as a PDF
Contents
1. Save data mat save the data of current view in a matrix format Save data dat save the data of current view in general xyz column format Read disp read in new graph parameters from a DIS file Save Graph as PS save the current graph in Adobe s Postscript format Save Graph as EPS save the current graph in Adobe s Encapsulated PS format Save Graph as PDF save the current graph in Adobe s PDF format Save Graph as WMF save the current graph in Windows metafile format Save Graph as GIF save the current graph in GIF file format These menu items bring up a standard Windows file selection dialog that is used to provide the file name for I O operations All data files are in text format see chapter 5 The graphs are saved A4 landscape mode see Appendix A The preferred output format is PDF or PS because their resolution is much better than that of GIF and WMF files The PDF format is particularly useful because Adobe s Acrobat Reader can be used to take bitmap snapshots at desired resolution 3 2 Edit menu The Edit menu contains following items Inverse gt data set current results of the inverse FFT as new initial data Subtract inverse subtract the inverse results from the original data Add inverse add the inverse results to the original data Color scale change the color scale rainbow gray scale etc Padding mode change the method used to pad and taper data Data type define potential field data generic gravi2. 1790 1780 2840 2850 2860 2870 2880 2890 X 28 Spacing 1 00x 1 00 km Appendix C 2D frequency amplitude spectrum 2 D Fourier transform analysis Transformed data 0 4 0 3 Q 2 0 1 Z 0 0 0 4 0 2 0 0 0 2 0 4 3 166 0 881 1 405 logl0 F 3 690 5 976 Generic data Dimensions 64x 64 km Spacing 0 01563 x 0 01563 1 km Nyquist 0 50000 0 50000 1 km Wave length 2 00 2 00 km Appendix D Difference between original data and inverse transformed data 2 D Fourier transform analysis Difference 1830 gt 1810 1800 1790 2840 X 2850 2860 2870 2880 0 010 0 005 0 000 Field ppm 0 005 29 Generic data Dimensions 64x 64 km 1 00 x Spacing 1 00 km Unprocessed inverse data 0 010 Appendix E Low pass filtered frequency spectrum Generic data 2 D Fourier transform analysis Dimensions 64x 64 km Transformed data Spacing 0 01563 x 0 01563 1 km Nyquist 0 50000 0 50000 1 km 0 4 Wave length 2 00 2 00 km Low pass filtered 0 3 0 2 0 1 ray Z 0 0 0 1 0 2 0 3 0 4 0 4 0 2 0 0 0 2 0 4 Kx 3 981 1 491 0 998 3 487 5 976 logl10 F Appendix F Low pass filtered data and its difference to original data 2 D Fourier transform analysis Difference 2 D Fourier transform analysis Inverse data j 1830 1830 1820 1820 gt 1810 gt 1810 1800 1800 1790 1790 T T T T 7 2840 2850 2860
3. between 65 and 128 This will then be adjusted to 128 when Padding button is pressed Note that the dimension of the grid can be decreased only using re interpolation by Plot matrix button a normal paddin b shifted paddin b padding tapering p 8 p 8 padded values tapered values padded values original data original data area area original data area padded values tapered values 0 0 N N 00 N 0 0 N Figure 4 1 Schematic view of a padding without shift b padding with shift and c shifted padding with tapering The new dimensions N and M are powers of two Padding and tapering depend on the selection made with the Padding mode item in Edit menu The padding modes are a Zeros which actually use the median of the outmost data points instead of zeros for padding b Linear extrapolation will extend with the outmost data point c Mean value based will use the nearest points inside some increasing search radius and d Gradient based which uses the mean value and the derivative of the field to extend the data to the padding zone The Shift yes no option is used to define if padding is added only to the top and to the right of original data area or if padding is made all around the data area which looks like the data were shifted Important Best results from Fourier transform processing are obtained using gradient based padding with shift The other padding options are kept primarily for teaching purposes so that the
4. created automatically The program then builds up the GUI shown in Appendix A None of the program controls widgets however are active before data has been read in The data processing and analysis is performed in few successive steps that are discussed next The preliminary step is always to check that the input data file is in correct format so that it can be read in see chapter 5 on file formats After reading in the data one should apply Plot matrix Padding and Plot FFT buttons to perform interpolation and padding and to compute the 2D frequency spectrum Only after the FFT has been computed one can perform frequency filtering Edit menu and frequency domain data processing Process menu 4 1 Reading in the data The first thing to do after starting up the program is to read in the data using the Read data DAT or Read matrix MAT menu items Note that Geosoft XYZ formatted files can be read as normal DAT files provided that the file header is added by hand See chapter 5 for more information about file formats After successful data input the program automatically interpolates the data on a regular grid and plots a contour map as seen in Appendix A The initial grid sampling is based on the number of data values and the spacing between the first two data points in the file The grid dimensions i e the number of grid nodes in x and y directions N and M will appear in Dim x and Dim y text fields in the control pane At
5. fields are defined as the negative gradient of the scalar potential The quality of some frequency domain filtering operations is strongly affected by the smoothness and continuity of the data To circumvent computational artifacts in the inverse transformed data it is often necessary to perform low pass filtering before any other processing operations The Automatic low pass filter menu item activates automatic filtering every time the FFT is computed The inner and outer radii of the filter can be changed manually using the dialog that appears after using the menu item Note that the radii are not defined as distances but as a ratio a value between O and 1 of the minimum Nyquist frequency of the data Automatic values are used if the radii are set to zero If the values are omitted totally 1 e a blank line is given the automatic mode is deactivated Low pass and high pass filtering are the two most typical frequency domain processing operations Considering 2D data low pass filtering removes nullifies values outside certain radius from the origin of the frequency amplitude spectrum see Appendices C E and F On the contrary high pass filtering removes low frequency data around the origin of the spectrum Together low and high pass filtering can be used as a band pass or notch filtering The direction filtering which is applicable to 2D frequency data only can be used to remove or to enhance linear features in the original data see A
