Tests of Spatial and Temporal

This is a project to demonstrate the feasibility of transferring a successful technology from seismic exploration for oil and gas to the problem of imaging the shallow seabed|10-20m|and small scatterers|7.5cm radius. In this rst study, we demonstrate that we have adapted our current code to this imaging problem and we provide tests of our choice of spatial and temporal parameters. Our simple example places a single scatterer at a water/sediment interface. Synthetic data is generated for a survey in which a towed array of receivers collects data from a series of impulsive sources, set o at regular intervals. The parameters we use are a source bandwidth of 20-80Khz, sources set o at 5m intervals, a 10m receiver array with receivers spaced every 4cm along the array. The sample rate in time is .005msec. Fourteen shots are set o , so that the transverse coverage of the subsurface extends over 80m. We then show that even a modest decrease of the bandwidth, or spatial or temporal sampling rate degrades the image. It should be noted that the sampling critieria we demonstrate here are not peculiar to our imaging method; they apply to any method whose objective is to resolve the boundaries of a target of this size. We then provide a simple example of simultaneous velocity analysis and target detection. In this method, we nd a \best t" of velocity and target location from the travel time records of two surveys of the previous type carried out along parallel lines. This is an admittedly primitive example that is only suggestive of more sophisticated methods of the type listed in the references. The latter use full waveform processing to produce an image of the scattering object, not just its location. They also use the structural features of the re ecting surface for the velocity analysis rather than just the far eld approximation used in this example; i.e., treating the target as a point scatterer for velocity analysis. INTRODUCTION This is a project to demonstrate the feasibility of transferring a successful technology from seismic exploration for oil and gas [Bleistein, 1975, 1976, 1987a, 1987b; Bleistein, N., Cohen, J., and Hagin, F., 1987; Docherty, 1987, 1988; Dong, W., 1990; Hsu, C., 1991; Liu, Z., and N. Bleistein, 1991, 1992, 1993; Sullivan, M.F., 1986; Sumner, B., 1989] to sea mine countermeasures. More speci cally, we discuss the adaptation of an exploration geophysics technique for collecting and processing data to the problem of detecting small scatterers in shallow water and shallow sediments. The objective of the discussion here is to establish bounds on the parameters of the data gathering experiment. Those parameters include bandwidth, spatial sampling rate and temporal sampling rate. It should be noted that the sampling critieria we demonstrate here are not peculiar to our imaging method; they apply to any method whose objective is to resolve the boundaries of a target of this size. 2 In the experiments that we propose, an array of receivers is towed behind a boat and acoustic sources are set o at regular intervals by the same boat. See Figure 1. In this manner, a synthetic aperture of data is created in the target area. In each * ................. Fig. 1. Model of an acoustic survey based on exploration geophysical surveying techniques. experiment, the downward propagating waves are scattered by the ocean bottom, re ecting layers in the seabed, and all other objects whose re ection properties di er from their host medium. The inverse methods that we have developed in the context of exploration geophysics allow us to create a re ector map of the region of interest. A major di erence between the current use of our inverse methods and the application here is scale, both temporal and spatial. In exploration geophysics, the range to the re ectors being imaged are hundreds or thousands of meters; here the range is a few tens of meters. The re ectors of interest in the former problem are of the same scales; here we seek scatterers of a few centimeters in diameter. Frequencies in exploration geophysics are measured in a few tens of Hz, at best; here the frequencies need to be tens of KHz|in fact, our reference examples below for adequate sampling in space and time uses a Ricker wavelet, Figures 2 and 3, with a bandwidth of 20-80KHz and a center frequency of 40KHz. Also, source and receiver spacings in exploration geophysics are measured in tens of meters; here we use a few meters separation between sources and a few centimeters between receivers. The bandwidth and both sample rates for data collection are dictated by the scale of the objects to be detected. The spatial sampling rate should be measured by the spacing of midpoints between source and receiver, which, in this case is 2cm. This is less than half (nearly a quarter) of the radius of the scatterer, which is 7.5cm; the 3 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Time (ms) 0 0.5 1.0 A m pl itu de Ricker Wavelet Fig. 2. Ricker wavelet in the time domain. 