A Plane Wave Equivalent Model for Forward Scatter Shadow Ratio in Spherical Wave and Its Application in Shadow Profile Retrieval

The forward scatter shadow ratio (FSSR), a novel parameter in forward scatter radar (FSR) systems, has been introduced recently for a plane wave. In contrast to conventional parameters like the forward scatter cross section (FSCS), FSSR utilizes the total electric field, providing a direct indicator of signal deviations from the direct path. This study overcomes the limitations of the current FSSR model by conducting a comprehensive mathematical and numerical analysis for a spherical wave with the Kirchhoff diffraction formula. By incorporating a point source and moving targets, this approach extends the application scope of FSSR. A plane wave equivalent model is derived and compared to the point source model, demonstrating its accuracy in various scenarios. The optimization method for target shadow profile retrieval in moving targets is discussed using the plane wave equivalent model. It is suggested that employing dual receivers can eliminate the ambiguity associated with target localization and enhance shadow profile retrieval precision.


I. INTRODUCTION
Recently, a new parameter to describe the target in the forward scatter radar (FSR) system with a plane wave as the signal source, namely the forward scatter shadow ratio (FSSR), was introduced [1], [2], [3].As a parameter relevant to target detection, size estimation, classification and shadow profile imaging, FSSR is defined as the ratio of the total received power density to the direct path signal (DPS) power density.Differing from the commonly used parameter of forward scatter cross section (FSCS) [4], [5], which is adopted from the concept of radar cross-section (RCS) in mono or traditional bistatic radar, FSSR uses the total electric field rather than the forward scatter radiation, and is a direct indicator of how far the total received signal deviates from the DPS.Unlike FSCS, using FSSR doesn't require the removal of the DPS, which is especially challenging in cases involving passive illuminators of opportunity (IOs) [6], [7], [8].
The associate editor coordinating the review of this manuscript and approving it for publication was Guido Lombardi .
We have proposed that FSSR can be used for target shadow profile retrieval through an optimization method.Represented by a finite number of rectangular strips, the target shadow profile can be retrieved using discrete observations of FSSR [1], [2].Compared to SISAR [9], this approach only discretizes the target using rectangular strips but without any further approximation.It is able to retrieve both the upper and lower limits of the strips (as opposed to heights only in SISAR).Furthermore, it is based on an optimization method and thus can easily accommodate FSSR samples from different locations.
There are two main limitations with the current investigations on FSSR and its application in target shadow profile retrieval.Firstly, the current definition of FSSR is given in a model where the incident signal is assumed a plane wave.The general scenario for a realistic FSR system where the DPS is from a point source is still to be investigated.Secondly, the current analyses of FSSR are based on the Fresnel diffraction formula [10], which is an approximation of the Rayleigh-Sommerfeld diffraction formula when the range of observation is large compared to the target size.An analysis with the Kirchhoff diffraction formula, in line with the existing studies of SISAR [11], [12], [13], is needed.
This paper extends the definition of FSSR to spherical wave, i.e., the transmitter is regarded as a point source.Furthermore, FSSR is analyzed using the Kirchhoff diffraction formula, in which the target is moving across the transmitterreceiver baseline.A comparison between the model using the Fresnel formula and that using the Kirchhoff formula is given, and a plane wave equivalent model for the point source model is derived.Let D 1 and D 2 be the distances from the target's projection on the baseline to the point source and to the receiver, respectively, the plane wave equivalent model can be summarized as follows: where R is the distance between the target and the receiver in an equivalent system with a plane wave as the signal source, under the condition that both the target's dimension and its distance from the baseline are significantly smaller than D 1 and D 2 .Using numerical analysis, it is shown that the plane wave equivalent model generates very marginal errors for both the far-field and the close to near-field scenarios.With the plane wave equivalent model, the existing conclusions using the Fresnel diffraction formula can be applied to FSR systems with a point source.
The optimization method for target shadow profile retrieval is discussed within the framework of the new model in which the target is moving across the baseline.The target shadow profile, represented by a finite number of rectangular strips, is retrieved using discrete observations of the FSSR.The plane wave equivalent model is applied to enable the utilization of the nonlinear least-squares algorithm [2].Different targets with various sizes and shapes are used to examine the shadow profile retrieval capability of the proposed method under both far-field and close to near-field scenarios.The numerical results indicate that using two receivers instead of one helps eliminate the ambiguity in locating the target and improves the accuracy of target shadow profile retrieval.
The rest of this paper is organized as follows.The FSSR model based on the Kirchhoff diffraction formula is described in Section II.In Section III, the plane wave equivalent model is derived and then analyzed numerically.The optimization method for target shadow profile retrieval with one and two receivers is described and then numerically analyzed in Section IV.Finally, Section V concludes this paper.