6. 2870 2880 2840 2850 2860 2870 2880 A x I r T j 57 403 39 563 21 723 3 883 13 957 6 307 3 429 0 551 2 327 5 205 Field Field 30 Appendix G Direction filtered frequency spectrum Generic data 2 D Fourier transform analysis Dimensions 64x 64 km Transformed data Spacing 0 01563 x 0 01563 1 km Nyquist 0 50000 0 50000 1 km Wave length 2 00 2 00 km Direction filtered 1 N 0 4 0 2 0 0 0 2 0 4 5 514 2 643 0 228 3 099 5 970 logl0 F Appendix H Direction filtered data and its difference to original data 2 D Fourier transform analysis 2 D Fourier transform analysis Inverse data Difference 1830 1830 1820 1820 gt 1810 gt 1810 1800 1800 1790 E 1790 2840 2850 2860 2870 2880 2840 2850 2860 2870 2880 X xX 53 915 36 265 18 614 0 964 16 687 17 166 9 296 1 426 6 444 14 315 Field Field 31 Appendix I Upward and downward continued data h 2 km 2 D Fourier transform analysis Inverse data 1830 1820 1810 1800 1790 I I 2840 2850 2860 2870 2880 X 46 355 34 312 22 270 10 227 1 816 Field 2 D Fourier transform analysis Inverse data 1830 1820 gt 1810 1800 1790 2840 2850 2860 2870 2880 X 168 690 97 428 26 166 45 097 116 359 Field Appendix J First vertical and horizontal x gradient 2 D Fourier transform analysis Inverse data fi 1830 1820 gt 1810 1800 1790 U I I 2840 2850 2860 2870 2880 X 37 774 30 3
7. 40 22 907 15 473 8 040 Field 2 D Fourier transform analysis Inverse data 1830 1820 1810 4 1800 1790 2840 2850 2860 2870 2880 xX 34 689 28 995 23 300 17 606 11 911 Field 32 Appendix K Second vertical and horizontal y gradient 2 D Fourier transform analysis 2 D Fourier transform analysis Inverse data Inverse data 1830 1830 8 1820 1820 gt 1810 gt 1810 1800 1800 1790 1790 j j 2840 2850 2860 2870 2880 2840 2850 2860 2870 2880 X X 39 358 31 592 23 825 16 058 8 292 31 676 27 262 22 847 18 433 14 018 Field Field Appendix L Horizontal gradient and total gradient analytical signal 2 a 7 ae mee 2 D Fourier transform analysis Inverse data 2 ane gt i Inverse data 1830 1830 1820 1820 gt 1810 n vas 1810 1800 1800 1790 1790 i i i 2840 2850 2860 2870 2880 2840 2850 2860 2870 pean X X 0 031 3 775 7 519 11 262 15 006 0 388 4 321 8 253 12 186 16 118 Field Field 33 Appendix M Tilt derivative and pseudo magnetic field k 0 01 SI Ap 0 2 g cm and To 52000 nT 2 D Fourier transform analysis 2 D Fourier transform analysis Inverse data Inverse data 1830 1820 1820 gt 1810 gt 1810 1800 1800 1790 1790 T i 2840 2850 2860 2870 2880 2840 2850 2860 2870 2880 X X 1 564 0 782 0 001 0 780 1 562 508 127 265 436 22 746 219 944 462 635 Field Field Appendix N Program GUI with 2 nd vertical gra
8. Blakely 1995 for example 2 Installing the program This version of FOURPOT is a 32 bit application that can be run on Windows XP Vista system with a graphics display of at least 1024x768 pixel resolution Memory requirements and processor speed and are not critical unless very large data sets are used The program has simple graphical user interface GUI that is used to handle file input and output to perform the computational operations and to visualize the data and the results The user interface and the data visualization are based on DISLIN graphics library The program requires these two files FOURPOT EXE the executable file an DISDLL DLL dynamic link library for the DISLIN graphics The distribution file FOURPOT ZIP also contains a description file README TXT this user s manual FOURPOT_MANU PDF and some example data files DAT To install the program uncompress e g Winzip or 7Zip the distribution file and move the resulting FOURPOT folder to preferred place e g C TOOLS To be able to start the program from a shortcut that locates in a different directory or from the desktop you should move or copy the DISDLL DLL file into the WINDOWS SYSTEM or SYSTEM32 folder or somewhere along the system path 3 Menu items 3 1 File menu The File menu contains following options Read data read in irregularly spaced data in default file format Read matrix read in regularly spaced data in matrix format
9. ETMAT subroutine included in the DISLIN graphics library see DISLIN user manual This subroutine works best if the data are already regularly sampled and the dimensions coincide with the original grid Therefore it is recommended that highly irregular data are interpolated on a regular grid using a more advanced third party software e g Golden Software Surfer and more suitable algorithm e g minimum curvature Naturally if evenly discretized data are passed to FOURPOT one should take care that the interpolation Plot matrix is made using the same grid sampling as that of the original data so that as minimal distortion is made by GETMAT as possible 4 3 Padding and tapering Padding and tapering are essential parts of successful Fourier transform processing The Padding button will automatically add extra columns and rows around the interpolated data matrix so that the grid dimensions N and M will increase to the next even power of two see Figure 4 1 and Appendix B The padding can be performed with or without tapering Tapering means that the padded data are such that the level and the derivative of the data are preserved Sometimes the original grid dimension may already be so close to some power of two that the tapering will not have enough time to suppress data discontinuity To further increase the 12 padded dimension from 64 to 128 for example one simply needs to change the current value of Dim X or Dim Y to some number
10. FOURPOT Fourier transform based processing of 2D potential field data Version 1 1 User s guide amp Markku Pirttij rvi July 2009 University of Oulu Department of Physics Geophysics Table of contents Table of contents 22 2 nasaietinivsed ti en nsi Ea eee cede el een adie 2 1 troduction a Fe evs EE E va Ss as E E EE E io 3 PAAT GING PROSTAR EE EE E E a E 4 3 Menu items eys ee nea fet aah A ado deel E A A A E 5 e E E i U TEE E EEE 5 J2 Edit MENU coe ina a E E a a aE A ae ve a en ote a 5 3 3 Process TOC ehes a a a a E a pa AAAS 7 As Using the program ereire eseria ar a E a E E E E AEE E aS S aea aiaa it 11 4 1 READ in TS CALA denrea e e a A a E a E E 11 4 3 Padding and tapering s ccisssicavecideiatavapcanirscsdeasthaaasaacs tus cceda EEE rea iE E E ES 12 AA Fo rrer transfor o ees ede Pe ie acts lsd a a ence ee 14 4 5 Inverse Fourier transform and the difference 2 0 0 0 eeesseceeeeeceeececeeeeeceeneeeeteeeenseees 15 AcOc Frequency tenne sol oo eein lat etal a tener Ae cade et cate ke 15 4 6 1 Low pass and high pass filtering cee eeeseceeeeceecceceeeeeceeeeecseeeecseeeeeseeeeees 16 4 6 2 Directional filtering cia casiacsscsscedanaceea sade siete ninini i 17 4 6 3 Interactive and manual filtering ssssssesseesesseeesseessseesseesseesseresseeesseessersseesseee 17 4 1 Radial amiplitide Spectruinsccisaveesicdesisosacivd vavessterseaesaseacesealegeausedengacesondeeconsvceaate lebanese 19 4 8 Color mapping rang
