The utility of acoustic-to-seismic coupling systems for land mine detection has been clearly established with potential as either a primary or confirmation sensor system. They can detect very low metal content mines that are difficult for ground-penetrating radar detection. For most applications, only the magnitude of the surface velocity is used to construct recognition algorithms. Recently, we introduced phase-based features in the classification scheme, significantly lowering false alarm rates at given detection probabilities. In this paper we introduce modeling equations that explain the phase features. We also describe the image processing techniques applied to velocity data collected in the time domain.
 Recent developments in land mine detection based on the acoustic-to-seismic coupling phenomenon have demonstrated the feasibility of detecting both metallic and low-metallic mines [Sabatier and Xiang, 2001; Xiang and Sabatier, 2000, 2002, 2003; Rosen et al., 2000; Lafleur et al., 2000; Keller et al., 2002; Hocaoglu et al., 2002; Wang et al., 2002]. In this method, vibration velocities at the ground surface being insonified by loudspeakers are measured by laser Doppler vibrometers (LDVs). Land mine detection is achieved by observing the velocity difference between regions that do and do not contain buried mines. Most of the work done in this area processes the magnitude component of the velocity. More recently, we also found that we were able to improve detection and reduce false alarms by combining the magnitude and phase information [Wang et al., 2003].
 Using driven harmonic oscillators to model acoustic land mine detection was suggested by Donskoy et al. [2001, 2002] and has been used by other researchers to explain various phenomena of acoustic land mine detection [Yu and Gandhe, 2002; Hocaoglu et al., 2003]. In this paper we use similar models to provide a physical explanation for the phase feature; that is, there exists a valley in the phase of the observed surface velocity over land mines at a higher frequency relative to the magnitude maximum. Although the feature was investigated to some extent previously, no physical interpretation was provided [Wang et al., 2003].
 We present experiments on two sets of data in this paper. The first set consists of data points of complex velocities at specific two-dimensional spatial locations and frequencies. This data set is acquired with a downward looking scanning LDV, which is described in the work of Sabatier and Xiang . The image processing and feature extraction algorithms used to process this data set are described in previous papers [Wang et al., 2002, 2003]. In addition to the land mine detection results, we also expand the discussion on using the harmonic oscillator model to understand the magnitude and phase signatures observed in the data. The second data set is acquired with a cart equipped with a set of loudspeakers and 12 LDVs. Additional description of the cart can be found in the work of Burgett et al. . The ground vibration velocity is recorded directly in the time domain while the cart is moving. We describe for the first time our image processing algorithm used to extract the phase information directly from time domain data. We find that similar relations between magnitude and phase signatures are observed in both frequency and time domain data, and examples are given to illustrate how phase signatures can be used to improve the detection of buried mines.
2. The Harmonic Oscillator Model
 We assume the coupled harmonic oscillator illustrated in Figure 1a which can be described by the following equations:
Here F, m, k, r, and u represent the driving force, mass, spring constant, damping factor, and displacement, respectively, and the subscripts “1” and “2” correspond to the two masses and springs as indicated in Figure 1a. Let us assume a sinusoidal driving force and displacement,
We can obtain the solution by substituting equations (3) and (4) into equations (1) and (2). Since we are interested in the surface vibration velocity, we only list the solution for u1 here:
We now define the following ratio of the surface velocity and the driving force:
which is the inverse of the impedance of the system. Here the subscript “CHO” stands for coupled harmonic oscillator. The magnitude and phase of gCHO versus frequency are plotted in Figure 1b. Here we can see that the phase changes from positive to negative as frequency increases past the first resonance and exhibits a minimum (or valley) between the two resonant frequencies. While the second magnitude peak often does not appear within the frequency range of the data, this phase valley after the first magnitude peak is the main feature that we utilize to improve land mine detection, as discussed in later sections.
 The phase plot in Figure 1 is relative to the driving acoustic force at the ground surface. However, the phase of the driving acoustic force at the ground surface is not directly available. Instead, what we can actually measure is the phase difference of ground surface regions that do and do not contain land mines. If we know the phase in background (no-land mine) regions relative to the driving force with a similar model, we can deduce from this information the phase in mine regions relative to that of the driving force. This in turn will allow us to interpret the phase signatures in the measured data with the plot in Figure 1.
