing model Press Run Save to save the model Creating a grid Before any estimations can be made we need to define at which locations the estimations should be made To create an image of the estimated values it is most common to create a grid This is done under File gt Create Grid File In the Create Grid File window first a New Grid File has to be created In the File and Variable Selector window choose or create a directory and press New File Choose a new name and press OK twice Arriving back at the Create Grid File window you can select an Auxiliary File to help in choosing the extents and the dimension of the grid Check Graphic Check to help Since we are working with 2D data select 2D Grid File ISATIS will automatically use approximately the minimum X and Y coordinates as Origin values Next to Mesh you give in values that represent the distance between 2 nearest points of the grid in the X resp Y direction Nodes Number represents how large the grid is in the X resp Y direction You can rotate the grid to better cover the dataset which can in turn reduce the amount of superfluous grid points Just press Rotation In the 2D Grid Rotation Definition window the tilt of the g
73. section line which indicates stable images See fig 33 As expected the filtered images of the short range component exhibit a major flaw The short range component mostly contains the artifacts caused by the driving lanes However we are only interested in the buildings and structures The variability that is associated with these structures are found at lag distances that are comparable with the distance between driving lanes As the mostly local structures are less represented in the area that is analyzed than the omnipresent driving lanes the component of the structures is not identifiable in the variogram Also it should be noticed that the fact that the structures are a local and not a global source of variability institutes a transgression of the intrinsic hypothesis of the Kriging system which supposes the structure of variability is a global constant However filtering the short range component still enables us to better identify the buildings than Ordinary Kriged images allowed so the filter is partly successful fig 13 3 2 6 Long range The big difference between the scales of the short range component and the long range component lead to many difficulties It is practically impossible to follow the recommen dations following the Meigem dataset see Ch 3 1 6 as the ranges of the short and long range components are very different it is impossible to both use a search window radius at least equal to the range of
74. sirable you can overwrite an existing variogram by selecting it from the list that will appear if experimental variograms already exist in the project Press OK and then Save If the dataset is very large say N datapoints it is possible that it takes a very long time to calculate the experimental variogram as the number of pairs rises with the square of the amount of points In that case it might be useful to work with a subset of the original dataset by selecting a selection in the File and Variable Selector window Available selections for a dataset or grid file see later are found in the upper left side and are distinguished from files and folders as they are identified by an S in front of their names Just press their name and it will highlight Then press the variables you want to work with on the upper right side More information on the selections we used can be found in the paragraph on selections B Fitting a variogram Statistics Variogram Fitting When arriving at the Variogram Fitting window the first thing you should do is select an experimental variogram at the top of the window click Experimental Variograms This takes you to the Experimental Variogram window which should show all available experimental variograms in the current project Select the correct variogram and press OK make fitting easier check Fitting Window in the Variogram Fitting window You can choose to select an alread
75. t SW 1 5 dist 0 2 short SW 2 dist 0 2 30 20 10 0 10 30 20 10 0 10 rho 0 998 rho 1 000 t Ims M S 5 Fi m c oa ct M uon ol i 1 to in se lin E EE E e e m e p ct o 38 3 e 9 to 30 20 10 0 10 30 20 10 0 10 short SW 1 5 dist 0_2 short SW 2 dist 0 2 c 1 5 m vs 2 5 m d 2m vs 2 5 m Figure 33 Carnuntum Magnetic susceptibility scatterplots comparing short range filter range 1m sill 0 005 MSU search window radius of 2 5 m 90 Image Image X m X m 3 5 2900 pelea pale pareng 3 5 637500 637550 637600 637650 637700 637750 3 0 3 0 2 5 2 2 0 2 0 5330300 5330300 5330300 5330300 1 5 s 1 0 1 0 0 5 lt 4 5330200 5330200 n 5330200 5330200 g P5 0 5 0 5 1 0 1 0 5330100 5330100 1 5 5330100 5330100 ix 2 0 2 0 2 5 2 5 3 0 3 0 5330000 5330000 5330000 5330000 637500 637550 637600 637650 637700 637750 637500 637550 637600 637650 637700 637750 m ju X m X m i zx jus ei zr a minimum distance 20 m mS m b minimum distance 3 m mS m Distance m Distance m E o m E om a a T go p E 2 E H e E pi i 2 E on E 4 E El m a E pus e E o EE se Lal ces D zn 50 TS 100 125 un D zn 50 TS 100 125 un Distance m Distance im c d Figure 34 Carnuntum Electric conductivity long range filter range 90m sill 15 3 mS m 1
76. tatistics dealing with spatially distributed datasets It has many applications among which mining soil science hydrology environmental control and agriculture It distinguishes itself from classical statistics in the underlying assumptions regarding the data In classical statistics the data are generally assumed to be independent after removal of a trend which means that it does not take into account any correlations between different data In geostatistics it is normally assumed that the data are correlated as nature generates only in rare cases phenomena without spatial correlation The first major goal of a geostatistical analysis is to model this spatial correlation The most used methods in geostatistics are prominently but not restricted to interpolation methods Not all methods make use of the available correlational data such as trend surfaces and inverse distance weighting but those that do generally give much better results The most important method based on spatial correlation is Kriging 2 2 2 The variogram In order to make estimations based on correlation between spatial data one has to know how the variable at the location to be estimated correlates with this variable in the sur rounding area Sometimes this information is given but most of the time the correlation has to be estimated from the collected data In order to do Kriging some assumptions regarding the data have to be made Sup pose the variance
77. te the model with prior knowl edge The resulting variogram is shown in figure 22 Both the short range component and the residual component a derivative of the long range component for small search windows are estimated using a search window of 2 5 m radius and a minimum distance of 0 2 m The image of the short range component shows almost exclusively the influence of the driving lanes while the image of the resid uals shows almost none of the driving lane artifacts fig 24 and fig 35 Besides the fact that the outlines of the building are now much more clear the new image shows a region in the part of the domain below the Roman aqueduct that show small rectangular features This place is a graveyard with grave monuments that were not apparent in previous images of the short range component This method shows that it is possible to extract structures that are not clearly rep resented in the variogram due to the fact that they are limited in space of amplitude even when there are strong perturbations as long as these perturbations can be modeled approximately 40 Distance m D 10 20 30 40 5 7D BD 50 100 gt MAL MPAGO TIPA Variorram TYH Figure 22 Variogram of the Carnuntum magnetic susceptibility dataset with a decaying sinusoidal small range component MSU right zoom for small lag values line fitted model dots experimental variogram The usefulness of this trick is of course limit