11. T data 25 Note that the format of the FOURPOT parameter file is likely to be changed in the future Also note that the character can be used to define superscripts exponent _ character is used to generate subscripts and the character is used to move the text back to the baseline 7 References Bhattacharyya B K 1967 Some general properties of potential fields in space and frequency domain a review Geoexploration 5 127 143 Blakely R J 1995 Potential theory in gravity and magnetic applications Cambridge Univ Press Claerbout J 1976 Fundamentals of geophysical data processing With applications to petroleum prospecting McGraw Hill Book Co Ruotoistenm ki T 1987 Estimation of depth to potential field sources using the Fourier amplitude spectrum Bulletin 340 Geological Survey of Finland PhD thesis 8 Additional information I made the first version of FOURPOT in 2003 when I worked at the Geological Survey of Finland for the 3 D crustal model project funded by the Academy of Finland The original idea was to utilize the depth analysis methods of Ruotoistenm ki 1987 In 2005 when I started as a lecturer of applied geophysics in Oulu the objective became more educational and I have used FOURPOT in the teaching of gravity and magnetic data processing The Fourier transform is based on the FFT algorithm FORK by Jon Claerbout 1976 Look for his webpage via Stanford Exploration Project http sep st
12. anford edu for more details Thanks to Richard Stuart for the comments on smooth frequency filtering More information about the geophysical use of Fourier transform methods in potential field processing can be found in Blakely 1995 for example The FOURPOT program is written in Fortran 90 style using Intel Visual Fortran version 11 1 The graphical user interface is based on the DISLIN graphics library version 9 4 by Helmut Michels For more information please visit http www dislin de Because the DISLIN graphics library is independent form the operating system FOURPOT could be compiled on other operating systems Solaris Linux Mac OS X without large modifications At the 26 moment however the source code is not made available and I do not provide any support for the software However if you find the results erroneous or if you have suggestions for improvements please inform me 9 Terms of use and disclaimer The FOURPOT software is totally free for personal and scientific use If you find it useful please send me a postcard If you decide to use results computed with FOURPOT in publications please use this user manual and the web page _ at http www cc oulu fi mpi softat as a reference because I have not yet published this work in any journal The FOURPOT program is provided as is The author MP and the University of Oulu disclaim all warranties expressed or implied with regard to this software In no eve
13. any time the original data can be plotted again with the automatic grid spacing using the Plot data button 4 2 Interpolation All input data are assumed to be irregularly sampled Even if the original data were already evenly discretized it will be re interpolated before it s passed to FFT Therefore after reading in the data the user must always apply the Plot matrix button which is used to perform the 11 required interpolation using the grid dimensions N and M provided on the Dim x and Dim y text fields Note that the grid spacing dy and d does not need to be equal in x and y directions The Plot data function is used merely to visualize the original data and it does not affect the current grid dimensions If the grid dimension changes from the default or initial values because of interpolation or padding one can revert back to original automatically computed sampling using the Def size button The Aspect ratio button on the other hand is used to reset either the Dim x or Dim y dimension so that the original shape of the data area is preserved This means that when a new value of Dim x or Dim y is given the other value is set so that the original aspect ratio remains when Aspect ratio button is applied Important The horizontal and vertical dimensions N and M of the data matrix passed to the FFT algorithm must be powers of two e g 64 128 256 512 etc The interpolation discussed above is performed using the G
14. are plotted as zeros on the log scale Naturally this means that nulled values are plotted incorrectly they are really zeros 4 8 Color mapping range center and levels The Range and Center scale widgets can be used to change the color mapping of the data maps and 2D frequency spectrum The Levels widget is used to define the number of contour levels the default is 21 levels By default the color scale eg rainbow or reverse rainbow is evenly distributed between the minimum and the maximum data values of the current map The Range widget allows changing the range of the color scale by between 5 and 125 of the original minimum and maximum values Decreasing the value is useful if the maximum data value of the map is larger than the mean data level due to outlier for example Increasing its value is useful if the maps get saturated at their minimum or maximum limit values 20 The Center scale widget changes the middle point of the color scale between 85 and 85 of the current data range Increasing the center value will emphasize small data values and increases the amount of colors at the beginning part of the color scale Decreasing the center value will emphasize large data values and increases the amount of colors at the end of the color scale Important After changing any of the abovementioned scales one needs to press the Update plot button to make the changes visible Because the scale widgets give different resu
15. dient after low pass filtering and modification of the color scale 72 Fourier analysis v 1 0 by M Pirttij rvi 2009 og File Edit Process Exit Plot data Def size r Generic data Plot FFT 2 D Fourier transform analysis Dimensions 64x 64 km ae Inversesdata Spacing 1 00x 1 00 km Piot dif g Dimx 64 Low pass filtered Any et 1830 2 nd vert gradient Aspect R Field parameters Height units 0 00 Incl deg 75 0 Decl dea 10 0 Fiter parameters 1820 Ring 1 m 3 46 Ring 2 m 287 Sect 1 deg 0 000 Sect 2 deg 0 000 Width deg 0 000 gt 1810 Range 2 1800 1790 te 2840 2850 2860 2870 2880 X 35 648 29 685 23 722 17 760 11 797 Field 34