 The following is a simple model that addresses the relative phase between the driving force and the surface vibration in background regions. The ground is modeled as an infinite number of identical thin layers, each corresponding to a mass and spring as shown in Figure 2a. The equations that describe the system in Figure 2 are
The subscript n is the index of the layers. For the purpose of simplicity, we ignore the difference between layers in real ground and use the same m, r, and k for each layer; hence no subscript is used for m, r, and k in equations (7) and (8). We assume a sinusoidal driving force as in equation (3) and assume that the motion of the masses forms a downward propagation wave:
Here q is the complex wave vector of the propagating wave, d represents the thickness of each layer, and A is the vibration amplitude at the ground surface. We do not include the wave propagating upward because, with the assumption that all layers are identical, we do not expect any reflection of the wave from layer interfaces in the ground caused by nonuniform ground acoustic properties.
Our interest, however, is the ratio between surface vibration velocity and driving force. This can be obtained from equations (10) and (12) as
Here the subscript “BK” stands for the background (no-land mine) region. We can greatly simplify equation (15) using equation (14):
A representative plot of the magnitude and phase of gBK is in Figure 2b. The slight decrease of magnitude with increasing frequency is due to the damping factor r. If we assume r to be small, the magnitude should be fairly constant. We are, however, more interested in the phase of gBK, which represents the phase difference between the driving force and surface vibration velocity in non–land mine regions. This is given by
This phase difference is zero at zero frequency or if we assume there is no damping (r = 0).
 The difficulty is that we do not actually have directly measured values of r and k, which can vary significantly in different environments. In Figure 3 we try to illustrate the effect of ΔθBK on our measurement of surface velocity phase differences between land mine and background regions. In Figure 3a the solid curve, which is the same as the phase plot in Figure 1b, represents the velocity phase in the land mine region. The dashed curve is ΔθBK assuming r = 0. The curve in Figure 3b is the difference between the two curves in Figure 3a and represents the phase difference between the land mine and background regions. In Figure 3c the solid curve is the same as that in Figure 3a, but the dashed curve is ΔθBK assuming much higher attenuation, with (ωr/k) = 1 at ω = 1000 (or approximately 160 Hz). This creates a frequency-dependent phase shift in the background region. Figure 3d is the difference between the two curves in Figure 3c. We can see that the curve in Figure 3d exhibits behaviors very similar to that of Figure 3b. As a result, we conclude that, when the phase of the driving force at the ground surface is not available as is in our case, we can use the background velocity phase as a good approximation that allows us to still retain the essential phase features used in land mine detection.
3. Analysis of Frequency Domain Data
 The data described in this section were taken over the gravel and dirt regions of a mine lane at a U.S. Army site. The image processing algorithms used for this data set have been described previously [Wang et al., 2002, 2003], with some modification in the phase estimation steps. In the following we will describe the steps involved in obtaining the phase information.
 The original data for each ground patch, which is 2.5 × 2.1 m2 in size, consist of a three-dimensional array of measured complex surface vibration velocities at specific locations and frequencies. In Figure 4a we show the phase of one such ground patch. Each small rectangle, which contains 25 × 21 pixels, is a view of the patch at the specified frequency, with horizontal and vertical being the cross-track and along-track directions, respectively. This ground patch contains a land mine near the bottom left corner where some phase “disturbance” is visible in a few frequencies. For comparison, the phase of a ground patch with no land mine is shown in Figure 4b. We use P(x, y, ω) to represent the phase at a specific pixel (x, y) at a given frequency ω. Here x and y are the pixel coordinates in the horizontal (cross-track) and vertical (along-track) directions, respectively.
 The processing steps are illustrated in Figure 5 with data of 8 consecutive frequencies belonging to the ground patch in Figure 4a. The first step is to remove the phase variations, which are mainly in the along-track direction, present in both patches. These phase variations are caused by the different sound propagation delays from the sound source to different spatial locations. In order to extract phase shifts caused by the land mines, we first try to estimate the background phase, denoted by PBG(x, y, ω), which is subsequently subtracted from P(x, y, ω). First, we start with the values of P(x, y, ω) between −π and π. Because the phase is approximately uniform in the horizontal (cross-track) direction, we start estimating the background phase by calculating the median phase along horizontal scan lines. Let us define
The function adjust_phase[.] is used to add or subtract 2π from the values to ensure the results are always between −π and π. There is a π phase shift between P1′(y, ω) and P2′(y, ω). Because near where the phase wraps around there is usually anomaly in the median phase calculation caused by phase noise in the data, we want to calculate the background phase only from regions where the phase wrap-around does not occur. The π phase shift between P1′(y, ω) and P2′(y, ω) makes them wrap around at different places, so that we can have
We can understand the criterion of selecting between P1′(y, ω) and P2′(y, ω) this way: If P1′(y, ω) and all its neighbors in y are closer to zero than to −π and π, we select P1′(y, ω) because it is not near a phase wrap-around. Otherwise, we select P2′(y, ω). This gives the median of horizontal scanlines without the anomalies caused by phase wrap-around.
