IRIS-TR-0060 01-11-1999GROUND PENETRATING RADAR DATA PROCESSING: ASELECTIVE SURVEY OF THE STATE OF THE ART LITERATURE.Luc van Kempen, Hichem Sahli

IRIS: Information Retrieval and Interpretation SciencesA member of: ETRO(Electronics and Information processing)IMEC(Interuniversity Micro Electronics Center)Address: Luc van Kempen, Hichem SahliVUB(ETRO), Pleinlaan 2, 1050 Brussels, BelgiumTel.: ++32-2-629 2858Fax.: ++32-2-629 2883Electronic mail: lmkempen,hsahli@etro.vub.ac.be

Ground Penetrating Radar dataprocessing: a selective survey of thestate of the art literature.

L. van Kempen, H. SahliVrije Universiteit Brussel - Faculty of Applied SciencesETRO Dept. IRIS Research groupPleinlaan 2, B-1050 Brussels - BelgiumE-mail: lmkempen,hsahli@etro.vub.ac.beIRIS-TR-0060

AbstractThe main purpose of this text is to give an overview of the recently published algorithms and methodsin the �eld of GPR data processing for anti-personnel mine detection. Although some of the papersmentioned do not directly address this topic, the ideas and methods explained in them can be generalizedfor the application of detecting man made objects in subsurface layers.Two main subsections will be distinguished: The processing of vertical slices called B-scans, and thatof horizontal slices called C-scans.Within these subsections, several processing aspects will be addressed, and each time a brief introduc-tion is followed by an overview of the described methods.

Contents1 General Introduction 32 B-scan Processing 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Source Signal Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Hyperbola estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Soil parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.2 Permittivity and conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.3 Velocity of propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5.4 X-T-V data matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6.2 Kircho� Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6.3 Some First Results on GPR Data (Obtained by our own tests). . . . . . . . . . . . 172.6.4 Application to electro-magnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . 172.6.5 Phase Shift Migration [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6.6 FK Migration [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6.7 Comments and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 C-scan processing 233.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Array Beam Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.3 Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.4 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.5 Identi�cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.6 Averaging and �ltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4.3 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.4 SAR Polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5 Karhunen Loeve transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5.2 Application and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5.3 First results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.6 Clutter removal and C-scan reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

CONTENTS 23.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.6.2 Restoration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.6.3 C-scan reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.6.4 Comments and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Chapter 1General IntroductionThe main purpose of this text is to give an overview of the recently published algorithms and methodsin the �eld of GPR data processing for anti-personnel mine detection. Although some of the papersmentioned do not directly address this topic, the ideas and methods explained in them can be generalizedfor the application of detecting man made objects in subsurface layers.Two main subsections will be distinguished: The processing of vertical slices called B-scans, and thatof horizontal slices called C-scans.Within these subsections, several processing topics will be addressed, and each time a brief introductionis followed by an overview of the described methods.

3

Chapter 2B-scan Processing2.1 IntroductionWhen one considers a line of equidistant one dimensional signals called A-scans one gets a two dimensionalstructure (B-scan) that characterizes the properties of the subsurface as a function of depth. The generalpurpose of processing these data structures is to extract these subsurface properties.We will start with some preprocessing techniques as �ltering and deconvolution. These steps arenecessary to enhance the quality of the data, so that the extraction of signal features is easier and moreaccurate in further steps. Secondly we will discuss a method for extraction of the hyperbolic features inthe B-scans. Then some ideas on how to estimate the physical properties of the soil will be proposed. Wewill end with an overview of migration techniques.2.2 FilteringOne of the major problems in processing GPR data is the removal of the air ground re ection. In [11] anumber of di�erent algorithms to do so were compared and evaluated on synthetic data, which consistedof a di�raction hyperbola, and a number of background clutter signals. Five algorithms were tested,namely:� Subtraction of the average A-scan. When calculating an average A-scan over the whole B-scan,and subtracting this from each individual A-scan, one only succeeds in eliminating the perfectlyhorizontal background clutter signals. This is the most classic and widely used way to eliminatethe problem of the air-ground re ection, but in many cases it is not very e�cient.� Application of a horizontal high pass Butterworth �lter. The paper shows that this will reduce thedi�erent background signals e�ectively, but will also modify the di�raction hyperbola substantially,which reduces the e�ciency of this method.� FK-�ltering. This was not only the most time consuming method, but the results are not evenremotely satisfactory.� The application of the Discrete Wavelet Transform. This seems to be the most interesting method.The transform was applied in the horizontal direction, and the coe�cients of the lowest octaves(which correspond to horizontal energy) were set to zero, after which the inverse transform wasapplied. The wavelet used in this case was a Daubechies wavelet with 20 coe�cients. Nothing issaid about the calculation cost of this method, but the �rst results seem to be satisfactory. Someadditional �ne tuning on the choice and the length of the wavelet can be done.� Windowed average subtraction. This is an adapted version of the general average subtractionmethod. Here the average A-scan is not calculated over the whole B-scan, but only over theA-scans within a window around the signal that is treated. This subtraction can adapt to slowlyvarying signals so that slightly oblique surface re ections can be eliminated.On the synthetic data that was presented to the �ve algorithms, the two latter (wavelet and slidingaverage) gave the best results in eliminating the various background signals and preserving the di�raction4

CHAPTER 2. B-SCAN PROCESSING 5hyperbola. O� course the results of applying these methods to real data remains to be seen. The slidingwindow averaging will only result in good background elimination if the number of A-scans that is presentover the width of one di�raction hyperbola is large (or in other words if the size of the sliding window canremain small with respect to the hyperbola size). The wavelet �ltering has to be investigated further.In [22] some processing is suggested to improve resolution of near surface targets. Although theGPR used for this experiment is of a lower frequency (100 MHZ) and the dimensions of the consideredobjects are thus larger, the resulting data is quite comparable with that of a higher frequency GPR andsmaller objects. First the deeper clutter in the B-scan is considered to be the result of a low frequencycomponent called wow. To reduce this, every A-scan is �ltered. A residual mean �lter with a 25 pointwindow (residual means that the �ltered data is subtracted from the raw data) is compared to a bandpass�lter (here 20-500 MHz) and to a residual median �lter. The length of this median �lter is derived fromthe spectrum of the signal. If the low frequency component is clearly separable form the higher frequencysignal, a reasonable cut o� frequency can be chosen. The cut o� period is then divided by the samplinginterval to get the number of data points. This way each trace is �ltered with its own �lter length. Thislast method was shown to reduce the clutter quite e�ectively. However the choice of the cut o� frequency,which was evident in the shown data is not that self evident in all GPR data.Another problem studied in [22], was the time zero aligning. The method that is proposed to compen-sate for sloping surfaces or time drifts in the measurement system is to align the A-scans according tothe large air-soil interface peak. This was done manually. An automated version can be considered, butis not mentioned in the paper.Another idea presented here is called eigen �ltering. This can be used to eliminate the large air groundre ection. The idea is to use the singular value decomposition (SVD) to decompose the image intoorthogonal components (Eigenimages), ordered by decreasing coherence. The image can be partiallyreconstructed omitting, some of these eigen images. Omission of the �rst one already eliminates thebackground re ection, but even better results are obtained when "band-passing" the eigen images (usingthe second up to the fourth ones). This seems to yield a very good result on the shown data, but theapplication on other GPR data remains to be studied.Another method applied to �lter B-scan images is proposed in [24]. Here the main goal is to reducethe speckle noise that is present in the image. The approach considered is a nonlinear multi-scale one.The multi-scale pyramid decomposition and reconstruction used here, decomposes the image um�1 atscale m � 1 into a coarser image um by low-pass �ltering and down-sampling. The detail image dm isobtained by subtracting an interpolated version of um from um�1.In the reconstruction step the interpolated image is passed through an edge detector. This will give amask with edge locations that is multiplied with the detail image of the same scale. Finally the sum ofthis modi�ed detail image and the interpolated image will give the reconstructed image to be passed onto the reconstruction step at the next scale.For the low-pass �lter a non-separable �lter de�ned by:Gns(!1; !2) = 8<: 1; j!1j+ j!2j < Bns1=2; j!1j+ j!2j = Bns0; otherwise (2.1)where Bns is �xed at 2�=3, is compared with a separable one de�ned by:Gs(!1; !2) = G(!1)G(!2); G(!) = 8<: 1; j!j < Bs1=2; j!j = Bs0; otherwise (2.2)and the latter one seemed to give better results.For the edge detection the Laplacian of Gaussian (LoG) is used, which corresponds to a bandpass�lter whose passband is determined by the spatial dispersion �G of the underlying Gaussian pulse. Afterapplying this �lter the zero crossings of the output are considered edges if the brightness gradient exceedsa threshold T .

CHAPTER 2. B-SCAN PROCESSING 6The method was applied to a real GPR image with Bs = Bns=p2. The parameter �G is chosen equalto 3. The threshold T seems not to be critical. The paper does not mention any values for the �lterfrequencies, but those have to be adjusted to the used data. The speckle noise reduction is best withthe separable �lter, but the elimination of the air soil interface is done in the traditional way (averagesubtraction).The Wiener �lter is proposed as a B-scan �ltering method in [25]. The Wiener �lter is an adaptivedigital �lter. The fact that it is adaptive means that it will calculate its parameters, according to thedata present locally in the image. The theory is based on the principle of mean squares estimation andis summarized as follows:The posed problem can be written as: y(n) = s(n) + v(n) (2.3)with y(n) a signal composed by an original signal s(n) and a noise signal v(n) (all values are zero forn < 0).Out of this signal we will try to calculate an estimation of s(k) called s(n). This will be written in theform: s(n) =Xk h(k)y(n� k) (2.4)where h(k) is independent of n. Our problem is now to �nd the values of h(k). These values will becalculated while minimizing the approximation error:J = E[e2(n)] = E[(s(n)� s(n))2] = E[(s(n)�Xk h(k)y(n� k))2] (2.5)Minimization will lead to: @J@h(i) = E[2(s(n)�Xk h(k)y(n� k))de(n)dh(i) ] (2.6)Since de(n)dh(i) = �y(n� i) (2.7)it goes that: E[(s(n)�Xk h(k)y(n� k))y(n� i)] = 0 (2.8)This can be rewritten as: Rsy(i) =Xk h(k)Ryy(i� k) (2.9)We see that the values for the �lter parameters are dependent on the autocorrelation function of thesignal Ryy and the cross correlation function Rsy between the signal and the original. Solving thisequation for h(k) can be done by taking the transform of the equation:H(z) = Ssy(z)Syy(z) (2.10)On the basis of the data one can calculate the signal spectrum Syy, and estimate the noise spectrumSvv. If we also assume that the noise and the original signal are not correlated we can write that:Ssy = Sss = Syy � Svv (2.11)so the �lter parameters can be calculated.