16. e center and levels eeceessecesceceeeeeceeeeeceeceecseeeeceteeeesaeers 20 Se Pile formats eiea tc acres somata ds lesen E acs cred Soenala es Gadel onan aediwlaciad E ES 22 5 1 Column tormatiod data Hes acisas etniniai ee ade eas 22 5 2 Regularly gridded matrix files i ccGuNoeS aon eu aoa aes 22 6 Graph OPH Ons aaa mendes aha E E E N ae 24 Of IRE LOS CCS ras teas at ia n a a a E a beatin iene aaae ias 26 8 Additional information i j ij2c eccches ei siete at lesa ieee eielnad sie heel eee eee 26 9 Vets OF use ANG Gise AMC 45 252 0c accine gthosskanvzetietanad t ii ii 27 10 Contact formation ia ssssistesiriesisior isce Dae inie S En ES S EES a Ea ee 27 Appendices A N Keywords Potential fields Fourier transform FFT 2D frequency filtering data processing 1 Introduction The FOURPOT program is designed for frequency domain processing and analysis of two dimensional 2D potential field arising in particular from geophysical gravity and magnetic field measurements The data can be irregularly or regularly sampled The frequency domain operations include high pass low pass and directional filtering upward and downward continuation pole reduction of magnetic data 1 st and 2 nd degree vertical and horizontal x and y directed gradients total horizontal gradient total gradient analytical signal and tilt gradient Pseudo gravimetric and pseudo magnetic fields can be computed also Considering a continuous function f x w
17. e provided but the cut off range can be omitted if the right mouse button is pressed when its value is asked for Important After interactive editing the given filter rings and sectors will be shown by dotted lines on the frequency spectrum The wave lengths and the angles will appear in the Ring 1 Ring 2 Sect 1 Sect 2 and Taper text fields in the left control pane Their values however are not stored if multiple filtering are made in this version of FOURPOT Also note that together low and high pass filtering can be used to accomplish band pass and notch filtering The Ring Sect and Width text fields however will show the parameters of the last filtering only Therefore the user should remember to memorize or save the parameters of the frequency filters because they may be needed to reproduce the processing results afterwards After the user gets accustomed with interactive filtering one should learn manual filtering which is more accurate than interactive filtering Manual filtering is made by providing the wave lengths in units of dimension of the filter rings low pass and high pass filtering or the angles of in degrees the filter sectors and width of the cut off range directional filtering in directly the corresponding Ring 1 Ring 2 Sect 1 Sect 2 and Width text fields in the left control pane Then by selecting the desired menu item from the Edit menu one can perform low pass high pass or direction filtering Some default para
18. e reduction and total gradient can be performed in a row by using the Jnverse Data menu item after the first operation and performing new FFT and the second Fourier operation Alternatively one can save the current inverse results and restart the program using the saved results as new input data 14 4 5 Inverse Fourier transform and the difference After the Fourier transform and the 2D frequency spectrum has been computed one can perform frequency filtering tasks in the Edit menu and or any of the frequency domain processing tasks in the Process menu The Plot inverse button can be used to display the current inverse results that is to say the either the inverse FFT of the current frequency filtered and or processed spectrum When tasks are selected from the Process menu the inverse results will be shown automatically Successive utilization of the Plot inverse and Plot data buttons can be used to visualize the changes that are made to the data due to the frequency domain operations As an alternative the Plot diff button can be used to visualize the difference between the original interpolated data and the data obtained from the inverse FFT after the most recent Fourier operation If no operations have been made the inverse will be and should be almost equal to the original data Note that pressing the Def size or Padding buttons will invalidate the current frequency spectrum and inverse results and the user needs to perform the frequ
19. e x and y axis relative to the size of the remaining origin shifted width and height of the plot area The total size of the plot area is always 2970x2100 pixels landscape A4 The fifth parameter defines the aspect ratio of the graph area when widescreen mode is used The remaining parameters are reserved for future use The 4 th line defines the initial values of the magnetic and gravity field components used in pole reduction and pseudo field computations These parameters include the inclination and declination in degrees from horizontal plane and true north and intensity in nano Teslas of the magnetic field the density contrast in grams per cubic centimeter and susceptibility dimensionless in SI units The 5 th line should be left empty The following 15 lines define various text items of the graphs These include the main title of the graph the 6 possible sub titles of the graph the axis labels and color scale label of the data in spatial domain the axis labels and color scale label of the data in frequency domain the axis labels of the graph of the radial spectra The maximum length of each text string is 70 characters The 16 th line between the two text blocks should be left empty The last 23 lines define various text items max 70 characters that are shown in the information text next to the graph These include the possible filtering operations made to the FFT data and the processing operation made to the inverse FF
20. ency filtering operations again Note also that after Padding the Plot matrix function becomes inactive because the N and M values are affected by the padding to the next power of two To be able to re grid the original data one needs to aplly the Def size button 4 6 Frequency filtering Important Unlike the other processing operations frequency filtering affects the current amplitude spectrum on which other processing functions are performed and from which the inverse transform is computed This means that frequency filtered data are always passed to further processing operations without the use of Inverse data menu item As a matter of fact low pass filtering is often an essential preliminary step for successful operation of the gradient filters Like the rapid changes in the data can cause problems when its FFT is processed rapid changes in FFT spectrum can cause oscillations in the inverse transformed data This is known as the Gibbs phenomenon Therefore instead of using an abrupt box car shaped filter function a smooth bell shaped sine and cosine functions are applied over certain cut off range In low pass and high pass filtering this means in practice that instead of a single radius of a single filter ring one needs to provide the radii for the inner and outer ring of the cut off 15 range separately Between the two rings the filtering is made using a quarter sine or cosine functions Likewise in directional filtering o