 Since there still exists some phase variation along horizontal scanlines, we add one additional term, which is the median difference between P(x, y, ω) and P′(y, ω) along vertical scanlines to obtain the background phase:
We then replace each pixel's value according to the following equation:
For the phase data in Figure 5a the median phase along horizontal scanlines in equation (20) is in Figure 5b, and the estimated background phase PBG(x, y, ω) is in Figure 5c. The modified phase in equation (22) is displayed in Figure 5d. We then smooth it by applying a 3 × 3 × 1 median filter in the spatial dimensions followed by a 1 × 1 × 5 median filter in the frequency dimension; the smoothed images are shown in Figure 5e.
 In Figure 6 we show the processed magnitude and phase of a ground patch where there is a land mine near the bottom left corner. Plotted in Figures 7a and 7b are the magnitude and phase versus frequency at approximately the center of the land mines in Figure 6. The plots in Figure 7b demonstrate behaviors similar to those in Figure 1b. As we can see two magnitude peaks, the phase, after reaching a minimum after the first magnitude peak, starts to increase before the second magnitude peak and then decreases again around the second magnitude peak. This indicates that the harmonic oscillator model is useful in helping us understand the behavior of measured phase data even though it is a very simple model.
 In Figure 8 we give an example of how phase signatures can help with the detection of land mines and elimination of false alarms. Figures 8a and 8b are the magnitude and phase of a ground patch where the land mine signature, which is located near the bottom right corner, is weak. The spatial locations marked by “M,” “FA1,” and “FA2” are where our land mine detection algorithms [Wang et al., 2003] identify as the land mine and false alarms, respectively. The magnitude and phase versus frequency for these three ground locations are plotted in Figure 9. We can see that all three have similar maximum values of magnitude, making the distinction between them difficult. However, there is a clear phase valley for the actual land mine near 230 Hz, hence distinguishing it from the false alarms. There is no such valley for FA1, and only a very shallow phase valley exists for FA2 near 210 Hz.
4. Analysis of Time Domain Data
 This time domain data set is acquired with the cart that has 12 LDVs. The data contain ground patches of approximately 0.8-m wide (cross-track) and 2-m long (along-track) in size. The loudspeakers are driven by the combination of 41 sinusoidal signals (80–280 Hz in 5 Hz step), each with a randomly selected phase. While the moving speed of the cart platform and data sampling configurations are adjustable, we present here only those data recorded with the following conditions: the platform moves at 1 cm/s, and every second a trigger signal tells the system to store 2000 raw data points from each LDV. We will call the 2000 raw data points a “spatial sampling unit.” All the data from a single LDV are called a “channel.” The time between data points is 0.3828 ms, corresponding to approximately 2600 samples per second.
 The ground surface velocity is obtained by calculating the Fourier transform of each spatial sampling unit. The data in each spatial sampling unit give rise to one complex value of surface velocity for each frequency. Let us call this value V(x, y, ω). Here x and y are the pixel coordinates in the horizontal (cross-track) and vertical (along-track) directions, respectively. There are 12 x values corresponding to the 12 LDVs. We can obtain the magnitude and phase of surface velocity at a given spatial location and frequency by
Here A(x, y, ω) and P(x, y, ω) are the magnitude and phase, respectively. G(x, y, ω) is the Fourier transform of the driving acoustic force as measured by a microphone under the loudspeakers and recorded in the same way as the LDV data. Figure 10 shows the magnitude and phase versus y and frequency of two channels of the same ground patch. One of the two channels passes over a mine and the other does not. The magnitude is always displayed in logarithmic scale from now on. Please note that, because the driving acoustic force is composed of 41 frequency components, we also only display results of the Fourier transform in steps of 5 Hz. The acoustic response of the same ground patch is also imaged in Figure 11, where each small rectangle is either the magnitude or phase at the given frequency. The spatial resolution of these images is 1 cm horizontally (along-track), which is the distance between consecutive spatial sampling units, and 6.7 cm vertically (cross-track), which is the distance between adjacent LDVs. Only the first 20 frequencies are included in Figure 11 as the land mine signatures, which are near the center in these images, are only visible in this frequency range.