78. ter of computers will decrease real time calculations significantly 2 4 1 Short manual and ISATIS specific methods The ISATIS manual specific for handling the used dataset can be found in Appendix B The search window Just like any software allowing Kriging estimations ISATIS allows the user to define the search window The search window is the area around the location to be estimated from which the points used for interpolation can be included The search window is an ellipse and the user provides the length of its main axes fig 4a Minimum distance between points ISATIS provides the user a useful setting for dealing with dense datasets When doing a Kriging estimation one can ask ISATIS to choose a configuration of points so that each point in this configuration has a minimal 15 distance provided by the user to any other point in the configuration fig 4b This allows the variable to be estimated to use information from a large area while limiting the amount of observations to be used This method is different from other methods that reduce sample sizes such as estimation on a regular grid that is an interpolation from an irregular grid or thinning out the sample size before applying the estimation method When using this method no information is thrown away a priori and each location to be estimated makes use of a different subsample A constant in the subset selection is that ISATIS always includes the datapoint
79. ters we can change the type of variogram and we can change the variogram if a classical variogram from Om nidirectional isotropic to Directional anisotropic First we will keep the variogram anisotropic In the bottom of the window is the variogram definition Clicking on any value will open a new window called Directions Definition Here we can change the values that define the experimental variogram see 2 2 2 Also it is possible to display the pairs using a button on the bottom of the window 61 When using the option of Directional variogram in the Variogram Calculation Parame ters window the lower part of this window show new options We can define the amount of different directions and a reference direction Now the Directions Definition window shows an extra option Tolerance on Direction Each direction can be selected individ ually and adapted Pressing OK on the bottom of either the Variogram Calculation Parameters or the Directions Definition window will recalculate the variogram A di rectional variogram will show multiple individual variograms each corresponding with a single direction To make a variogram model the experimental variogram should be saved Application Save in Parameter File in the window of the graphic In the Save in Parameter File window press NEW Experimental Type in a New File Name ISATIS will au tomatically put it in an appropriate direction If de
80. the sampling area is chosen too small to capture the range of the underlying model This non stationarity of the second order can still fall under the assumptions of the intrinsic hypothesis as long as the increase is of less steep than A nested model Any set of variograms can be summed and again form a variogram y h 4 yp h ya h This so called nested variogram can be used to describe complex patterns over multiple spatial scales An often recurring combination is that of a nugget effect and a simple model The components of a nested variogram need not necessarily represent any physi cally occurring phenomena but if it does it will facilitate the interpretation of the vari ogram and its resulting estimations An example of a nested variogram is shown in figure 6a further ahead where 2 spherical components are combined to fit the experimental anisotropic models After the calculation of the experimental directional variograms it is often possible to find the two orientations in which the spatial variability has its largest continuity and smallest continuity respectively the experimental variogram with the largest range and the shortest range mostly these two directions are perpendicular If the sill of all directional variograms are the same and only the range changes as a function of the orientation we are confronted with geometric anisotropy The range of the variogram in any direction 6 can be found by first calcu
81. wer weight depending on the variogram This phenomenon is called the screening effect This is the result from the information contained in A The observation closest to xo will contain the most information on so it takes priority over observations in its close neighborhood Kriging has an inherent declustering effect search neighborhood The neighborhood around xo where the observations are taken into account to interpolate Z xo is called the search neighborhood Usually the shape of this neighborhood is taken to be a circle in case of an isotropic variogram or an ellipse if the variogram is anisotropic The dimensions of the neighborhood are generally taken to be the ranges of the variogram as observations lying at further distances are considered to be uncorrelated The maximum number of neighbors to be used for interpolation can be limited to a small number usually lt 50 Observations at a large distance will generally receive a small weight both due to a smaller correlation with Z xp and the screening effect from points closer to A minimum number of neighbors is usually also specified between 2 5 When not enough points are in the search neighborhood the interpolation is not performed and the non interpolated points are usually reported as missing data If observations are very inhomogeneously distributed such as highly clustered data it might be advantageous to divide the search neighborhood into segments On t
82. y existing model under Model Initialization by clicking Source Model and choosing a model You can make ISATIS do the fitting automati cally but we will choose to do the fitting manually Press Manual Fitting and then press Edit under the options that have appeared Always use the Fitting Window to check the correctness of your model 62 Arriving at the Model Definition window one can select whether or not we work with an anisotropical directional model by checking Anisotropy We can add a Nugget Effect and Add Structure Each new structure will initialize with default values Change the structure type by pressing the button next to Structure Type with traditional options like Spherical Exponential and Gaussian next to some other more unusual var iogram components One can change the range and the sill of the component under the Structure Type button When using anisotropy two values for the range are present one for the U direction and one for the V direction You can check which directions U and V represent by pressing the button next to rotation The directions that U and V represent can be adapted under Azimuth by changing the slidebar or by submitting the value manually You can check the valability of your model any time by pressing the Test button in the bottom left part of the Model Definition window The current model will be shown in the Fitting Window When

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