21. er padding and tapering effectively remove so called Gibbs phenomenon as well as ringing and other artifacts in the inverse transformed data Padding will be discussed more in the next chapter FOURPOT does not consider what kind of data that are passed to it Strictly speaking pole reduction and pseudo gravity for example are applicable to magnetic data only For teaching and testing purposes the Data type menu item allows defining the actual data type In practice this has effect only on the information text next to the graph and the scaling of the radial amplitude spectrum see chapter 4 7 The Miscellaneous menu item contains following items Contour Image change between contour map and image pixel map modes Show Hide points show hide the actual data points in original data view Meters Kilometers swap units between meters and kilometers 6 Reverse sign reverse the sign of the original data Autom low pass filter automatic low pass filtering when activated Note The information text next to the graph will show the dimensions incorrectly unless the Meters lt Kilometers menu item is utilized Although knowledge about the spatial dimensions is usually not required the pseudo gravity and pseudo magnetic field need to know the real dimensions for the amplitude of the transformed field to be more or less truthful The Reverse sign item can be useful when working on the potential of the original field since often potential
22. ernational geomagnetic reference field The Pole reduction item is used to transform magnetic field data as if it were measured on a magnetic pole This helps estimating the true dip and strike directions of the targets Important When performing pole reduction the inclination and the declination of the magnetic field are read from the Incl and Decl text fields The values are provided as angles in degrees Inclination J 90 is taken from horizontal plane and it is positive downwards northern hemisphere and negative upwards southern hemisphere Declination D 90 is positive in clock wise orientation from the true north direction The pole reduction is unstable operation if the inclination is very small near the equator The vertical and horizontal x and y gradients are used for the visual enhancement of some features in the data Both first and second degree gradients of the data f f x y can be computed Here x direction represents the horizontal axis and y direction the vertical axis of the mapped data The vertical gradients are particularly useful in enhancing the lateral dimensions of anomalous sources df dz for magnetic data and d f dz for gravity data The less usual gradients d f dxdy d f dxdz d f dydz can be used to compute the off diagonal components of the gravity tensor provided that the gravity potential is first computed Examples of gradient operations are shown in Appendix J and K The horizon
23. h estimates are quite different Depending on the data the normalization might not be useful at all so one should prefer to general data type Please refer to Bhattacharyya 1967 and Ruotoistenm ki 1987 for more information about the use of radial amplitude spectrum for depth determination Note that the depth interpretation option has not been fully tested in this version of FOURPOT Interactive line fitting is accomplished by pressing the Define line button when radial amplitude spectrum is displayed The mouse pointer changes into a cross hair cursor over the graph area The user should select the start and end points of the line over the scattered data plot of the radial amplitude spectrum by pressing the left mouse button The editing mode is ended by pressing the right mouse button The graph will be redrawn with the newly edited line on it and the slope of the line will be displayed on the information text on the right side of the graph Multiple lines can be added to the same graph pressing the Define line button again The most recent line can be deleted from the graph pressing the Delete line button Note Low pass and high pass filtering affect the computation of the radial spectrum Because of the cut off range of the filters the radial spectrum may look quite different from the one shown in Fig 4 3 for example Also note that the vertical axis of the radial spectrum and 2D spectrum as well is logarithmic and that the nulled values
24. ing magnetic field must be provided in the dialog that appears after using the Pseudo gravity or Pseudo magnetic menu items Also as discussed in previous chapter the actual units of the spatial dimensions must be defined in order to compute the amplitude of gravity effects An example of pseudo magnetic field is shown in Appendix M Note that the validity of the pseudo magnetic and pseudo gravity field computations has not been verified in this version of FOURPOT The potential of the gravity and magnetic field data can be computed also In the latter case the current values of field inclination and declination are used to account for the direction of the inducing magnetic field The Exit menu has two items The first one can be used restart the whole program in a mode that is more suitable to widescreen or normal 4 3 displays The second menu item is used to confirm the exit operation Before exit the user should save processed data because they are not saved automatically Errors that are encountered before the GUI starts up can be found from the FOURPOT ERR file When operating in GUI mode run time errors arising from improper parameter values for example are shown on the display screen 10 4 Using the program When the program is started it reads graph parameters and some additional settings from the FOURPOT DIS file see chapter 6 for more information If this file does not exist default parameters will be assigned and the file is
25. ith continuous first derivatives the Fourier transform F k and its inverse transform can be written as e g Blakely 1995 F k ae f x dx and f x md fet Fk dk 2m co 0o Considering discrete and 2D data the governing equations are nk ml N M aE 1 Fose NE fig and UT k 1 1 n 0 m 0 N 1M 1 nk ml 3 a AINW F where N and M are the number of data values in x and y directions The transform is complex which means that it has both amplitude and phase spectra FOURPOT computes the discrete 2D Fourier transform using the fast Fourier transform FFT algorithm The Fourier transform represents a sum of sine and cosine terms with different spatial frequencies k and ky that are defined by data sampling dy and dy in x and y directions The highest spatial frequency is the so called Nyquist frequency e g max k 0 5 d The lowest frequency is based on the data coverage e g min kx 0 5 max x min x Considering that the inverse of the spatial frequency represents wave length A I k zero frequency means infinite wave length i e constant level of data Because of the properties of the Fourier transform symmetry linearity shift and derivate properties several computational operations can be performed in Fourier transformed frequency kx ky domain more efficiently than in the spatial x y domain For more detailed information about Fourier transform methods in potential field analysis please see