0 2 4 6 8 10 12 14 Frequency (x10khz) 2 4 6 8 10 A m pl itu de Ricker wavelet in frequency domain Fig. 3. Ricker wavelet in the frequency domain. 4 40KHz center frequency gives us a 3.75cm wavelength in water, again half the radius. For high frequency methods to be valid, it is necessary that 4fL=c 1: Here, f is the frequency in Hz, c is the propagation speed and L is any of the length scales of the problem. Here the most crucial length scale is the radius of the scatterer. For 40KHz and a propagation speed of 1500m/sec, with L = 7:5cm, 4fL=c = 4 and for the larger speed of 6000m/sec in the interior of the \solid" scatterer, 4fL=c = 1. Thus, are choice of parameters is just adequate in the worst case, more than adequate for water velocity. The purpose of this report is to show that with adequate parameters|frequency, spatial and temporal sampling rates|a scatterer of this scale can be detected and that when the requirements on any of the parameters is violated, the image degrades. The image becomes progressively worse as any of the parameters depart further from its prescribed constraint. Our examples also use our current two-and-one-half dimensional (2.5D) computer codes. This theory assumes a two-dimensional variation in scattering parameters but accounts for the geometrical spreading e ects of three-dimensional propagation. Thus, for this formalism, only a single line of data need be gathered, but all scatterers are assumed to be cylindrical, extending out of the vertical plane through the data gathering line. This formalism su ces for the demonstration of sampling parameters that we present here. Development of three dimensional processing codes will be carried out under follow-on funding. These model experiments are carried out under the assumption of a known (constant) propagation speed. However, we ultimately plan to extend the method to determine one or two parameters characterizing the propagation speed along with target imaging. Here, for demonstration purposes, we also present a simple example of that sort. We consider a case in which a towed array survey is carried out on two parallel lines and a single scatterer is to be detected. The propagation speed is assumed to be an unknown constant. We exploit the fact that the target is in the far eld and can be treated as a point scatterer. We generate a synthetic set of travel times for re ections from the source to the scatterer to the towed array. We simultaneously determine the location of the point scatterer and the propagation speed of the host medium by an optimization technique. This technique is derived here and the results of some simple tests are reported. This simple example is far less sophisticated than the velocity analysis techniques we have available [Liu and Bleistein, 1991, 1992, 1993]. There, we use the full wave form for processing the data and produce an image of the scatterer in the derived velocity eld. Furthermore, those methods do not treat the re ector as a point scatterer, allowing velocity analysis on the sediment boundary re ectors as well as the on the smaller targets. 5 DETECTION OF SMALL SCATTERERS In this section, a benchmark model and its inversion will be presented. We will then proceed to degrade the inverted image by violating the basic requirements of high frequency imaging and adequate sampling. In doing so, we establish the need for the criteria we used in the benchmark result that produced a quality image of the scatterer. The Basic Model 0 80 10 20 40 Fig. 4. Model of an acoustic survey modeled on exploration geophysical surveying techniques. The basic experimental con guration is shown in Figure 4. We also show an expanded view of the 2m 2m region that will appear in our inversion output. The acoustic survey is carried over a range of 80m above a circular scatterer situated at the water sediment interface 10m deep. The dimensions of the experiment are as follows: cable length, 10m; separation between the source and nearest receiver, 2m; receiver spacing, 4 cm. sources set o every 5m. The propagation speed in water is taken to be 1500 m/sec; in the sediment it is 2000 m/sec; in the scatterer it is 6000 m/sec. The source is the Ricker wavelet of Figure 2 or 3 in the frequency domain. In Figure 5, we show the synthetic data for an experiment like the one considered here. The horizontal coordinates are shot numbers; the synthetic data are displayed as \wiggle traces" vertically, much like the single traces in Figures 2 and 3, closely packed together in groups of 250, one for each receiver in