 The flowchart in Figure 12 lists the steps of processing the time domain data. The background removal part of the magnitude data consists of subtracting the median magnitude value of a channel from all the magnitude values of that channel. This step is done separately for each channel because the 12 LDVs do not all have the same characteristics. The smoothing part of the magnitude data consists of first averaging over a 21 × 3 × 1 window (i.e., over the x and y coordinates) and then applying a 1 × 1 × 5 median filter (i.e., over frequency ω). The size of the spatial averaging window is selected to give approximately the same physical size in both along-track and cross-track directions.
 The purpose of the “align phase” step for the phase data is to minimize the overall phase difference between channels, probably due to the timing difference between the electronics of the 12 LDV. This step is implemented as follows. First, we select a channel with the minimum temporal phase variation; that is,
We represent channels in the equations with x because there is a one-to-one correspondence between them. Here x* represents the x that gives the smallest value within the bracket at the right-hand side of equation (25). Using minimum temporal phase variation helps prevent the selection of channels with high phase noise. Once a channel is selected, the phases of the data points in the other channels are adjusted so as to minimize the mean square phase difference with this selected channel:
In the actual implementation we adjust Δθ in increments of (1/8)π and select the value that gives the smallest mean square phase difference. After the “align phase” step, the phase data go through background removal and smoothing operations similar to those used for the magnitude data: subtraction of each channel's median phase from that channel's phase data, averaging over a 21 × 3 × 1 window (i.e., over the x and y coordinates), and then a 1 × 1 × 5 median filter (i.e., over frequency ω). In Figure 13 we illustrate the processing of the phase for all the channels at a single frequency (135 Hz in Figure 11). We can see that there appears to be an overall phase difference between, say, channels 5 and 6 in Figure 13a in the phase data before processing. These phase differences and phase wrap-around make the land mine signature here harder to see. Figure 13b is the result of the “align phase” step, and Figure 13c is the result after all the smoothing. In Figure 13c we can clearly see the land mine near the image center.
Figures 14–16 display three examples of processed magnitude and phase data. In Figure 14, there is a land mine near the center of the ground patch. This land mine is highly visible with the LDV, and we can see clearly a phase minimum near 125–145 Hz after the magnitude maximum near 95–105 Hz. Figure 15 is a ground patch that contains no land mine but contains a false alarm detected by the Planning Systems Inc.'s GPR near the center of the patch. Here we see no distinguishable sign of any mine-like object where the GPR false alarm is supposed to be in either the magnitude or the phase images. This is a good example of how acoustic land mine detection can be used with GPR to reduce false alarms and make land mine detection more robust [e.g., Gader et al., 2003]. Figure 16 is an example of a land mine that is more difficult to detect with the LDV system. The land mine is located near the center of the ground patch. In the magnitude images we can see only a very faint signature that spreads approximately from 100 to 150 Hz. However, this magnitude signature is so weak that, if we lower the detection threshold of magnitude in order to detect this land mine, we are likely to include many false alarms as well. In this case the clearly visible phase minimum near 220 Hz at the same spatial location as the weak magnitude signature can help us decide that there is actually a land mine there.
 In this paper we showed how the simple mass and spring models can be used to explain phase-based phenomena in acoustic-to-seismic coupled measurement systems that we had earlier observed heuristically in measured frequency domain data. With a similar mass and spring model of the ground, we also explained the feasibility of using measured background region phase in place of that of the actual driving force while extracting phase signatures for land mine detection. Additionally, we described for the first time the image processing algorithms utilized by our group in processing new time domain data collected on a cart containing 12 LDVs. From the models and processed images, we have shown that fusing phase information with that of magnitude holds great potential for making a robust land mine detection scheme for an acoustic-driven system.
 This work was supported by the U.S. Army CECOM in part under contract number DAAB15-00-00-C-1023. The authors would like to thank Kelly Sherbondy, Ian McMichael, and Erik Rosen for their support. They would also like to thank James Sabatier, Ning Xiang, and Richard Burgett of the University of Mississippi for their technical assistance and for making the data available to us.