26. k to unprocessed but frequency filtered data Upward continuation makes the data appear to have been measured higher above the surface of the earth or the plane of measurements It is used to estimate the low frequency regional trend of the data Likewise downward continuation makes the data appear to have been measured below the plane of measurements i e inside the earth It is used to enhance the high frequency content of the data and to estimate the depth to the top of the targets Unlike upward continuation downward continuation is not a stable procedure If the plane of continuation is located below the actual potential field sources the results become erratic Examples of upward and downward continuation are shown in Appendix I Important When performing upward or downward computation the height difference or elevation is read from the Height text field in the control pane on the left side of the graph area see Appendix A The height value is always positive that is to say the difference is computed downwards or upwards depending on the selected task The dimension of height is the same as for the data itself e g kilometers or meters depending on sampling Unlike gravity field the static magnetic field of a symmetric body e g vertical prism exhibits non symmetric anomaly shape because the inclined direction of the inducing magnetic field The behavior of Earth s magnetic field can be estimated using the IGRF model int
27. line of the data file and asks the parameter NOP ICO1 ICO2 ICO3 values from the user 5 2 Regularly gridded matrix files The matrix format MAT expects that the data are stored in a regular grid without x and y coordinates The matrix format is provided for easier functionality with programs such as Matlab and Maple The format of a matrix file DAT is illustrated in the example below 22 64 64 0 43160E 09 0 43320E 09 0 43270E 09 The header line defines the number of data points in x and y directions NP and MP The following NOP NP x MP lines contain the actual data values Note that because the matrix format does not contain coordinates the sampling frequencies and corresponding wave lengths shown by FOURPOT are not correct if they are not provided correctly in the dialog that appears after the matrix data are read in The dialog window is used to define the x and y coordinates of the origin and the x and y step between the matrix elements Note that if the x and or y steps are negative the axis gets reversed Moreover the data are mirrored in x or y direction if the last two parameters from zero to one in the auxiliary dialog Since the matrix file does not contain coordinates it is important to know the order in which the data are stored The origin is always located in the bottom left corner By default the data is stored in column wise fashion from bottom to top and from left to right Using a generic pr
28. lts for different graphs one can reset the color mapping parameters to their defaults using the Reset scale button Note that FOURPOT was not designed to be used as a plotting program To prepare the final results the user is advised to use plotting programs like Golden Software s Surfer 21 5 File formats Potential field data can be read into FOURPOT in two file formats a generic column formatted data file DAT and b regulary gridded data matrix MAT 5 1 Column formatted data files The format of a DAT file is illustrated below 4096 1 2 4 0 00 0 00 0 8665106E 01 0 33489 10 00 0 00 0 8558651E 01 0 44134 20 00 0 00 0 8365253E 01 0 63474 The data file can contain multiple columns and the header line at the first line is used to define the number of lines NOP and the indices of the columns that contain the x and y coordinates ICO1 ICO2 and data ICO3 FOURPOT ignores empty lines and lines starting with 1 L or characters Thus provided that NOP is large enough ICO refer to correct column indices FOURPOT can also read Geosoft XYZ files X YZ illustrated below 5000 1 2 4 Test data LINE 1 0 00 0 00 0 8665106E 01 0 33489 10 00 0 00 0 8558651E 01 0 44134 20 00 0 00 0 8365253E 01 0 63474 LINE 2 0 00 10 00 0 8165106E 01 0 23489 10 00 10 00 0 8258651E 01 0 34134 20 00 10 00 0 8465253E 01 0 53474 Note The header line can be omitted in which case the program shows the first
29. meters are used if the text fields are equal to zero 18 4 7 Radial amplitude spectrum The Radial spectrum button which is active only when the 2D frequency spectrum is active will compute the amplitude spectrum as a function of the radial frequency The radial amplitude is the mean of the 2D Fourier amplitude spectrum A F l Re F Im F ve along rings with radius k ky k The amplitude spectrum has traditionally been used to estimate the depth to the bottom of potential field sources This is established by fitting linear lines to the decaying amplitude curve on a semi logarithmic scale An example is shown in Figure 4 3 2 D Fourier transform analysis Value log F F i kr Figure 4 3 Radial amplitude spectrum with two manually fitted lines The slopes give estimates for the depth to the bottom of the potential field sources In frequency domain the Fourier transform of a potential field can be formulated as FxCe giving log F C h k Thus the depth to the top of an anomaly source A is equal to the tangent or the slope of the linear parts of the amplitude spectrum For general data type the coefficient C 1 and the vertical axis is the logarithm of the amplitude spectrum loglFl For gravity data the coefficient C 1 k k k and in case of magnetic data C 1 k k selected by the Data type item in Edit menu Because of the data type the slopes and therefore also 19 the dept