the array. In the model output shown here, we have scaled up the size of the scatterer by a factor of 10 and scaled down the frequencies by a factor of 10. We need the temporal 6 0 10 20 30 40 50 T im e (m s) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Shot Number Shot Gathers Fig. 5. Synthetic data range of 10-50msec to show the full range of time responses for the problem of either scale. On the other hand, for the smaller scatterer, over this range, the responses would appear one-tenth as broad as they do in this gure and would be much harder to see. Furthermore, for the scatterer scaled down by a factor of 10, the separation between the responses from the front and back face of the scatterer would also be compressed by a factor of 10 and would not be visible on a plot of this scale. Thus, plotting a prototypical example as we have here allows us to show all of the dynamic features of the problem: total temporal range of responses and signal responses from all of the re ectors of the model. The inversion of the data of Figure 5 is shown in Figure 6. One can see the circular scatterer and the sediment interface, as well as some artifacts of the processing. Note that both the upper and lower surfaces of the scatterer are imaged. The model data takes proper account of the refractions at the water-metal interface and so does the inversion code. The peak amplitude of the output on the re ectors varies with re ection strength|hence the circular scatterer appears darker than the sediment boundary, since the normal re ection coe cient of the former is .60, while for the latter, it is .14. The most apparent artifacts are the dim near-vertical images below the scatterer. These are so-called \di raction smiles." They arise because the survey does not generate specular returns from the complete circular boundary, but \misses" zones near horizontal propagation, that is, near the equator of the scatterer. So, in fact, the 7 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) Inversion Fig. 6. Inversion of the synthetic data of Figure 5 scatterer image actually is missing some small segments near the equator. However, the theory predicts di raction smiles coming o tangent to these edges. There are \many" of them|250, to be precise|one for each xed o set between a source and a receiver. In order to \see" this near vertical section of the scatterer in a medium of constant velocity, one would require a survey of \in nite aperture" rather than the one here. However, the 80m range here| 40m above the scatterer|is so broad that the segments of scatterer that are missed here are negligible and not even apparent because of the continuations through the di raction smiles. In a depth dependent medium, it is possible to achieve horizontal propagation, and even turned rays, at nite o set. Thus, in a depth dependent medium, we can see more of the steeply dipping parts of the scatterers than we can here in a constant propagation speed ocean. It would be possible, then, in such a medium to obtain images with signi cantly smaller di raction smiles. E ects of decreasing bandwidth In Figures 7 through 11, we show the e ects of progressively halving the center frequency and the bandwidth. (Note that this technique retains the percentage bandwidth, that is, (fmax fmin)=(fmax + fmin) remains constant. At rst the scatterer boundaries merely become \fuzzier," but then, the circular scatterer degenerates to a 8 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) fpeak=10000hz (5000~20000hz) Fig. 7. Test of Bandwidth 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) fpeak=5000hz (2500~10000hz) Fig. 8. Test of Bandwidth 9 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) fpeak=2000hz (1000~4000hz) Fig. 9. Test of Bandwidth 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) fpeak=1000hz (500~2000hz) Fig. 10. Test of Bandwidth 10 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) fpeak=500hz (250~1000hz) Fig. 11. Test of Bandwidth fuzzy point and, nally, to a \smear" that would be indistinguishable from background clutter and noise in a real ocean environment. In Figure 10, note that half-wavelength, c=f at the center frequency of 1000Hz is .75m, which is about the thickness of the band representing the sediment boundary. This is a qualitative indication of what is happening in this series of experiments. This set of examples support the use of the bandwidth of our benchmark wavelet| 20-80KHz for imaging a scatterer of the indicated size in water. Temporal sampling Here we show the e ect of diminishing the rate of temporal sampling. Note that the original sample rate of .005msec is equivalent to a Nyquist frequency of 100KHz, above all but a negligible part of the bandwidth of the Ricker wavelet in Figure 3. Now, we show the