CHAPTER 2. B-SCAN PROCESSING 7The �ltering of the data was done in a matlab environment, which �lters the B-scans in blocks. Thebest values for the parameters (block size for �ltering and noise estimation) were chosen according to theempirical SNR de�ned as:SNR = average energy in object signalsaverage energy in background signals � 1 (2.12)The results were surprisingly good (compared to other methods) since the background and measurementnoise were almost completely attenuated, without distorting the signal itself.An example of the �lter results by the Wiener �lter can be found in �gure 2.1

Figure 2.1: The results of the application of the optimized Wiener �lter to a B-scan. Bottom: A detailedview of one of the A-scans, before and after �ltering, top: The B-scan before and after �ltering.2.3 Source Signal DeconvolutionIn order to increase the resolution of the GPR source signature, deconvolution in GPR data was appliedin [10]. The deconvolution was performed using complex spectral division.The recorded signal r(t) can be represented as a convolution of the source signal s(t), the responsefunction of the ground g(t) and the response of the receiver h(t): r(t) = s(t) � g(t) � h(t). Fouriertransforming to the frequency domain gives: R(f) = S(f)G(f)H(f).The main aim of deconvolution is to remove the e�ect of the source waveform from the recorded data:R0(f) = R(f)=S(f) = G(f)H(f). Due to the band limited nature of the emitted wave and the e�ect ofnoise, deconvolution is an ill posed operation. To regularise this, a white noise is added, middled by a socalled water lever parameter �. The new equation will become:R0(f) = R(f)S�(f)=(S(f)S�(f) + �) (2.13)where S�(f) is the complex conjugate of S(f). The value of � was set to 1% of the maximum sourcesignal power. Inverse Fourier transforming the function R0(f) will result in the deconvolved function

CHAPTER 2. B-SCAN PROCESSING 8r0(t), which (if the receiver response is assumed a constant) is equal to the ground response g(t), whichwas searched for.One of the major problems in applying this method is the knowledge of the exact source signal. Inthis article this is experimentally estimated in two ways: �rst by measuring and averaging the directwave between the source and the receiver, when both are coupled to the ground, and second by liftingboth into the air and measuring the transmitted signal in that case. The second method gives no usefulresults because the e�ects of coupling to the ground on the emitted wave are not accounted for. The �rstmethod results in a useful wave, which used in the deconvolution is able to improve interpretation of thedata (in this case only geologic structures were investigated). When looking at the power spectrum ofthis source signal, however, the peak energy occurs at a frequency that is 30% lower than the nominalfrequency of the antenna. The main conclusions of this article are that source signal deconvolution canbe useful in interpreting the data, but that an accurate way to measure the source signal still has to befound.2.4 Hyperbola estimation2.4.1 IntroductionWhen scanning over a re ective object with a Ground Penetrating Radar, the B-scan will show a hyper-bolic structure in the re ection. This hyperbola contains information about the re ective object itself (tobe used as features in a pattern recognition program), the position of the object (can be used in C-scanprocessing and aligning operations) and the soil structure (the velocity of propagation in the soil can becalculated from such a hyperbola as shown in the next section). However, to be able to use the hyperbolafeatures for all these applications, the �rst step has to be the extraction of the hyperbola from the B-scanimage.2.4.2 TheoryIn [23] a method for determining linear structures (as the air soil interface) and hyperbolas is introduced.It is based on a modi�ed version of the Hough transform. A brief overview of the theory is given.Standard HT for straight linesAny point (Xi; Yi) lying on a line satis�es � = Xi cos �+ Yi sin � where � and � are the polar coordinatesof the line. These polar coordinates are then discretized. For each bright pixel (Xi; Yi) in the image allallowed values of � are scanned and each time the corresponding � is calculated. This will result in asinusoidal curve in the Hough space (�, � space) and these curves will intersect for all points on the sameline. The coordinates of the intersection will give the detected line. Each occurrence of �, � is counted,and the high spots in the distribution will de�ne the detected lines. So, both for the selection of thebright pixels, as for the selection of detected line, a threshold has to be applied.HT for hyperbolasIf D is the depth of the scattering object, Sc its horizontal position, and S the horizontal position of theantenna, the Time Of Flight (TOF) of the wave is:TOF = 2(D2 + (S � Sc)2)1=2=v (2.14)with v the velocity of propagation of the electromagnetic waves in the medium. Since both axes aredisplayed with discrete values one can write: TOF = j�T , S = i�x and Sc = ic�x (�T;�x are thehorizontal and vertical scanning steps). This discretisation will lead to the following parametric equationof the hyperbola: j2 = �+ �(i� )2 (2.15)with the three parameters �, � and given by:

CHAPTER 2. B-SCAN PROCESSING 9� = 4D2=�T 2v2 (2.16)� = 4�x2=�T 2v2 (2.17) = ic (2.18)By picking three distinct bright points in the image [PL = (iL; jL); PM = (iM ; jM ); PN = (iN ; jN )] onecan obtain the values for �, � and : = [(i2N � i2L)(j2M � j2N )� (i2M � i2N )(j2N � j2L)]=2[(iN � iL)(j2M � j2N )� (iM � iN)(j2N � j2L)](2.19)� = (j2M � j2N )=[(iM � iN )(iM + iN � 2 )] (2.20)� = j2L � �(iL � )2 (2.21)NOTE that in the formula in the article an error was discovered: the factor two in the denominator of was placed in the wrong position. The corrected formula is shown here.The accumulation of votes of the triplets �, �, , corresponding to possible hyperbolic arcs occurs ina 3D array while scanning over the bright pixels of an image. The high spots in the accumulator space(�, �, ) provide the parameters that identify the most likely hyperbolic arcs.This is a very calculation intensive method. Therefore the Randomized Hough Transform (RHT) isproposed. This accumulates votes only to curves passing through triplets of points selected at random.This is repeated a �xed number of times (500 to 6000). If we automatically discard triplets that generatehyperbolas with upward pointing tails, (which have no physical meaning) the number of calculations isfurther reduced.2.4.3 ResultsThe results shown in the paper [23] are on synthetic data, and one real image (a DeTeC image). Theresults seem very good, the hyperbolas are well detected. The method remains to be tested on otherimages, and also remains to be seen what happens if the average subtraction is not applied.2.5 Soil parameter estimation2.5.1 IntroductionIn many processing algorithms based on GPR data such as migration (see 2.6) or tomography (see 3.2),the knowledge of the physical properties of the local soil is needed. The most important of these propertiesis the velocity with which the signal will propagate in the material. This will highly in uence the shapeof the di�raction hyperbola in the B-scans and thus also any use of this hyperbola thereafter. Since thisvelocity (v) is dependent on two other properties (permittivity (�r) and conductivity (�)), an estimationof these parameters can also be useful.Since the knowledge of these properties can improve the use of GPR for mine detection both in theacquisition stage (use of the appropriate acquisition parameters) as well as in the processing stage, somearticles focus on obtaining an accurate estimate of these parameters, based on antenna characteristicsand simple local tests.2.5.2 Permittivity and conductivityIn [1], the local permittivity and conductivity of the soil are being estimated.The �rst step is to produce the antenna monograph. This is a two dimensional graph in the conduc-tivity - permittivity plane, that will show the resonant frequency and resonant resistance curves. Themonograph is calculated once for each antenna in lab conditions.The procedure that is followed is called the Spectral Domain Moment Method, and results in solvingfollowing matrix equation: [E] = [G][I ], where [E] is the electric �eld applied to the antenna, [G] iscomputed by a complex integration of Green's function with antenna geometry, and [I ] is the unknowncurrent density. When [E] is applied as a testing function (driven by the voltage V ), and [G] is calculatedthrough the correct formula's, this equation can result in [I ]. The input impedance can then be foundby dividing the applied voltage with the current (calculated from the current density).