30. ne needs to provide three values that define the upper and lower sector range and the width as an angular difference of the cut off range The filter rings and sectors and the cut off ranges are illustrated in Figure 4 2 a 2D low pass filter b 2D direction filter all values cut off cut off nulled range ranges all values g preserved lo upper and lower sector outer ring digo k 0 k 0 k 0 Figure 4 2 Schematic view of a cut off rings and b sectors of 2D filters and the smooth bell shape filter function of c low pass d high pass and e band pass filters 4 6 1 Low pass and high pass filtering In low pass filtering all data outside the outer ring high frequencies are nullified and all data inside the inner ring are preserved see Figure 4 2a and c On the contrary in high pass filtering all data inside the inner ring low frequencies are nullified and all data outside the outer ring are preserved see Figure 4 2d The radius of the rings are defined as wave length A which is the inverse of radial frequency k ktk Where k and k are the axes of the 2D frequency graph The unit of the wave length is either meter or kilometer depending on the selection made using the Meters lt Kilometers item in the Edit Miscellaneous menu 16 Note If the sampling intervals are different in the x and y directions the spatial frequencies kx and ky will be different also Because a single value of wave length is
31. nt shall the author or the University of Oulu be liable for any indirect or consequential damages or any damages whatsoever resulting from loss of use data or profits arising out of or in connection with the use or performance of this software 10 Contact information Markku Pirttijarvi Department of Physics Tel 358 8 5531409 P O Box 3000 Fax 358 8 5531484 FIN 90014 University of Oulu E mail markku pirttijarvi at oulu fi Finland URL http www gf oulu fi mpi 27 Appendix A FOURPOT GUI after data input 2D Fourier analysis v 1 0 by M Pirttijarvi 2009 Ga File Edit Process Exit Plot data Def size eee Generic data Plot FFT 2 D Fourier transform analysis Dimensions 52x 48 km ok Verse igi dat Ompinal data Spacing 1 00x 1 00 km Dim x 52 Dimy 48 1830 Field parameters Height units 0 00 Incl deg 750 lee 10 0 1820 iter parameters Ring 1 m 0 00 Ring 2 m 0 00 Sect 1 dea 7000 Sect 2 deg 0 000 Width deg 7000 gt 1810 fear i 1800 E gt l a Update plot Reset scale Depth spectra Flediel Spectr 1790 Denne Ine Te Ine 2840 2850 2860 2870 2880 X 56 648 38 820 20 991 3 163 14 665 Field Appendix B Data map after padding with shift and tapering Generic data 2 D Fourier transform analysis Dimensions 64x 64 km Interpolated data 1840 1830 1820 gt 1810 1800
32. ogramming language notation we would have do i 1 np do j 1 mp read or write f i j end do end do When matrix data are read in the order can be changed so that for each column the rows will be read by giving a negative value for the number of points in x direction NP NP Furthermore if MP lt 0 the matrix file is considered to have multiple data columns which will be read row by row the file really looks like matrix If NP gt 0 and MP lt 0 the rows are considered to be the y columns of the mapped data as in the default case If NP lt 0 and MP lt 0 the rows are considered to be data rows but the origin is still in lower left corner Using a generic programming language notation we would have in this case do j 1 mp read f i j i 1 np end do 23 6 Graph options Several graph parameters and text strings can be changed by editing the FOURPOT DIS file This allows one to localize the graphs into another language for example Note that if the format of the FOURPOT DIS file is important If the format of the file becomes invalid one should delete the file and a new one with default parameter values will be generated automatically next time the program is started The file format is illustrated below Fourpot ver 1 10 parameter file 36 32 24 0 0 0 0 2 1 0 1 0 0 0 350 600 0 90 0 80 0 91 0 00 0 00 0 00 75 0 10 0 52000 0 1 00000 0 01000 0 0 0 0 2 D Fourier transform analysis Original data Interpolated data T
33. ppendices G and H The frequency filtering can be accomplished either graphically with the mouse or manually using the given numeric values Frequency filtering will be discussed more in the next chapter 3 3 Process menu The Process menu contains the items for data processing operations Upward continuation Downward continuat Pole reduction dF dz gradient dF dx gradient dF dy gradient d F d gradient d F d gradient d F dz gradient d F dxdy gradient d F dxdz gradient d F dydz gradient Hori gradient Total gradient Tilt gradient Pseudo gravity Pseudo magnetic performs upward continuation performs upward continuation performs reduction to magnetic pole computes vertical gradient of the data computes horizontal x gradient of the data computes horizontal y gradient of the data computes second vertical gradient of the data computes second horizontal x gradient of the data computes second horizontal y gradient of the data computes xy gradient of the data computes xz gradient of the data computes yz gradient of the data computes horizontal gradient computes total gradient analytic signal computes tilt gradient computes pseudo gravity field from magnetic data computes pseudo magnetic field from gravity data Gravity potential computes gravity potential from gravity field Bouguer data Magnetic potential computes magnetic potential from total field TMI data Undo process reverts bac
34. r program widgets become inactive and the mouse cursor changes from normal arrow into a cross hair above the graph area note only above the graph area The 2D frequency spectrum is shown together with auxiliary polar coordinate system grid The user is given instructions to provide first the inner and then the outer ring by pressing the left mouse button above the graph To validate the last value and to end the editing mode one must press the right mouse button The distance from the origin determines the radius of the filter rings Likewise in directional filtering the user is instructed to provide the upper and lower angle of the filter sector and the width of the cot off range 17 The width can be defined as a angular difference from the lower or upper sector and the filter cut off range will be adjusted symmetrically at both sides automatically Note that the ring and sector values can be provided in reverse order in which case they will be automatically arranged so that the inner ring is smaller than outer ring or the upper angle is larger than lower angle Values can be omitted by pressing the right mouse button instead of the left one In low pass and high pass filtering omitting the first value cancels the whole interactive editing mode Omitting the second value will make the two rings the same which means that the sine cosine bell function is replaced by a box car filter In directional filtering the upper and lower angle must always b