e ect of progressively decreasing the sample rate in Figures 12 through 15 through the values, .01msec, .02msec, .04msec, .1msec, respectively. Note that at .01msec, Figure 15, the Nyquist frequency is 50KHz, suggesting some aliasing of the wavelet in Figure 3. As might be expected, the results are not too di erent from the benchmark output in Figure 6. On the other hand, at the lower sampling rates, we start to see the e ects of aliasing in the \ghost" images of the circular scatterer and the at re ectors. There is also a secondary e ect, namely, the loss of resolution, comparable to that in the previous set of gures. The reason is that reducing the sampling rate also reduces the maximum frequency in the data|through aliasing| 11 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) Migration dt=0.01ms Fig. 12. Test of dt Sample Rate as well as the e ective center frequency of the data|hence, an e ect similar to the bandwidth reduction of the previous examples. What can be concluded from this set of gures is that while a modest decrease in sampling rate from our benchmark is tolerable, signi cant change in sampling will produce both aliased images and lower resolution images. Spatial sampling Here we show the e ect of diminishing the rate of spatial sampling rate. As noted earlier, 4cm spacing between receivers provides 2cm sampling of midpoints between source and receiver. Now, we show the e ect of progressively decreasing the sample rate in Figures 16 through 18 through the values, 6.4cm, 8cm, and 10cm, respectively. In these gures, another form of \ghosting" is apparent, when compared to Figure 6. There is an additional e ect of a relative \ attening" of the upper half of the re ector, less so, on the lower half of the re ector. This is a subtle e ect of the aliasing of the transverse wave number that requires some understanding the relationship between wave vector and dip angle of the image. Basically, the normal to the re ector is giving by the wave vector, (kx; kz) = (kx;p!2=c2 k2 x): The e ect of aliasing is to reintepret kx as a value folded over its Nyquist value prescribed by the sampling rate. Further, the value of kz is increased since it depends on the subtraction of k2 x from the total wave number. The inclination angle of the normal to the re ector is 12 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) Migration dt=0.02ms Fig. 13. Test of dt Sample Rate 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) Migration dt=0.04ms Fig. 14. Test of dt Sample Rate 13 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) Migration dt=0.1ms Fig. 15. Test of dt Sample Rate thereby rotated towards the vertical, making a atter image. For larger values of c; as in the metal, smaller values of k1 produce the normal vector. This occurs because a larger value of c produces a smaller value of !2=c2, bounding the usable values of kx for producing the normal vector. Thus, the values of kx used to image the lower part of the re ector tend to be smaller than the values used to produce the upper part of the re ector. The result is that there is more \foldover" in the kx values imaging the upper part of the re ector. As a simple example, let us consider the specular return at 45 and neglect the o set between source and receiver. Then, for a given frequency, f , the wave numbers that produce the specular return are given by kx = kz = 2 f c cos 45 ; while the Nyquist wave number is given by kNyq = dx; with dx the sampling rate. It is fairly straightforward to check that at 40KHz and a propagation speed of 1500m/sec, kx 37 : For a spacing of receivers of 4cm the sampling rate, which is the spacing of midpoints between source and receiver, is 2cm and kNyq = 50 , larger than kx. On the other hand, for 6.4cm spacing|3.2cm sampling rate, the Nyquist wave number is 31 , less than kx. 14 In the rst case, one should expect accurate imaging. In the second case, the transverse wave number of 37 is folded over the Nyquist frequency to a value of 25 ; with a corresponding value of kz 47 : The result of this is a normal that makes an angle of arctan kz=kx 62 : Thus, the tangent to the re ector at 45 has been rotated to 28 : On the other hand, it should be noted that inside the scatterer, the value of c is four times as large and therefore the horizontal and vertical wave numbers that produce the specular at 45 are one fourth of the numbers above, namely, about 9 . This is well below Nyquist for kx; no aliasing occurs and the re ector is properly imaged on the lower half. Thus, we see that even a mild change in spatial sampling, from 4cm to 6.4cm makes a dramatic change in the image recovered, where the distortion associated with aliasing is even more signi cant than the ghosting associated with aliasing. With the two half images