CHAPTER 2. B-SCAN PROCESSING 10To obtain the monograph, the voltage V (and thus the test function [E]) is chosen and the matrix [G]is calculated for di�erent values of permittivity and conductivity. For each of these values the matrixequation is solved, and so the corresponding resonant impedance is calculated. The resonant frequency isthat frequency for which the reactance part of the impedance becomes zero, and the resonant resistanceis the resistance at that frequency.When on the speci�c location the resonant frequency is measured (measuring the impedance at di�erentfrequencies, and selecting the frequency with zero reactance) and the corresponding resonant resistance isdetermined, it will be su�cient to search for the intersection of the corresponding curves in the monograph,who's coordinates will give the local permittivity and conductivity.In practice this method leaves the problem of measuring the antenna's monogram, as well as the insitu measuring of the resonant frequency and impedance. These are things that remain quite di�cult tobe done in practice.2.5.3 Velocity of propagationA number of simpler, more easy to use methods to estimate the velocity of propagation v in the mediumare presented in [2].� Locating objects of known depth. If the depth zknown of a reference object is known, and the twoway travel time tpick is measured up to an error terr (tpick = tacc � terr) out of one of the A-scans,then: 2zknowntpick + terr � v � 2zknowntpick � terr (2.22)This will give an upper and lower boundary for the estimated velocity. The travel times can beestimated out of the number of samples in the one dimensional signal between the signal emissionand the �rst re ection peak, and the known sample frequency with which the signal was acquired.� Geometric scaling. If a reference object is used that re ects in all directions, and that is positionedat a known horizontal location (XY coordinates known), but at an unknown depth (Z coordinateunknown), the scanning movement will lead to a di�raction hyperbola. The length of the wavepathw will be w2 = 4(x2+ z2), where x, the horizontal distance between the object and the antenna, isknown and z, the depth of the object, is unknown. The expression for the two way travel time t(x)(which yields the hyperbola) is:t(x) = wv =r4x2v2 + 4z2v2 =r4x2v2 + t20 (2.23)Where t0 is the vertical two way travel time. If we can extrapolate the hyperbola out of the B-scanimage, we know for every x, the time t(x). This formula will then easily su�ce to estimate thepropagation velocity v.The problem of estimating the di�raction hyperbola out of the measured data was not at alladdressed in this paper, but it can be possible to do so using the Hough transform as described in2.4.� CDP recording. Here one will vary the o�set x between the emitter and receiver antenna and use aplane re ector at unknown depth. The wavepath between emitter and receiver can in this case beexpressed as w2 = x2+4z2, where x is the horizontal distance between the emitter and the receiver(this is twice the x of the previous method), and z the depth of the re ector. The formula of thetwo way travel time t versus the o�set x can then be given as:t(x) = wv =rx2v2 + 4z2v2 (2.24)This equation is quite similar to the one in the previous section. Again the velocity v can beestimated from at least two A-scans (if more scans are used the accuracy of the estimation willimprove).In practice the placement of a large plane re ector can cause di�culties. This is not discussed inthe paper.

CHAPTER 2. B-SCAN PROCESSING 11� Use of standard velocities. If the relative dielectric permittivity �r of the survey area is known (see2.5.2), v can be approximated by v = cp� , where c is the standard speed of light in vacuum.If, however the conductivity � of the local soil is also known a more correct formula for the localvelocity can be used: v = cp�r�rs 2p1 + (�=!�0�r)2 + 1 (2.25)where �0 = 8:8854x10�12As=Vm, � the magnetic permeability (usually chosen 1), � the conduc-tivity (S/m), and ! the frequency.The knowledge of the parameters permittivity and conductivity becomes thus the new problem. [1]tried to estimate them locally, but if the main goal is to know the local propagation velocity, oneof the three previous methods would be much more interesting. If the local properties of the soilare known (by lab tests or by general knowledge) these formulas can be useful.2.5.4 X-T-V data matrixThe method presented in [18] uses migration techniques to estimate the propagation velocity as a functionof depth. The main idea of this method is to use the fact that migration, performed with the correctpropagation velocity v will concentrate the energy present in the di�raction hyperbola in its apex. If thisprocess is then repeated for a number of possible velocities, the velocity which gives the best results canbe selected.A B-scan is considered as a two dimensional x� t radar re ection, with x the horizontal position, and tthe one way re ection. This image is then processed by migration in the wavenumber-frequency domain(FK migration, see 2.6) for various propagation velocities v. This will result in a three dimensional datastructure: the x � t � v data matrix. This will thus be a three dimensional intensity structure calledI(x; t; v).The re ected intensity will show a sharp peak in the v direction at the position of the hyperbolicapex of the re ection. In addition, the velocity v at which this intensity peak occurs coincides with thepropagation velocity between the ground surface and the depth of the hyperbolic apex.In practice for each value of space xi and time tj , the following values are calculated:� average intensity: ave(xi; tj) = 1n nXk=1 I(xi; tj ; vk) (2.26)This is the intensity for each position (x; t) after averaging over all the velocities v. (Here i; j andk range from 1 to n).� peak intensity (For one position (x; t) the maximum intensity over the v direction in absolute value.)p(xi; tj) = � max1�k�n I(xi; tj ; vk) ave(xi; tj) � 0�min1�k�n I(xi; tj ; vk) ave(xi; tj) < 0 (2.27)� peak value:pv(xi; tj) = � p(xi; tj)�maxfave(xi; tj); I(xi; tj ; v1); I(xi; tj ; vn)g ave(xi; tj) � 0p(xi; tj) + maxfave(xi; tj); I(xi; tj ; v1); I(xi; tj ; vn)g ave(xi; tj) < 0 (2.28)This expresses how much this peak intensity di�ers from the average intensity at that particularpoint.The reason for adding the �rst and last intensity values in the next formula is not explained in thearticle. A personal interpretation of their presence is given in this paragraph. The peak intensitywill be used to detect possible apexes. If a point (xi; tj) represents an apex the curve in functionof v should theoretically rise from the �rst velocity v1 where the hyperbola is under migrated, to amaximum at vp where the hyperbola is correctly migrated, and then fall back to the last velocity

CHAPTER 2. B-SCAN PROCESSING 12vn where the hyperbola is over migrated. If now the curve will rise at one of the extrema in v, sothat that value will be higher than the average, this will indicate that that particular point does notcompletely correspond to the model of a hyperbola apex. So including those values in the formulawill eliminate those points. A particular example of such a point is the point at one end of theunder migrated hyperbola. At the �rst extremum in v the energy at that point will be large. Asthe migration continues this energy will drop steadily to the other extremum. This way we have acurve with a large peak intensity, but since it does not represent an apex, the peak value should besmall. This is achieved by introducing the extremum values in the formula.If a certain point represents a hyperbola apex the peak intensity will be maximum at that velocity(vp) where the energy of the complete hyperbola is concentrated in the apex (xs; ts). Here the di�erencebetween the peak intensity and the average will be large. Therefore the points (xi; tj) at which the peakvalue pv(xi; tj) is larger than threshold T1 are extracted and interconnected regions are obtained. Withineach region the points with maximal peak intensity p(xs; ts) are selected as possible hyperbola apexes.To eliminate erroneous candidate apexes one uses the fact that in the well migrated image the hyperbolais focused at its apex. Thus for all the possible apexes (xs; ts) the following binary image is created:Ibinary(xi; tj ; vk)jvk=vp(xs;ts) = � 1 jI(xi; tj ; vk)jvk=vp(xs;ts)j � p(xs; ts)� T10 jI(xi; tj ; vk)jvk=vp(xs;ts)j < p(xs; ts)� T1 (2.29)In this binary image the regions around the candidate apex points are selected where the local intensityis larger than the candidates peak intensity minus the threshold T1. If these areas are too large thecandidate has to be rejected. So �rst the candidate area value will be normalized by dividing it by thepeak value pv(xs; ts), and comparing this result with a second threshold T2. Those values smaller thanthis threshold are then considered as real hyperbola apexes.In summary this method allows to simultaneously detect the hyperbolas, extract their apex positionand to estimate the propagation velocity. If this is done for hyperbolas at di�erent depths the result willbe a propagation velocity distribution as a function of depth (or time). One drawback of the method isthe calculation complexity, which is quite large.Finally extracting the intensities along the estimated v(t) curve out of the x � t � v data matrix willresult in the reconstructed image.Results: the method was applied to a B-scan taken with a GPR at 400 MHz, over an area with a largenumber of pipes at several depths and constructed of several materials.The method results were compared to those of standard migration and were found to be much better.The method was applied to an image if pipes at di�erent depths and of di�erent sizes. In the resultingimage all pipes can be seen and their horizontal position as well as their depth can be determined. Atthe same time a curve propagation velocity vs propagation depth was constructed.2.6 Migration2.6.1 IntroductionMigration is an image processing technique that until now has mainly been developed in the �eld ofseismic data processing. There the purpose is to reconstruct, on the basis of re ection data, acquired atthe surface, the whole re ecting structure that is present in the sub-surface, and is causing said re ectiondata.The main aim of migration techniques (both in time- and frequency-domain) is to give an idea of theexact physical position and shape of the re ectors in the subsurface. The ideal migration will be anaccurate transformation from the position-time domain to the position-depth domain.In doing so, much emphasis is laid on the reconstruction of plane re ectors with a certain dip (inclinationdownward). This is indeed very important in the seismic point of view, since these re ectors are of maininterest.What we will attempt is to apply these methods (after adaptation) to data coming from GPR sensors.Here the type of re ectors that are of main interest are the point scatterers. The idea of using thesemethods originates in a certain observed similitude in the non-migrated data. Indeed, a point scattererwill cause the presence of a hyperbolic structure in the image. This is due to the width of the emittedbeam "antenna footprint". This will cause the antenna to receive re ections of point scatterers that arenot directly below it. Since this re ection is not perpendicular to the antenna surface, the time that the

CHAPTER 2. B-SCAN PROCESSING 13

Figure 2.2: The problem: the re ected energy of the object is spread out over the di�erent receiverpositions leading to the measurement hyperbola. The idea of migration is to recombine this energy inone position.beam will need to travel to the object and back will be larger than when the antenna is directly abovethe object. This is why in the X-time diagram the hyperbolic structure appears, with the apex at theposition of the object.The e�ect of point scatterers in the seismic approach is also the presence of a hyperbola, but here theunderlying physical principle is very di�erent. Indeed, in this case the emitted beam is not widening, butwill remain quite narrow. When this beam reaches a point scatterer, it will be di�racted in a wide area.So in this case the beam widening does not appear in the emitted beam, but in the re ected one. Notethat to be able to record this type of data one has to have one emitter in a �xed position, and a scanningreceiver.So although the e�ect on the resulting data seems quite similar, the underlying physical principles aredi�erent.The next sections will introduce some of the existing approaches to this problem, each with theirparticular properties. First a number of techniques, developed for seismic processing are summarized,then a possible cross-over to electro-magnetic migration is mentioned, and the paragraph will be endedwith some discussion. All this is based on [12], [13], [14], [15] and [16].2.6.2 Kircho� MigrationThis type of migration is based on the di�raction summation principle. The basic idea is to try tocalculate the wavefront at a certain time and position, when it is known at another time and position.The known data is recorded at all times, at depth z = 0, and what we want to calculate is the originatingwavefronts at time t = 0 at all depths z.We have to emphasize that in migration the moment of re ection is considered time t = 0 and thatthe position of re ection is regarded as the source. The movement of the wave towards the re ector isnot at all considered, and also multiple re ections are excluded.In order to mathematically comprehend this type of migration we need to introduce some concepts.(At some points in the text the terminology is borrowed from seismic and vibration methods)