35. ransformed data Inverse data Difference Radial ave spectre X Y Field KX Ky log10 F Kr log F Low pass filtered High pass filtered Direction filtered Upward continued Downward continued Pole reduced dF dz gradient dF dx gradient dF dy gradient d 2SF dz 2S gradient d 2SF dx 2S gradient d 2SF dy 2S gradient d 2SF dxdy gradient d 2SF dydz gradient d 2SF dydz gradient Horizontal gradient Total gradient Tilt gradient Pseudo gravity Pseudo magnetic Gravity potential Magnetic potential Unprocessed data 24 The two first lines are used as a header to identify the file and version number The 1 st actual data line defines three character height values The first one is used for the main title the second one is for the axes labels and the third one is used for the plot legend The 2 nd line defines some integer valued options The first one sets the screen mode between screen normal 3 4 aspect ratio and wide screen mode The second one defines the color scale The third and the fourth parameter define padding and tapering modes The fifth parameter sets either contour or image map mode The sixth parameter defines the use of automatic low pass filtering mode The extra parameters are reserved for future use The 3 rd line defines the x horizontal and y vertical distance integer values of the origin of the main graph in pixels from the bottom left corner of the page and the length of th
36. tal gradient h x y d f dx d f dy total gradient a x y d f d d f dy dfa also known as analytical signal tilt gradient t x y arctan df dz h x y and the horizontal gradient of tilt gradient s x y d dx d t dy are useful operations that can used in visual interpretation of magnetic and gravity maps They are used especially to produce the location of lateral contacts of different geological units Examples of these gradient operations are shown in Appendix L and M Note The horizontal gradient of tilt gradient s x y can be computed manually as follows First the user needs to compute the tilt gradient t x y and use Inverse gt data menu item to replace the original data Then one computes the 2 nd horizontal x and y gradients d t dx and d t dx and saves their results into two separate files Finally using a spreadsheet program one computes the amplitude of the horizontal tilt gradient s x y d dx d v dy The gravity and magnetic potentials and hence the fields of an anomalous target with the same dimensions but different petrophysical properties density and magnetic susceptibility are interrelated via Poisson s relation The so called pseudo gravity field can be computed from the measured magnetic data Likewise pseudo magnetic field can be computed from measured gravity data For these operations some assumed values of density susceptibility and the intensity of the induc
37. ty magnetic Miscellaneous define some additional settings Low pass filter perform GUI directed or manual low pass filtering High pass filter perform GUI directed or manual high pass filtering Direction filter perform GUI directed or manual direction filtering The use of Jnverse data menu item can be used to put the current inverse results as new input data for successive frequency domain operations For example tilt gradient can be computed freom the upward continued data Likewise the Subtract inverse menu item allows computing the residual field directly as the difference between the original interpolated data and the upward continued or low pass filtered data On the contrary the Add inverse menu item is used to add the inverse results to the original data Although this item is not very useful it can be used to enhance high frequency content of the data for example The Padding mode menu item is used to demonstrate the way padding and tapering is made Padding is an operation where null values are added around the original data area see Appendix B The purpose of padding is to extend the data N and M to even power of two e g 64 128 256 512 etc as the FFT algorithm requires this Tapering is an operation where the padded values are made such that they prevent rapid amplitude changes at the border of the data area This means that the data and their derivative are made more or less continuous in padding Togeth
38. used as filter dimension both in x and y directions the filter rings will appear as an ellipses instead of circles in the 2D frequency graph 4 6 2 Directional filtering Directional filtering is equal to the so called fk filtering commonly used in seismic and GPR data processing Directional filtering removes or enhances linear features with certain angle in the data In directional filtering the data are removed inside two sectors that are symmetric around the origin see Figure 4 2b The filter removes the frequency data inside a sector defined by two angles that correspond to the upper and lower limit of the sector The angles are defined in degrees in counter clockwise direction taken from the positive k axis In addition the cut off range is defined by providing a third parameter as an angular difference either from the upper or the lower sector the width of the cut off range will be the same on both sides Because of the symmetry of the frequency spectrum the directional filter will be mirrored around the origin automatically Thus one needs to define only single sector either above the k axis or left to the k axis 4 6 3 Interactive and manual filtering Filtering can be done either a interactively using the mouse or b manually by providing the numerical values of the filter cut off ranges The selection is made when the corresponding menu item is chosen in the Edit menu When interactive mode is chosen most of the othe
39. users can see the effect and importance of padding and tapering 13 4 4 Fourier transform After sampling interpolation and padding one can use the Plot FFT button to compute the FFT and to see the 2D frequency spectrum see Appendix C At this point one should remember that the frequency spectrum F is complex i e it has both real and imaginary parts F Re F i Im F Thus the 2D Fourier graph actually represents the 10 base logarithm logio of the amplitude spectrum A Re F Na Im F jej as The origin k 0 k 0 of the graph is located at the middle of the graph and the horizontal and vertical axis range between the Nyquist frequencies the value of which are shown in the information text next to the graph Lowest frequencies or largest wave length long variations of data are located in the middle of the graph and highest frequencies or shortest wave lengths short variations of data are located far from the origin Continuous linear features in the data appear as linear features in the FFT spectrum as well the angle of slope is inversed though Only after the FFT has been computed one can perform the frequency filtering and other frequency domain processing The inverse FFT and the difference between the original gridded data can be plotted using Inverse FFT and Plot Note that each Fourier operation is made on the interpolated and padded data matrix Plot matrix Padding Multiple Fourier operations e g pol
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