more clearly separated in Figure 16, it is also easier to see the e ect of limited aperture, that is, the e ect of not having in nite transverse range in receivers. Both half images are more clearly seen not to close at the equator of the scatterer. Further decrease in sampling in Figures 17 and 18 only exaggerate the e ects seen in Figure 16. In those gures, attening of the lower surface occurs, as well. The main point of this discussion is that there are two types of degradation that occur with inadequate transverse sampling. First, there are ghost images|appearing as additional ripples|on the image. Second, there is image distortion, with steeper dipping segments of the re ector aliased on to atter images due to a decrease in the interpreted transverse wavenumber and a corresponding increase in the vertical wave number. SIMULTANEOUS TARGET IDENTIFICATION AND VELOCITY ESTIMATION Here, we will show a simple example of simultaneous estimation of the velocity in water and target location. This method is based on an optimization procedure carried out on the travel times associated with the scattering from a small target in the far eld. 1 The basic con guration is shown in Figure 19. Two towed array surveys are carried out on parallel lines. (Of course, this could be done with one boat running each line in turn.) We model the target as a point scatterer in a medium of constant propagation speed. The objective is to determine that speed, along with the location of the point scatterer. For simplicity of exposition in the derivation below, we take the source and receiver positions to be coincident. However, the actual procedure tested numerically included travel times from all source receiver pairs in the towed arrays, as in the imaging experiments described in the previous section. 1A target of diameter 7.5cm at a range of 10m is safely in the far eld. In fact, the criterion that R=r > 3 is su cient, with R the range and r the radius of the target. 15 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) dx=0.064m Fig. 16. Test of dx Sample Rate 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) dx=0.08m Fig. 17. Test of dx Sample Rate 16 9.0 9.5 10.0 10.5 D ep th ( m ) 39.0 39.5 40.0 40.5 41.0 Midpoint (m) dx=0.10m Fig. 18. Test of dx Sample Rate The objective is to obtain a best estimate the propagation speed and target location that t the travel time data. We use a least squares estimator to do this. For coincident source and receiver, the travel time is given by t(x; y; x0; y0; z0) = 2q(x0 x)2 + (y0 y)2 + z2 0 c : (1) In this equation, x and y are the coordinates of the ship on the surface, (x0; y0; z0) are the coordinates of the target, and c is the propagation speed. In this survey, two parallel lines imply two values of y, say, y1 and y2 while the motion of the ship along these lines gives us a suite of values of x, say xi. (For noncoincident source and receiver, we could describe the set of source and receiver positions as xi and xi hj , respectively, with all sums over i below replaced by double sums over i and j. Traditionally, in oil exploration, these coordinates are described by xi hj , with xi the midpoints between source and receiver and the hj the half-o sets. Either of these will work for identi cation.) Suppose that Ti is the measured data at the ith receiver position, with 1 i n corresponding to y1 and n + 1 i 2n corresponding to y2; then, the objective function is f( ) = 2n Xi=1 2i ( ): (2) 17 Fig. 19. Experiment design for simultaneous determination of velocity in host medium and target location. The hyperbolas represent the graphs of the time responses of coincident source and receiver experiments. 18 In this equation, i( ) = t(xi; y1; x0; y0; z0; c) Ti; i = 1:::n; (3) i( ) = t(xi; y2; x0; y0; z0; c) Ti; i = n+ 1:::2n; and = (x0; y0; z0; c): (4) We seek the value for which f( ) is the minimum. To nd this minimum, we use the Gauss-Newton method, as follows. 1. Choose a start point, f( 0). 2. Calculate 0 = (A 0A0) 1A 0 ( 0); (5) 0 = 0; A0 = "@ i( ) @ j # = 0 ; (6) ( 0) = 1( 0); 2( 0); :::; 2n( 0) ; (7) and the superscript denotes the transpose of the matrixA0, or the vector, ( 1( 0); 2( 0); :::; 2n( 0). 3. Let 1 = 0 + 0 (8) Then 1 is the rst correction to the approximation, 0 = (x00; y0 0; z0 0 ; c0). 