CHAPTER 2. B-SCAN PROCESSING 14Point SourceThe pressure �eld due to an impulsive point source that erupts at t0 and is located at r0 in an homogeneousvelocity medium is given by Green's function:G(r; tjr0; t0) = �(t� t0 � jr � r0j=c)jr � r0j (2.30)where c is the velocity of propagation. Here the primed (unprimed) variables indicate the source(receiver) variables. The source is the re ector in the subsurface, and the receiver is the probe scanningthe surface. Equation 2.30 describes an expanding sphere in x-t space. If the source explodes at t0 = 0and r0 = (0; 0; 0), then the sphere expands from the origin with speed c, while the amplitude is attenuatedby the geometrical spreading factor jr � r0j.An imploding point source point sink is one where the spherical wavefront collapses to the sourcepoint as time increases; The �eld due to an imploding source is represented by the backwards Green'sfunction: G(r; tjr0; t0) = �(t� t0 + jr � r0j=c)jr � r0j (2.31)Huygens PrincipleFor forward propagating wavefronts, Huygens' principle states:Each point on a wavefront at time t can be thought of as a secondary point source. Thefuture wavefront at t+dt can be reconstructed by taking the envelope of exploding wavefrontsat t+ dt originating from the secondary point sources.For backward propagating wavefronts, a corollary to Huygens' principle can be stated:Each point on a wavefront at time t can be thought of as a secondary point sink. The pastwavefront at t� dt, that is an earlier wavefront, can be reconstructed by taking the envelopeat t� dt of imploding wavefronts that collapse to the secondary sinks.In migration, we try to use data at the surface to reconstruct the past history of wave-�elds in thesubsurface. Therefore we will be using the backwards Green's function to continue the measured �eldsat the surface in the downward direction towards the subsurface scatter sources.Kircho� Migration TheoryThe problem with Huygen's Principle is that it predicts two secondary wavefronts, one inward propagatingand the other outward propagating. This contradicts reality where, in a homogeneous medium, a oneway propagating wave will not produce backward propagating wavefronts. If not, then we would beconstantly bombarded by echoes of our conversations with others. To cure this problem, Kircho� in the1800's suggested that the outward propagating secondary wavefront could be correctly predicted from theprimary wavefront by taking the envelope of a distribution of secondary monopole and dipole wavefronts.In this way it can be shown that the inward propagating wavefront disappears.Formally, Kirchho�'s equation derived from Green's Theorem says that the �eld P (x; z; t) can bepredicted by a surface and volume integration over the wavefront:P (r; t) = XS0 Xt0 G(r; tjr0; t0)dP (r0; t0)dn0 dS0dt0�XS0 Xt0 dG(r; tjr0; t0)dn0 P (r0; t0)dS0dt0+XV 0 Xt0 G(r; tjr0; t0)F (r0; t0)dV 0dt0 (2.32)where n0 is the normal vector on the surface S0, and is pointing in the direction of the wave propagation;G is the causal Green's function described earlier; and the interior volume integral contains the bodyforces represented by F (x; z; t) (see Morse and Feshbach Volume I for the derivation in the time domain).

CHAPTER 2. B-SCAN PROCESSING 15Here the �rst term represents the surface monopole sources, the second term the surface dipole sources,and the third the volume sources. The coordinate system for the above equation can be seen in �gure2.3.

Figure 2.3: The coordinate system of the theory.To migrate data we have to �nd a downward continuation of the recorded data into the earth. Thismeans that we will now look at the point on the surface as the source (exploding re ector) of the inversepropagating wave�eld. We assume that the exploding re ector is just outside the volume of integration.In that case the volume integral term in the equation becomes zero. Since we wish to reconstruct ancientwave�elds from current data we use the backwards Green's functions. The practical form of the downwardcontinuation operator can be derived in the following way:1. Homogeneous subsurface velocity is assumed, although for variable velocity a high frequency Green'sfunction can be used.2. Assume that the semi-sphere of integration goes to in�nity so that there are no contributions fromthe semi-sphere for �nite evaluation times. Also, the initial condition terms in this equation areassumed to be zero.3. For a planar surface, the image Green's functionG0 = G(x; y; z; tjx0; y0; z0; t0)�G(x; y; z; tjx0; y0;�z0; t0) (2.33)can be inserted into the equation to eliminate the �rst surface integral term becauseG0(x; y; z; tjx0; y0; z0 = 0; t0) = 0 (2.34)and dG0(x; y; z; tj; x0; y0; z0 = 0; t0)dz0 = 2dG(x; y; z; tjx0; y0; z0 = 0; t0)dz0 (2.35)

CHAPTER 2. B-SCAN PROCESSING 164. Since we assume a plane horizontal surface, dn0 = dz0 and the dGdn0 term in equation 2.32 becomes:dGdn0 = dGdz0 = 1jr � r0j d�(t� t0 + jr � r0j=c)dz0 +O( 1jr � r0j2 ) (2.36)The second order term inverse in jr� r0j can be neglected because jr� r0j is large. This assumptionmeans that the backward propagation model will be accurate at deeper levels where this is true.The remaining term can be evaluated with the identity:d�(t� t0 + jr � r0j=c)dz0 = �1c d�(t� t0 + jr � r0j=c)dt0 djr � r0jdz0 (2.37)where djr�r0jdz0 = z0jr�r0j = cos(�).Using this result to evaluate the dipole term in equation 2.32 gives:XS0 Xt0 P (r; t)dGdn = �XS0 Xt0 d�(t� t0 + jr � r0j=c)dt0 P (x0; y0; z0; t0) cos(�)cjr � r0j= XS0 _P (x0; y0; z0; jr � r0j=c+ t) cos(�)jr � r0jc (2.38)Equation 2.38 is the downward continuation operator for data recorded on a plane. The Kircho�migration is obtained by evaluating the continued �eld in equation 2.38 at t = 0:M(x; y; z) =XS0 _P (x0; y0; z0; jr � r0jc ) cos(�)jr � r0jc (2.39)which is evaluated using the MATLAB program below.%% KIRCHOFF MIGRATION PROGRAM%% nx = number of traces at z=0 along at recording plane% DP(x,z=0,t) = time derivative of pressure �eld trace at (x,0).% c = 1/2 the actual velocity%(dt,dx,dz) = sample intervals in time, o�set and depth.% for x = 1:nxfor z = 2:nzxx = x*dx;zz=z*dz;for itrace = 1:nxr2 = ( (xx-dx*itrace)^2 + zz^2 );t = sqrt(r2)/(c*dt);costheta = z/r2;M(x,z) = DP(itrace,0,t)*costheta + M(x,z);endendendM = M/c;Migration tricks to improve quality of migrated imagesHere are some tricks, but there are undoubtedly other tricks that are as or more e�ective than these.

CHAPTER 2. B-SCAN PROCESSING 171. Data tapering. Edges of your data set should be padded (for several wavelengths at least) withtraces identical to the edge traces, and these padded traces should be gradually tapered to zeroamplitude. Tapering decreases ringing in your migrated image at the expense of some resolutionloss. This idea is similar to apodization (gradual darkening of lens edge) of telescopic lenses, whichsuppresses optical di�ractions from the edges of the lens.2. Post-migration �ltering. The migration process can introduce high frequency noise into the image.Such noise can be eliminated by low-pass �ltering, where the low-pass frequencies are consistentwith the bandpass range of signal.3. While migrating subsequent B-scans, the model can be improved by iteratively changing the velocitymodel with each new migration image.2.6.3 Some First Results on GPR Data (Obtained by our own tests).The suggested algorithm in the previous paragraph was implemented and slightly altered (in its presentform it does not work). The parameters were estimated on the basis of the information present onthe website of the DeTeC group of the EPFL, where the used data originated. An example of thesepreliminary results can be found in the next �gure.

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(a) (b)Figure 2.4: An example of some preliminary migration results : (a) Before migration ; (b) After migrationIn the �gure one clearly sees that the hyperbolas present in the original image have much more convergedtowards their apex after migration. One can also see that the dimensions of the main re ection aftermigration are closer to the real dimensions of the object. So these simple tests show that further studyof a migration algorithm, based on the electro-magnetic variant of Kircho� migration seems certainlywarranted.2.6.4 Application to electro-magnetic wavesAlso an extrapolation of the theory towards electro-magnetic waves was made ([16]). For electromagneticwave propagation, the magnetic �eld ~H and electric �eld ~E must satisfy the di�usion equation.4 ~H � �0�n @ ~H@t = 0 (2.40)4 ~E � �0�n @ ~E@t = 0 (2.41)Let P (~r; t) be any scalar component of an observed Electromagnetic Field. The migrated �eld PM (~r; t),calculated out of such a scalar component P 0(~r; t) will have to satisfy following boundary conditions:

CHAPTER 2. B-SCAN PROCESSING 18PM (~r; �)jz=0 = � P 0(~r; T � �)jz=00 for 0 � � � Tfor � < 0 ; � > T4PM (~r; �)� �0�n @PM (~r;�)@t = 0 for z > 0PM (~r; �)! 0 for j~rj ! 1; z > 0 (2.42)These three equations describe a boundary value problem, which can be solved usingGreen's theoremand Green's functions for the di�usion equation.Before going deeper into the solution of the problem, let us �rst de�ne a few notations:� ~r : Cartesian coordinates of the observer (in two dimensions: ~r = (x; z). An example of a real lifeobserver is the receiver, i.e. the scanning device on the surface.� t : Real time of the receiver.� ~r0 : Cartesian coordinates of source (in two dimensions: ~r0 = (x0; z0). The source is the point ofre ection.� t0 : Real time of the source.� T : Period of receiver registration.� � = T � t : Inverse receiver time.� �0 = T � t0 : Inverse source time.� P : Any scalar component of ~E or ~H.� g(~r; tj~r0; t0) : Green's function for di�usion.� G(~r; tj~r0; t0) : Green's image function for g.On one hand we can write that the migrated �eld at source coordinates has to satisfy:r20PM (~r0; t0)� �0�n @PM (~r0; t0)@t0 = 0 (2.43)On the other hand we will de�ne a Green's function g that answers to the following condition:r20g(~r; tj~r0; t0) + �0�n @g(~r; tj~r0; t0)@t0 = �4��(~r � ~r0)�(t� t0) (2.44)If we now multiply 2.43 with g(~r; tj~r0; t0) and 2.44 with PM (~r0; t0) and integrate the result over spaceand over time from 0 to t+, we get the following equation:Z t+0 dt0 Z Z Z dV0 �PM (~r0; t0)r20g(~r; tj~r0; t0)� g(~r; tj~r0; t0)r20PM (~r0; t0)�+�0�n Z t+0 dt0 Z Z Z dV0 �PM (~r0; t0)@g(~r; tj~r0; t0)@t0 + g(~r; tj~r0; t0)@PM (~r0; t0)@t0 � = �4�PM (~r; t)(2.45)We will now use Green's theorem, which states that since:I Ugrad(V )dS = Z Z Z (grad(U)grad(V )dv + Z Z Z Ur2V dv (2.46)then: I [Ugrad(V )� V grad(U)]dS = Z Z Z [Ur2V � Vr2U ]dv (2.47)So when we apply this theorem to the �rst integral of equation 2.45 and perform the time integrationon the second this will result in:

CHAPTER 2. B-SCAN PROCESSING 19PM (~r; t) = 14� Z t+0 dt0 I dS0 �g(~r; tj~r0; t0)grad0(PM (~r0; t0))� PM (~r0; t0)grad0(g(~r; tj~r0; t0))�+ v4� Z Z Z dV0[PM (~r0; t0)g(~r; tj~r0; t0)]t0=0 (2.48)This is due to the fact that the causality principle dictates that g(~r; tj~r0; t0) = 0 when t < t0. We nowstill have to construct a Green's function that can be used in this case. We will de�ne ~R = ~r � ~r0, and� = t � t0. We can write that g(~r; tj~r0; t0) is the result of an one-, two-, or three-dimensional Fourierintegral: g(~R; �) = 1(2�)N Z ei~p~R (~p; �)dV~p (2.49)with to be de�ned, N will be 1,2 or 3 depending on the number of dimensions and ~p contains theFourier variables associated to ~R.On the basis of this equation we can say that:r20g(~r; tj~r0; t0)� �0�n @g(~r; tj~r0; t0)@t0 = 1(2�)N Z ei~p ~R ��p2 � �0�n�@ @� �� dV~p (2.50)because @t0 = �@�.Also one can state that: 1(2�)N Z ei~p~RdV~p = �(~R) (2.51)Application of equations 2.44 and 2.51 in equation 2.50 results in:�4� 1(2�)N Z ei~p~R�(�)dV~p = 1(2�)N Z ei~p~R ��p2 � �0�n�@ @��� dV~p (2.52)By assuming equality of the integrals we can deduce following formula for :4��(�) = p2 + �0�n�@ @�� (2.53)Solution of this di�erential equation for results in: = 4��0�n e�� p2�0�n��u(�) (2.54)where u(�) is the Heaviside step function:u(t� t0) = � 1 t� t0 > 00 t� t0 < 0 (2.55)Insertion of this de�nition of into formula 2.49 of the Green's function will result in:g(~r; tj~r0; t0) = 4�(2�)N�0�nu(�) Z ei~p~Re�� p2�0�n��dV~p (2.56)= 4�(2�)N�0�nu(�) � Z +1�1 eipxRxe�� p2x�0�n��dpx (2.57)� Z +1�1 eipyRye�� p2y�0�n��dpy (2.58)� Z +1�1 eipzRze�� p2z�0�n��dpz (2.59)This is of course assuming we work in three dimensions. If only two dimensions are available (forexample x and z the second integral becomes 1).

CHAPTER 2. B-SCAN PROCESSING 20When we concentrate on the exponent in the �rst integral we can change variables:ipxRx �� p2x�0�n� � = �� pxp�p�0�n � ip�0�nRx2p� �2 ���0�nR2x4� � (2.60)= � ��0�n �2 ���0�nR2x4� � (2.61)where � = px � i�0�nRx2� .The �rst integral will thus become:Z +1�1 e�� ��0�n �2����0�nR2x4� �dpx = e���0�nR2x4� � Z +1�1 e�� ��0�n �2�d� = �0�nr�� e���0�nR2x4� � (2.62)Performing the same transformation for all integrals results in following total Green's function:g(~r; tj~r0; t0) = 4��0�n � �0�n2p���N e��0�nR24� �u(t� t0) (2.63)For our implementation however we will use the image Green's function G(~r; tj~r0; t0) which will bede�ned as: G(x; y; z; tjx0; y0; z0; t0) = g(x; y; z; tjx0; y0; z0; t0)� g(x; y; z; tjx0; y0;�z0; t0) (2.64)In this case Gjz0=0 = 0 and grad0(G) = 2 dgdz0What we measure is the wavefront emitted by the source, but at the earth surface, thus at z0 = 0. Ifwe now introduce Green's image function in equation 2.48 with this constraint that z0 = 0 the resultingformula for the migrated �eld will become:PM (x; y; z; t) = �24� Z t+0 I dS0PM (x0; y0; 0; t0) dg(~r; tj~r0; t0)dz0 ����z0=0 (2.65)2.6.5 Phase Shift Migration [12]This type of migration takes place in the frequency domain. The general theory behind this approachcan be summarized as follows for seismic applications.Given is the data p(x; z = 0; t) that is measured along a plane at z = 0. Given is also the knownvelocity of propagation c = c(z) that can be either constant or variable with depth.The wave equation will then be: r2p(x; z; t)� 1c2 @2p(x; z; t)@t2 = 0 (2.66)Applying a Fourier transform that has the kernel ei!t results in:@2P (kx; z; !)@z2 = (�!2c2 + k2x)P (kx; z; !) (2.67)Where P (kx; z; !) is the Fourier transform of p(x; z; t).De�nition of kz = sgn(!)q!2c2 � k2x leads to the solution:P (kx; z; !) = Aeikzz +Be�ikzz (2.68)Assuming upward coming waves only we can state that A = 0. Since at z = 0 we have the measureddata B = P (kx; 0; !).The formula of the wavefront at depth z is then:P (kx; z; !) = P (kx; 0; !)e�ikzz (2.69)

CHAPTER 2. B-SCAN PROCESSING 21The migrated image M(x; z) will then be the inverse Fourier transform of the downward continuedwavefront at t = 0. M(x; z) = p(x; z; t = 0) =Xkx X! P (kx; 0; !)e�i(kxx+kzz) (2.70)An idea of aMATLAB code for this type of migration can be (just an idea, it remains to be checked):for iz = 1:nzz=iz*dz;for iw = -wnyquist:wnyquist;w=dw*iw;D=0;for ik = -iknyquist:iknyquistkx=dkx*ik;if abs(kx) < abs(w/c);kz = sgn(w)*sqrt((w/c)^2 - kx^2);D(ik) = e^(i*kz*z)*P(ik,iw);end;end;d = i�t(D);M(:,z) = M(:,z) + D(:);end;end;2.6.6 FK Migration [12]This method is a variant of the phase shift method described in paragraph 2.6.5 that is only valid for aconstant propagating velocity c.The trick is to change variables in the formula for the migrated image:! = �kzcs1 + k2xk2z (2.71)so that: d!dkz = kzc2! (2.72)With this change the formula for the migrated image becomes:M(x; z) = c2 Z dkz Z dkx kz! P (kx; 0; !)ei(kzz+kxx) (2.73)This way the migration can be performed by a two dimensional Fourier Transform of the transformeddata, scaled by kz! . This will seriously reduce the computational complexity.2.6.7 Comments and DiscussionThe �rst thing one has to emphasize is that the accuracy of the migration will depend on the knowledgeof the physical parameters of the soil. It is possible to measure them or even to approximate them onthe basis of tables of laboratory measurements, but the main requirement will remain that the propertiesof the soil must not change too drastically within one image. Indeed if, when scanning one B-scan, theelectro-magnetic properties of the soil change, general migration techniques will become useless.If it is possible to have an accurate estimation of the properties of the soil to perform this transformatione�ciently the results could be used in several steps of the pattern recognition algorithm. The migratedimage will give more accurate information about the exact position and depth of re ectors, which can beadded in the detection step. It would allow us to better pinpoint those regions in a scanned surface thatdeserve a closer look.

CHAPTER 2. B-SCAN PROCESSING 22Further it would be quite thinkable that the shape of the original re ector would be better representedin the migrated image. This type of information can be used in the recognition algorithm as a supple-mentary feature. This could especially be done when migrating all B-scans of a C-scan (or even better,extrapolating the migration algorithm to directly calculate the 3-D migration results) to give a view ofthe objects shape.O� course all this depends on the knowledge of the properties of the soil. In uence on the results fromnoise and clutter also have to be studied further. Nevertheless, it seems that this approach holds enoughpromises to warrant further study.