4. Repeat step 2 and 3 using 1, 2, etc., until j kj < , where is the prescribed spacing between successive approximations. The stability depends on the existence and condition number of (A 0A0) 1. For separated survey lines, the vectors in A0 are four linearly independent vectors of length 2n, that is, A0 is a 4 2n matrix, where 2n 4: The elements of A 0A0 are the dot products|like correlation coe cients except for normalization|of these independent vectors. Consequently, in this 4 4 matrix, the diagonal elements are the norms of the independent vectors, while the o -diagonal terms are dot products between di erent vectors of this linearly independent set. Thus, the matrix tends to be diagionally dominant. For example, the method was tested on the model of Figure 19 with x0 = 40:0m, y0 = 5:0m, z0 = 10:0, c = 1500m/sec. The two lines of data were collected at y1 = 10m and y2 = 20m, with towed array parameters as in the previous section 19 on each survey line. The rst guess we tried in this case was x00 = 10:0m, y0 0 = 10, z0 0 = 5:0, c0 = 1000. For this example, A 0A0 = 0:2501 0:0140 0:0135 0:0013 0:0140 0:1214 0:0103 0:0034 0:0135 0:0103 0:1154 0:0033 0:0013 0:0034 0:0033 0:1001 ; (9) with eigenvalues 0.25317, 0.126864, 0.0992903, 0.107676. We chose = 0.01; after 19 iterations, the result is x0 = 39:999m, y0 = 5:011, z0 = 10:008, c = 1499:826. Note that the least accurate of these data is the value of y0. This is not surprising: the \biaxial vision" of the survey in this direction is most crucial. Indeed, if we move y2 to y1, that is, if we take both survey lines to be identical, then the matrix A 0A0 is singular and the method breaks down. We will say more about this below. As a second test, we added 3msec of noise to the data in the rst test above. (Note that the minimum travel time in the data is about 15msec.) The Gauss-Newton method now yields the results, x0 = 39:348; y0 = 5:132; z0 = 10:218; c = 1492:442, again after 19 iterations. These values are not as good as with perfect data, but still acceptable. In summary, we have shown in a simple example, that simultaneous velocity analysis and target location are possible. In our more sophisticated velocity analysis techniques [Liu and Bleistein, 1991, 1992, 1993], we use the full wave form for processing the data and produce an image of the scatterer in the derived velocity eld. The singular case When y2 = y1, there is only one line of data, repeated; the matrix,A 0A0 is singular in this case and the method breaks down. Physically, this makes sense. With only one line of data in the x direction, all points on a circle of constant p(y y1)2 + z2 produce the same travel time. Here, we explain how the mathematics predicts this result. In particular, we show that at least one eigenvalue of A 0A0 becomes zero, making the matrix singular, hence, noninvertible. The fundamental mathematical idea is the following fact from linear algebra: the number of linearly independent columns of the matrices, A0 and A 0A0 are the same. 2 Thus, the latter matrix is singular whenever the former matrix has two linearly dependent columns. (Because A0 is not square, we cannot characterize linear dependence of columns of this matrix as making it singular.) Before proceding to the mathematical discussion, let us return to the de ntion of A0 in (6) and check the claim that two of the columns become singular when y2 = y1: 2Two columns of a matrix are linearly dependent if the ratio of the elements of each row is the same, making one column vector just a multiple of the other. 20 In fact, the discussion about equal travel times on circles in y and z indicates which two columns should be examined, namely, the ones corresponding to the y0 and z0 derivatives of i: These are the second and third columns. By using the de nition of i in (3) with the de nition of t in (1), we nd that @ i( ) @y0 = 2c y0 y1 ri1 ; @ i( ) @z0 = 2c z0 ri1 ; i = 1:::n; (10) @ i( ) @y0 = 2c y0 y2 ri2 ; @ i( ) @z0 = 2c z0 ri2 ; i = n+ 1:::2n: In these equations, ri1 = q(x0 xi)2 + (y0 y1)2 + z2 0; i = 1:::n; (11) ri2 = q(x0 xi)2 + (y0 y2)2 + z2 0; i = n+ 1:::2n: Note that the rst n rows de ned by (10) have the property that the ratio of left element to right element is (y0 y1)=z0 while the ratio of the last n elements is (y0 y2)=z0. Consequently, when y2 = y1, these two half columns have the same ratio; the elements of the columns have a common ratio and they are linearly dependent. Thus, we have established the claim that for the given experiment, coincident lines of data lead to a column matrix with linearly dependent columns. Now we will verify that this, in turn, will lead to a singular matrix, A 0A0: This follows from the identity, x A 0A0x =k A0x k2 : (12) Here, the right side is just the square of the length of the vector, A0x: Now, suppose that x is a null vector of the matrixA0; that is,A0x = 0: (In particular, for the matrix A0 of interest here, this will be true for the vector (0; z0; y0 y1; 0) ; when y2 = y1.) Then, premultiplication by A 0 does not change this result; that is, A 0A0x = 0; too. This means that x is an eigenvector of A 0A0 with eigenvalue zero and this matrix is singular. On the other hand, if A 0A0x = 0; then premultiplication by x does not change that equation and, from the identity, (12), x A 0A0x = k A0x k2= 0; from which it follows that A0x = 0; that is, whenever x is an eigenvector of A 0A0 with eigenvalue zero, it is a null vector of A0. From these two arguments, it follows that the eigenvectors of A 0A0 with eigenvalue zero and the null vectors of A0 are one and the same. So, now we see how the physics and the mathematics complement one another: when we lose biaxial vision, the matrix characterizing our ability to discriminate position becomes singular. Indeed, it becomes singular in a way that exactly characterizes the loss in biaxial vision. In fact, one can further show that for the two parallel lines \nearby" one another, the iteration process is ill-conditioned. When two lines are nearby, the ratios of rows in 21 (10), (y0 y1)=z0 and (y0 y2)=z0 are nearly equal. When the elements of two columnsof A0 are within O( ) of being constant, as they are here, then there exist vectors ofunit magnitude for which [A 0A0] 1x is O( 1), which implies ill-conditioning.CONCLUSIONSWe have demonstrated here acceptable bandwidth and spatial sampling param-eters for a seismic{exploration-like survey adapted to the problem of sea mine de-tection. We have also demonstrated how the image process breaks down when theparameters we originally chose are degraded. We have demonstrated the e ects ofspatial and temporal aliasing, as well as the e ect of decreasing bandwidth of theprobing signals. The bandwidth and the sampling rates in space and time are dic-tated by the radius of curvature of the scatterer to be detected. We claim that thesampling critieria we demonstrate here are not peculiar to our imaging method; theyapply to any method whose objective is to resolve the boundaries of a target of thissize.We have also presented a primitive example of simultaneous velocity analysisand target detection; in the series of papers by Liu and Bleistein, appearing in thebibliography, more sophisticated simultaneous velocity analysis and re ector mappingare presented. It is these more sophisticated methods that we plan to extend to thecurrent problem to determine one or two parameters in a characterization of thepropagation speed in shallow water and sediment.ACKNOWLEDGEMENTSThe authors gratefully acknowledge the support of the O ce of Naval Research,Ocean Acoustics Branch and members of the Consortium Projection on Seismic In-verse Methods for Complex Structures at the Center for Wave Phenomena, ColoradoSchool of Mines.22 REFERENCESBleistein, N., 1975, Direct image reconstruction of anomalies in a plane via physicaloptics far eld inverse scattering, JASA, 59, 6.Bleistein, N., 1976, Physical optics far eld inverse scattering in the time domain,JASA, 60, 6.Bleistein, N., 1987a, On the Imaging of re ectors in the earth, Geophysics, 52, 931-942.Bleistein, N., 1987b, Kirchho inversion for re ector imaging and sound speed anddensity variations: Worthington, M., Ed., Deconvolution and inversion: 1986EAEG/SEG Workshop Rome, Italy, Blackwell Scienti c Publishers, Oxford, 305-320.Bleistein, N., Cohen, J., and Hagin, F., 1987, Two and one-half dimensional Borninversion with an arbitrary reference: Geophysics, 52, 26-36.Docherty, P., 1987, Ray theoretical modeling, migration and inversion in two-and-one-half-dimensional layered acoustic media: Ph.D. thesis, Colorado School of Mines,Center for Wave Phenomena Research Report, CWP-051, Colorado School ofMines, Golden, CO.Docherty, P., 1988, CXZ: Fortran program for laterally varying velocity inversion,Center for Wave Phenomena Research Report, U9, Colorado School of Mines,Golden, CO.Dong, W., 1990, CXZCS: A 2.5D common shot inversion program in a c(x,z) medium.Center for Wave Phenomena Research Report, U13R, Colorado School of Mines,Golden, CO.Hsu, C., 1991, CXZCO: A 2.5D common o set inversion program in a c(x,z) medium.Edited by Z. Liu, Center for Wave Phenomena Research Report number U20,Colorado School of Mines, Golden, CO.Liu, Z., and N. Bleistein, 1991, Migration Velocity Analysis: Theory and an IterativeAlgorithm, Center for Wave Phenomena Report, CWP-104, Colorado School ofMines, Golden, CO, to appear, Geophysics.Liu, Z., and N. Bleistein, 1992, Velocity analysis by residual moveout: Center forWave Phenomena Report, CWP-123, Colorado School of Mines, Golden, CO.Liu, Z., and N. Bleistein, 1993, Velocity analysis by perturbation: Center for WavePhenomena Report CWP-135, Colorado School of Mines, Golden, CO.Sullivan, M.F., 1986, CCCO: Fortran programs for common o set velocity inversion,descriptions and instructions, Center for Wave Phenomena Research Report,U05, Colorado School of Mines, Golden, CO.Sumner, B., 1989, CZ: Fortran programs for 2.5-D strati ed velocity inversion, de-scriptions and instructions, Center for Wave Phenomena Research Report, U06R,Colorado School of Mines, Golden, CO.23