Chapter 3C-scan processing3.1 IntroductionA C-scan is a three dimensional data structure that is acquired, by scanning a regular two dimensionalhorizontal grid. We will �rst address the problem of tomography that will try to inverse the problem,and continue with a pseudo tomographic process called array beam imaging. We will discuss polarimetry,which is not really a processing technique, but a di�erent way of acquiring data. Finally we will showsome ways to compress the three dimensional structure in a two dimensional image.3.2 Tomography3.2.1 IntroductionThe general idea of tomography is to use the acquired data and a propagation model to solve the inverseproblem and to �nd the position and shape of the sources. In this case these sources will be the scatterersin the subsurface. Most of these methods are based on array scanning systems with a resolution smallerthan the detected objects.3.2.2 ReconstructionIn [4] a tomographic method is introduced for impulse radar operating in pulse echo mode. This methodwas designed to detect anomalies in solid concrete.The data was acquired with a low-power single-pulse video-band pulser with a resistively loaded dipoleantenna having an approximate beamwidth of 140o. This system produced a signal having a frequencycontent from approximately 500 MHz to 3.5 GHz. With this, the surface was scanned with 1.27 cmsample spacing.A preprocessing step consisting of standard background subtraction as well as pulse deconvolution, isapplied to the data before the actual processing.The main idea behind the processing is the plane-to-plane backward propagation method. If we considerthe measured data r(x; y; z) as the result of a synthetic two-dimensional planar array of antennas, andwe have the object distribution o(x; y; z) in the subsurface (this is what we want to estimate), then foreach wavelength � in the emitted signal we can write:r(x; y;�) = Z [o(x; y; z0;�)u(x; y; z0;�)] � g(x; y; z0;�)dz0 (3.1)Here � is a convolution, u(x; y; z0;�) is the emitted wave �eld, and g(x; y; z0;�) is the Green's functionfor a homogeneous background: g(x; y; z;�) = ejkr4�r (3.2)where r =px2 + y2 + z2 and wavenumber k = 2�=�.If the source is an isotropic radiator (the same amount of energy is emitted in all directions) at positionr0 this can be expressed as: 23

CHAPTER 3. C-SCAN PROCESSING 24u(x; y; z;�) = ejk(r�r0)4�(r � r0) (3.3)and the received wave�eld becomes:r(x; y;�) = Z �Z dr0[o(r) ejk(r�r0)4�(r � r0) ] ejk(r�r0)4�(r � r0)� dz0= Z o(x; y; z0) � g2(x; y; z0;�)dz0The wave propagation is modeled as a spatial linear �lter given as:H(fx; fy; z;�) = ( ej2�zp1=�2�f2x�f2y f2x + f2y = 1=�20 otherwise (3.4)where f2x and f2y are the spatial frequencies in the x and y direction. The spatial spectrum of theequivalent source S (the re ecting object is called source) is then given by:S(fx; fy; z) = Z R(fx; fy;�)H+(fx; fy; z;�)d� (3.5)where R is the spectrum of the received wave�eld and + denotes a pseudo inverse. The estimate forthe source (object) distribution is then given by the 2D inverse Fourier transform of S.To remove the dependency of g2 equation 3.4 is derived with respect to k:r0(x; y;�) = @@kr(x; y;�) = C Z o(x; y; z0) � g(x; y; z0;�=2)dz0 (3.6)with C a constant.This derivation has a physical meaning: di�erentiating with respect to k in the frequency domain cor-responds to multiplying by a factor �t, which will exactly compensate the factor r�r0 in the denominator,or in other words remove the loss due to spherical expansion of the incident wave�eld.So the actual �lter is applied to the gain compensated data and, the object distribution is calculated.In practice the data has to be recorded for several wavelengths. After a derivation with respect tothe wavenumber (inverse proportional to the wavelength) the data is �ltered by the proposed wavepropagation �lter. The resulting data is than the object con�guration.The result shown in this article [4] is quite impressive: a good three dimensional image of a scannedstructure is created. This method warrants further study for the application in detection in more clutteredenvironments.3.3 Array Beam Imaging3.3.1 IntroductionThe method described in this paragraph is called Array Beam Imaging (ABI) and is concentrated uponprocessing three dimensional radar data to achieve an accurate 3D visualization of the data. This is nota real tomographic method since it does not start from a propagation model it tries to invert. Insteadthis method uses only the data and some knowledge about the acquisition array.The main aim is to show the level of re ected energy from the interior of a medium, based on acoustic,seismic or electromagnetic waves. This method is based on [6].For Seismic waves, high frequency �elds (KHz range) allow the general shape characteristics of a landmine to be resolved.For electromagnetic waves �elds in the GHz (radar) range have the same wavelength (10 cm range)and therefore the same resolution capability as the before mentioned seismic waves.Although re ection amplitudes are equivalent for seismic and EM waves, the poor propagation of EMwaves under wet conditions will result in the use of seismic ABI methods for wet soil conditions, and EMABI methods for dry soils.There is an overlap in applicability and in those cases a "dual technology" approach is quite attractive.

CHAPTER 3. C-SCAN PROCESSING 25The wave�eld is generated by an array of controlled sources which are in contact with the scannedmedium. The wave�eld will propagate into this medium as an approximation of a plane wave front.When it encounters targets (objects present in the medium), part of the incident wave is backscattered.Recording the re ected wave �eld is done by an array of receivers that is spread out over the same areaas the emitting array. Out of each of the receivers one time signal is extracted per emitted wave.3.3.2 CalibrationOne can use calibration sources to estimate both attenuation parameters and propagation speeds of themedium.This is done by placing a number of calibration sources (these are emitting elements) near the receiverarray. The placement is at the same depth of the array, and at both sides of it. So when a signal isemitted towards the array it will travel through the upper layer of the soil. The distance between thesources and each of the elements of the receiver array is considered known.Since the distance between these sources and each of the receivers is di�erent we obtain a number ofsignals for di�erent source-receiver distances. When one plots the maximum amplitude in these signalsin function of these source receiver distances, one can see an exponential decay given by:I(r) = 1r� I0 (3.7)where I0 is the emitted intensity, r is the distance between emitter and receiver, and � is the attenuationcoe�cient of the local soil. Since I(r) is experimentally determined, the attenuation coe�cient can beestimated. One has to mention here that the proposed method will only give the average attenuationbetween emitter and receiver, and that for further analysis this attenuation is considered homogeneouslypresent in the whole scanned volume (e.g. no variance of � with the depth is considered).Also the e�ect of the small objects in the scanned volume on the total attenuation of the wave isconsidered negligible, and the total scanned volume is assumed homogeneous.When knowing the distance between source and receiver, and estimating the arrival time of the mea-sured signal, one is also able to estimate the propagation velocity of the medium. A more accurate valuewill be achieved, when repeating this estimation for all source receiver distances, and averaging them.Again one has to mention that the resulting propagation velocity will only be an average estimate forthe top layer in the soil. In the further processing it is assumed that the same velocity is homogeneouslypresent in the entire scanned volume.3.3.3 ProcessingTime shiftThe data signals coming from the di�erent receivers in the array are combined so that the array is usedas a lens to focus on a particular small volume element. The re ected energy from that volume elementcharacterizes its intrinsic re ectivity.The size of these volume elements can be in the order of a fraction of the smallest wavelength presentin the emitted wave. For GHz EM �elds this will allow to visualize objects of a few cm (within a scannedvolume of several cubic meters).The used focusing techniques are derivatives of the phased array beam forming method.The basic principle of this method is to calculate the correct time delay that has to be introducedbetween the signals of the di�erent receivers, so that the signals coming from one particular volumeelement will be added constructively. Indeed, when one looks at the two dimensional plane underneaththe array, for each point in this plane the distance towards each of the elements of the receiver arraycan be calculated. As homogeneous propagation velocity was assumed, the re ected signals emanatingfrom that particular volume element will arrive at the di�erent sensors in the array with a time shiftthat is proportional to these calculated distances. Since the calibration allowed to estimate an averagepropagation velocity, these distances can be easily recalculated into time shifts. These time shifts willthen be applied to the di�erent signals in order to focus on that particular volume element. This methodis called beam steering the array. Summing these shifted signals (called stacking) will enhance the signalscoming from the particular volume element, and will partially or wholly cancel re ections from otherpositions, as well as multiple re ections.

CHAPTER 3. C-SCAN PROCESSING 26As was mentioned this method is based on the phased array processing methods. However there arestill some large di�erences. First of all the goal of a phased array is to be able to scan a larger volumewith a smaller array. Therefore the emitted beam will be already constituted of time shifted emittedsignals so that it is emitted at a certain angle with respect to the array. Scanning will occur when thisangle is changed. In our case the emitted beam is launched perpendicular to the array, and is assumedto be a plane wave.The receiver part in phased array will also be processed with the same time shifts as the emitted partin order to be listening in the same direction as was emitted. Again varying the angle will result in onesemi circular image of the scanned area. In the Array Beam Imaging method one is not interested inviewing the re ections at a certain angle but in focusing the array to a particular volume element. Thiswill also result in time shifts between the di�erent signals, but these time shifts will no longer be linearlyincreasing from one receiver to the next (as they are for phased arrays) but will be directly dependenton the position of the focused volume element.Re ectivity calculationBased on the knowledge of the attenuation (the attenuation as function of frequency is achieved by passingthe same sliding frequency �lter on the data obtained by calibration, and then calculating an attenuationcoe�cient for each frequency) and the emitted energy Eemitt, one can calculate how much energy arrivesat a certain position in the scanned volume:E(r) = Eemitt 1r� (3.8)The measured re ected energy after focusing of a certain volume element, can also be multiplied by r�in order to calculate the real re ected energy at that time and position.The total re ectivity of the volume element is then calculated by simply dividing received and re ectedtotal energies reflectivity = reflected total energyreceived total energy (3.9)3.3.4 ImagingOnce we have a physically meaningful re ection coe�cients (The "intrinsic re ectivity") for each ofthe small volume elements, one can create a tomographic image by rendering these values in a threedimensional structure.Clutter will cause many strong re ections that will overlap in time. They can obscure nearby re ectingobjects by casting shadows. In an attempt to resolve this problem, a secondary processing step is applied.The general idea is to detect in the tomographic image all visible re ectors. Based on the position ofthese re ectors the above mentioned processing is inverted to get the parts of the original signals thatwere the origin of these re ectors. Eliminating these parts, produces a residual time series in which signalsfrom already imaged re ectors are removed. Relaunching the standard ABI process will then result insecondary re ector images of which all already detected objects were deleted, and so the obscured onesappear. This process can be repeated until no new objects are found.3.3.5 Identi�cationOnce the complete tomographic image is achieved it is �ltered with a �lter function "matched" to the3D shape of a mine. This �ltering is done in the spatial wave number domain. This will produce animage with sharp spikes at the center of re ecting objects. Normalization of this image gives a threedimensional probability density map.Analysis time is considered quite acceptable for demining purposes. Computer simulations show ahigh likelihood of detection under conditions expected in the �eld. In extreme clutter, performances arelowered, but can be improved by the dual �eld approach.Estimated price: a few thousand dollars for an image from 2 to 3 m on a side and 1m depth in a minuteor two.

CHAPTER 3. C-SCAN PROCESSING 273.3.6 Averaging and �lteringTo enhance the Signal to Noise ratio, a number of preprocessing steps can be introduced on the data,before starting the actual processing algorithm.The �rst simple possibility is to summate a number of acquisitions (called stacking). This requires arepeatable wave generation process by the source. This is performed directly at the acquisition stage andfurther processing is done on these stacked signals.Another type of processing that can be applied here is to �lter each time signal with sliding narrowfrequency �lter. This will result in the generation of a new time signal for each of the �lter positions (onetime signal is expanded to N time signals which each contain a small fraction of the frequency contentof the original signal).Since wave speeds and attenuations are frequency dependent, use of a broader spectrum in the emittedsignal can lead to blurring of the image. If the above mentioned calibration process is repeated for eachof the di�erent frequencies (the signals used for calibration are also passed through the same slidingfrequency �lter) the frequency dependency of the soil properties can be obtained. The ABI process canthus be repeated for each of these �ltered signals to create a tomographic image representing a fairlynarrow frequency band. Summation of these images will then result in a sharper, more complete totalimage.3.4 Polarimetry3.4.1 IntroductionAcquiring radar polarimetric data can enhance the contrast between di�erent parts of the scanned scene.This is however not a signal processing tool, it is simply another way of acquiring data.The main principle of polarimetry is quite di�erent in radar applications than in Infra red ones. Theactive nature of radar will allow emitting a polarized wave, and the receiver part can be tuned to besensible to a certain type of polarization.The re ecting objects will due to their shape and structure be sometimes more re ective for certainpolarization types than others. This will result in signals with more discriminating object informationthan the background.3.4.2 TheoryIn [7] the theoretical principle of radar polarimetry is presented. The basic formula that is used in thisapproach is the formula of the scattering matrix of an object.� Er1Er2 � = � S11 S12S21 S22 �� Et1Et2 � 1p4�r2 (3.10)In this formula the terms Et1 and Et2 represent the emitted, and Er1 and Er2 the received �elds. These�eld components will determine the polarization, which can be linear or circular. If linear polarization isused, the �rst component Et1 will be the horizontal polarization (H), and the second one Et2 the verticalone (V). If the emitted beam is completely horizontally polarized, the second component will be zero.When using circular polarization the component Et1 represents the left circular polarization (L) ant thecomponent Et2 the right one (R).The terms Er1 and Er2 will represent the re ected components. The same division in linear or circularcomponents goes here.The scatter matrix itself will show how much of a certain type of polarization is re ected by thescattering object. This can be measured by simply assembling the four di�erent constructions that arepossible. For the linear case the scattering matrix will become:� SHH SHVSV H SV V � (3.11)where:� HH: Horizontal polarization emitted, Horizontal polarization measured.� HV: Horizontal polarization emitted, Vertical polarization measured.

CHAPTER 3. C-SCAN PROCESSING 28� VH: Vertical polarization emitted, Horizontal polarization measured.� VV: Vertical polarization emitted, Vertical polarization measured.and for the circular case: � SRR SRLSLR SLL � (3.12)where:� RR: Right polarization emitted, Right polarization measured.� RL: Right polarization emitted, Left polarization measured.� LR: Left polarization emitted, Right polarization measured.� LL: Left polarization emitted, Left polarization measured.The scatter matrices for certain canonical shapes can be calculated as there are for linear polarization:� Sphere: Ssphere = � 1 00 1 � (3.13)This is simple to see because a spherical target will not disturb linear polarization in re ection. Sohorizontal polarization will be re ected as perfect horizontal polarization and the same is valid forvertical polarization.� Line target: Swire = � cos2� 12sin2�12sin2� sin2� � (3.14)Here � is the orientation angle of the wire. One can notice here immediately that HH and VVpolarizations will give di�erent results. The arriving polarization is decomposed in a factor alongthe line, and a factor perpendicular to it. Only the parallel factor will be re ected. If horizontalpolarization E is emitted the parallel factor will be Ecos(�). This is re ected and upon arrivalagain decomposed in its horizontal (Ecos2(�)) and its vertical (Ecos(�)sin(�)) components. Thesame is valid for vertical re ections.� Diplane (two sided corner re ector (relative orientation of 45o):Sdiplane = � cos2� 12sin2�12sin2� �cos2� � (3.15)Here � is the orientation angle of the diplane versus the incident beam.� Helix: Shelix = 12 � 1 �j�j �1 � (3.16)Note here that this transfer matrix between incident linear polarization and re ected circular one.Once a scatter matrix has been measured through four experiments it can be decomposed into a numberof these canonical matrices, and so one can look for objects with a speci�c shape.For example the decomposition can be done in sphere, diplane and helix. This is the SDH decomposi-tion: [S] = ej� �ej�sks[S]s + kd[S]d(�) + kh[S]h(�) (3.17)It can be shown that this type of representation can be directly calculated out of the circular polarizationscatter matrix as follows:

CHAPTER 3. C-SCAN PROCESSING 29ks = jSRLjif jSLLj < jSRRj : k+d = jSLLjif jSLLj > jSRRj : k�d = jSRRjif jSLLj < jSRRj : k+h = jSRRj � jSLLjif jSLLj > jSRRj : k�h = jSLLj � jSRRj� = 1=2(�RR + �LL � �)� = 1=4(�RR � �LL + �)�s = �RL � 1=2(�RR + �LL)Here jSij j is the magnitude and �ij the phase of the complex value Sij .Furthermore it is also suggested to create pseudo color images where each of the polarization combi-nations represents a di�erent color:red green blueLIN HV HH VVCIRC RL RR LLSDH ks kd khThe three types of representation were then compared on the basis of an airborne SAR image. LIN andSDH appear to have the best contrast between the di�erent targets. In the examples only the magnitudesof the di�erent components were used, the phases however might also give some additional information.[21] is a review paper about this type of decomposition in scatter matrices. It retakes the representationshown in [7] (This seems to be the most widely used presentation for this type of applications), and alsopresents other ways to mathematically represent the use of polarization (next to the scatter matrices, alsothe Mueller matrix and Stokes vector as well as the eigenvector analysis of the covariance and coherencymatrices are introduced). This is a purely theoretical paper, that can be used for reference, if somewhereone of the latter representations is used.3.4.3 Test ResultsIn [8] the polarimetric SAR was used to detect buried mines form the ground. Here no theory aboutpolarimetry is given, only the results are analyzed. Two experiments were set up: one with di�erenttypes of mines in a sand background, and one with one mine in di�erent backgrounds as sand, grass,gravel and mould. The three combinations of linear polarization (HV, HH and VV) were then shownand the resulting images were quite di�erent for the di�erent polarizations. The HH polarization seemedto give the best result in both experiments, although in the experiment with the various backgrounds(especially with grass) even this best result was not very good.3.4.4 SAR PolarimetryIn [9] a very brief overview is presented about the tests with an UWB polarimetric SAR. Also the VVand HH polarizations are compared. Although due to the relationship between the re ection coe�cientand Brewster's angle for vertically polarized waves the experiments should have shown more penetrationfor VV versus HH, still the HH images showed more contrast.3.5 Karhunen Loeve transformation3.5.1 IntroductionAnother idea to process C-scan data is the application of the Karhunen Loeve transformation. The theoryof this transformation is explained in [19].The main idea is to reduce the sequence of images into one or two images in which all the dynamicinformation of the sequence is concentrated. In other words the areas that change throughout the sequence(such as objects) will be accentuated.

CHAPTER 3. C-SCAN PROCESSING 303.5.2 Application and commentsFirst it would be logical to apply this transformation to the subsequent C-scans. Indeed if somewhere inthe subsurface an object is present the resulting image will show its position independently of the objectsdepth.A possible drawback of the method is that in very inhomogeneous soils the inhomogeneities can addup to a greater dynamic range than that caused by the presence of an object so that the object can bemasked. In this case the transform will remain indecisive.Advantages of the method are that it does not need any training or parameter estimation, and that itcan be performed directly on the not preprocessed data.Preliminary tests of the transform on the available data showed that indeed after applying the transformand averaging the �ve �rst transformed images (the original sequence had 512 images) the (x,y) positionof the object could be detected with a simple threshold.Possible ideas for further research could be the study of the covariance matrix used for the trans-formation. Indeed it is quite thinkable that this matrix already holds enough information to pinpointthe position of the object. A possible application would be to calculate this position in the three di-mensions by performing the transformation not only on C-scans, but also on the B-scans and the slicesperpendicular to the B-scans.An interesting thing to investigate could be the result of the Karhunen Loeve transform on the dataafter 3D migration was applied.Another problem is the presence of the air-ground interface re ection. A few tests were performed tosee whether the Karhunen Loeve transformation applied to the B-scans would be able to remove thisre ection. The answer was no, because the large re ection present in all the B-scans will cause a clusterof points in the dixel space far from the origin. The actual object re ections will cause another clustercloser to the origin. Since the Karhunen Loeve transform does not use any learning it is not able tonormalize the clusters like the Kittler and Young transform and so the cluster far from the origin will bepredominant in the transformation. So not the background re ection but the actual useful signal will beattenuated.3.5.3 First resultsThe Karhunen Loeve algorithm was adapted and implemented. The resulting program was executed ona number of test C-scans. The top 300 C-scans were used each time (since almost no useful informationcould be found in the lower ones).

Slice 50 Slice 100 Slice 150 Slice 200Figure 3.1: A number of slices out of the original C-scan.In the original sequence (�gure 3.1), there are some images where the object can be seen (still withinsome clutter, but most of the images give no or very little indication of the presence of an object.

CHAPTER 3. C-SCAN PROCESSING 31

First Transformed Second Transformed Third TransformedFigure 3.2: The three �rst transformed images.The transformed sequence (�gure 3.2), however shows a clear indication of the object presence in areduced background clutter environment.3.6 Clutter removal and C-scan reduction3.6.1 IntroductionIn [20] Daniels introduces a method to compress the di�erent slices of a C-scan into one image, whilereducing clutter and enhancing object presence. This method is based on the assumption that thecorrelation between subsequent slices is larger in those areas where objects are present than in thosewhere there are none. Besides that also two image restoration methods are mentioned.3.6.2 RestorationIn order to get a more focused view of a C-scan slice the application of an inverse �lter is suggested.What is essentially meant is that one tries to perform a deconvolution with the antenna footprint. This isthe major drawback of the method, since this antenna footprint has to be measured or at least estimated.Fortunately it seems that a circular Gaussian is a good estimate for this footprint, so that the experimentalestimation becomes easier.Another way is to use a two dimensional correlation �lter, in search for speci�c objects. This methodwill use a target mask (usually a subset of the main image) to assess the degree of 2D cross correlationwithin the image. This seems to give good results on a synthetic example, on real radar data the resultswill probably be less convincing.3.6.3 C-scan reductionNOTE: In [20] some of the formulas are not very clear. This section describes our own interpretation ofthe method, based on the paper. This interpretation seemed to be the correct one after discussion withthe author.The proposed method will encompass a number of steps. First for each slice in the C-scan its averagegray value is subtracted, and the absolute value is taken. If the original three dimensional C-scan datais denoted Axyz, where x; y and z range between 0 and N � 1, thenARxyz = jAxyz �AVz j AVz = (N�1X0 N�1X0 Axy)z (3.18)

CHAPTER 3. C-SCAN PROCESSING 32Since the main idea is that the object information will be spread out over several depths, and theclutter won't, a number of cross correlation vectors can be derived from the matrix AR such thatR(z; k) = Z Z AZ(z):AZ(z � k)dxdy (3.19)Here AZ(z) = fARxygz. The main idea is that for a certain depth z we wish to express the slice atthat depth resembles the slices at other depths. This is done by multiplying the slice at depth z withthat at depth z � k and then integrating over x and y. This will result in one value which expresses the"correlation" between slices z and z � k. If this is done for all possible k values one will obtain an Ndimensional vector which expresses the "correlation" between slice z and all the other slices in the set.If such a vector is calculated for each of the depths z one obtains an N �N dimensional matrix R(z; k)which will be used in further steps.Next an intermediate matrix ARM is calculated for each depth z:ARMz = [Rz;z�1:ARz�1]:[Rz;z:ARz ]:[Rz;z+1:ARz+1] (3.20)In this set of matrices the areas which show a certain "correlation" between subsequent slices will beextra highlighted. One could also extend the comparison to z � 2 to z + 2, or even further. This wouldhighlight objects that spread even further in depth. However if one would take the range to wide, thebackground would become prominent, and blur out the objects.The resulting submatrices are then weighted with a coe�cient kz derived from the statistical propertiesof each submatrix ARz . ARMWz = kz :ARMz (3.21)with: kz = ((0:5:mean(ARz)) + std(ARz)) (3.22)This coe�cient expresses how much energy and variance of it (this points to an object) is present ineach submatrix. The slices with a large energy and variance (the energy is concentrated in a certainregion) are this way extra highlighted.Finally the combination of all depths is done based on this weighted data set:CARMW = "N�1Xz=0 ARMWz#2 :"N�1Xz=0 peak(ARMWz)# (3.23)Where peak means the peak value in the matrix.3.6.4 Comments and resultsAlternatives to the method were suggested as deconvolving the matrices ARMWz with the antennafootprint before the �nal combination to the CARMW matrix in 3.23, normalizing the submatrices, andto subdivide the image into subregions and treat them separately. The regions are weighed sinusoidicallyin order to have a uent transition from one region to another, and the sinusoids are chosen so that theirsum is always one. Finally the results can be added to gain a clearer image.The results shown by Daniels on a set of radar data seem rather promising. A possible applicationcould be to use this method in the same way as the Karhunen Loeve transform and maybe even combinethe results of both as well for detection as for shape extraction.The proposed method was implemented on three C-scans of a PFM-1 mine, a PMN mine and a stone.Note that the raw data was used (so no prior background subtraction or �ltering). The results can beseen in �gure 3.3, and are quite good.

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Bibliography[1] "Estimation of the soil permittivity and conductivity by a GPR antenna"; Y. Wakita and Y Yam-aguchi; GPR'96: International conference on Ground Penetrating Radar; September 30 - October 31996; Sendai, Japan; pp 123-127.[2] "Some aspects on the estimation of electromagnetic wave velocities"; R. Fruwirth and R Schmoller;GPR'96: International conference on Ground Penetrating Radar; September 30 - October 3 1996;Sendai, Japan; pp 135-138.[3] "Resolution in radar tomography for wall or pillar inspection"; S. Valle and L. Zanzi; GPR'96:International conference on Ground Penetrating Radar; September 30 - October 3 1996; Sendai,Japan; pp 229-234.[4] "Three-dimensional ground penetrating radar imaging using multi-frequency di�raction tomogra-phy"; J. Mast and E. Johansson; Spie vol 2275: Proceedings on advanced microwave and millime-terwave detectors; July 1994. (http://www-dsed.llnl.gov/documents/imaging/jemspie94.html)[5] "An image reconstruction methos using GPR data"; Z. Wu and C. Liu; GPR'96: Internationalconference on Ground Penetrating Radar; September 30 - October 3 1996; Sendai, Japan; pp 241-246.[6] "Array Beam Imaging (ABI): Basic features", TRAC-NA, February 1996[7] "On the importance of utilizing polarimetric information in radar imaging and clasi�cation"; E.Krogager and W. Boener, AGARD SSP symposium, 22-25 April, 1996, Toulouse, France.[8] "Anti-personnel mine detection by using polarimetric microwave imaging"; A. Sieber, J. Fortuny, G.Nesti and M. Fritzsche, Spie proceedings vol 2496, 1995, pp 14-19.[9] "Imaging of buried and foliage-obscured objects with an ultrawideband polarimetric SAR"; D. Sheen,Spie proceedings vol 1942, 1995, pp 12-20.[10] "Source signature deconvolution of ground penetrating radar data"; F. Neves and J. Miller, GPR'96International conference on ground penetrating radar, sept 30 - October 3 1996, pp 573-578.[11] "Detection of buried landmines using ground penetrating radar"; M. Fritze, Spie proceedings vol2496, 1995, pp 100-108.[12] "GG552: Exploration and engineering in seismology", Jerry Schuster, Utah Tomography & Model-ing/Migration Consortium, University of Utah, Electronic Bookshttp://utam.geophys.utah.edu/ ebooks/gg552[13] "Seismic Data Processing", Ozdogan Yilmaz, Society of Exploration Geophysicists, 1987, Tulsa,USA[14] "Surface Penetrating Radar", David Daniels, The Institution of Electrical Engineers, 1996, London,UK[15] "Resistivity imaging by Time Domain Electromagnetic Migration (TDEMM)", M.S. Zhdanov, P.NTraynin, O. Portniaguine, Exploration Geophysics, 1995, Volume 26, pp 186-194[16] "Underground imaging by frequency-domain electromagnetic migration", M.S. Zhdanov, P.NTraynin, J.R. Booker, Geophysics, 1996, Volume 61, pp 666-68234

BIBLIOGRAPHY 35[17] "Methods of theoretical physics", P.M.Morse, H. Fleshbach, McGraw-Hill Book Company, 1953.[18] "Radar imaging of underground objects in homoheneous soil using X-T-V data matrix"; H.Hayakawa, GPR'96 International conference on ground penetrating radar, sept 30 - October 3 1996,pp 579-584.[19] "Dynamic Infrared image sequence analysis for anti-personnel mine detection"; L. van Kempen, M,Kaczmarec, H. Sahli, J. Cornelis, SPS'98, Leuven, Belgium, pp 215-218[20] "Surface Penetrating Radar Image Quality"; D.J. Daniels, MD'98 second international conferenceon th detection of abandoned land mines, 12-14 October 1998, Edinburgh, UK, pp 68-72[21] "A review of target decomposition theorems in radar polarimetry"; S. R. Cloude and E. Pottier,IEEE Transactions on geoscience and remote sensing, March 1996, vol. 34, pp 498-518[22] "Processing ground penetrating radar data to improve resolution of near surface targets"; K. Gerlitz,M. Knoll, G. Cross, R. Luzitano, R. Knight, Proceedings SAGEEP, Geophysics, 1993, pp 561-574[23] "Advanced image processing technique for real time interpretation of Ground Penetrating Radarimages"; L. Capineri, P. Grande, J. Temple, IEEE Transactions, 1998, vol. 9, pp 51-59[24] "The application of a nonlinear multiscale method to GPR image processing"; A. Pizurica, W.Philips, I. Lemahieu and M. Acheroy, Proceedings of the IASTED International Conference Signaland Image Processing, 27 - 31 October, 1998, Las Vegas, Nevada-USA.[25] "Feature extraction and classi�cation methods for ultra-sonic and radar mine detection."; H. Sahli,E. Nyssen, L. van Kempen and J. Cornelis, IEEE Cesa'98 conference, 1 - 4 April, 1998, Tunis,Tunesia, pp 82-87.