II. FSSR IN SPHERICAL WAVE
Consider a forward scatter scenario shown in Fig. 1, where a target is in close proximity to the baseline that connects the transmitter (TX) with the receiver (RX).The TX is a point source of a spherical wave.This spherical wave model is the same as that used in SISAR [9].Note that the model with a plane wave closely resembles the spherical wave propagation when the source is far away from the target [2].With the  distance from the source to the target taken into account, the spherical wave model offers a higher level of accuracy.Consider a Cartesian system with the transmitter at the origin and the baseline on the y-axis, as shown in Fig. 1.A second Cartesian system with target's center at its origin is also established, with axes x ′ , y ′ and z ′ in the same direction with axes x, y and z, respectively.For simplicity, the target is considered crossing the baseline perpendicularly and the point of intersection is at point K, which is D 1 from the transmitter and D 2 from the receiver.The FSSR (ε) of a target for specific locations of the transmitter and the receiver is the ratio of the total received power density (P tot ) to the power density (P inc ) of the direct path signal (DPS) measured at the receiver: in which E tot and E inc are the complex amplitudes of the total electric field and the direct path electric field at the receiver, respectively.Note that this definition is compatible with that in the plane wave introduced in [2] and [3].The total electric field is the outcome of the incident electric field interfering with the forward scattering field caused by the target.Using Babinet's principle and replacing the target with a complementary aperture that has the shape of the target shadow silhouette on the x ′ −z ′ plane, we can apply the Kirchhoff diffraction formula [10] and obtain the complex amplitude of the diffraction field E D when the target is located at (x, D 1 , z): where the imaginary unit is denoted by i, the wavelength of the signal is denoted by λ, and z ′ are all normalized with respect to the wavelength, so they are dimensionless.Some examples of the normalized value for different wavelengths can be found in Table 1.
The aperture function F x ′ , z ′ is given by where is the target shadow silhouette on the x ′ − z ′ plane, and The complex amplitude of the direct path electric field is .
Finally, the complex amplitude of the total electric field is given by Although both the expressions for the diffraction field in (3) and the direct path electric field in (7) have wavelength λ in them, from (2) we can see that λ will no longer exist in the ratio of FSSR.From ( 2) to ( 8), it can be concluded that FSSR is not only affected by the target shadow silhouette, but also the locations of the transmitter, the receiver and the target.Due to the forward scattering field from a target interfering with the DPS, FSSR deviates from one and changes in value while the target changes position with respect to the transmitter and the receiver.If a series of FSSR samples are measured at the receiver, they can potentially be used for target detection, classification, size estimation and shadow profile imaging.In Fig. 2, the numerically calculated FSSRs on a plane perpendicular to the baseline for two different scenarios are shown.In Fig. 2(a), the target is a 180 by 120 rectangle, which is on a plane 2 × 10 5 from the transmitter, and in Fig. 2(b), the target is a circle of diameter 100, and on a plane 3 × 10 5 from the transmitter.In both cases the transmitter and the receiver are 4 × 10 5 apart.

III. PLANE WAVE EQUIVALENT MODEL FOR FSSR IN SPHERICAL WAVE
In previous studies, the Fresnel diffraction formula [3] for a plane incident wave was used for the FSSR definition, where the FSR system in Fig. 3 was considered.The Cartesian system is defined where the receiver is on the y-axis and the incident wave is in the direction of the y-axis.The projection of the target's center on the y-axis is the origin.The Cartesian system of x ′ − y ′ − z ′ is defined the same way as that in Fig. 1.The diffraction field for the target at (x, 0, z) and the receiver at (0, R, 0) is given by  in which x, z, x ′ and z ′ are also all normalized with respect to the wavelength.The expression of the diffraction field in ( 9) is simpler than that in (3), and wavelength λ is eliminated in (9).Furthermore, from (2), ( 8) and ( 9), the FSSR is given by [2]:

A. THE PLANE WAVE EQUIVALENT MODEL
First of all, we approximate (3) as follows: under the condition that which can be interpreted as where L is the longest distance from any point described by F x ′ , z ′ to the baseline and D is the lesser of D 1 and D 2 .
In other words, compared with its distances to the transmitter and the receiver, both the target's dimension and its distance 134988 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
from the baseline are significantly smaller.From ( 8) and (11) we can obtain From ( 2) and ( 14), FSSR is given by Note that the condition in ( 13) can also lead to the second approximation: 16) We substitute R 1 and R 2 with ( 5) and ( 6), and make the third approximation as: in which the following condition needs to be satisfied Note that ( 19) is valid as a result of (13).Similarly, we also have Consequently, FSSR is given as follows: Let R defined as (1), (21) then becomes which is exactly the same as (10).The analysis above indicates that the FSSR model based on the Kirchhoff diffraction formula can be approximated as a plane wave equivalent model, which is based on the Fresnel diffraction formula.The equivalent model is premised on the assumption that both the target's dimension and its distance from the baseline are significantly less than its distances to the transmitter and the receiver.This assumption is considered valid for most FSR applications.A system with a point source can be regarded equivalently as a system with a plane wave, where the distance between the target and the receiver is given by ( 1).The plane wave equivalent model for the setups in Fig. 2 and the numerically calculated FSSRs using (22) are show in Fig. 4. Compared with Fig. 2, it can be seen that the FSSRs are essentially the same.

B. NUMERCIAL RESULTS OF PLANE WAVE EQUIVALENCE FOR TARGET CROSSING THE BASELINE
Consider the far-field parameter S [14]: in which H is the largest dimension (normalized with respect to the wavelength) of the target.In Fig. 5, we assume that the target is travelling perpendicularly crossing the baseline in the z direction and examine the FSSR calculated by the Kirchhoff diffraction given by ( 2) to ( 8) and the FSSR given by the equivalent model in (22).The distances of D 1 and D 2 are set at 3 × 10 5 and 10 5 , respectively, so R in the equivalent model is 7.5 × 10 4 according to (1).The target in Fig. 5(a) is a circle with a diameter of 100, so it is a far-field scenario with S = 0.2.Both the FSSR by the Kirchhoff diffraction (ε K ) and that given by the equivalent model (ε F ) are represented by the same black curve as they cannot be visually distinguished from each other.The difference between them (ε K − ε F ) is plotted with the red dashed curve.It can be seen that the difference between ε K and ε F is in the order of 10 −4 .In Fig. 5(b), the target is a circle with a diameter of 200 so it is a close to near-field scenario where S = 0.8.Compare to (a), the ranges of ε K and ε F are larger due to stronger shadowing effects, but the difference between them is still in the order of 10 −4 .Fig. 5(c) shows the results of an isosceles triangle target with base 100 (along the x direction) and height 150 (S = 0.45), and Fig. 5(d) is for a similar triangle with base 140 and height 210 (S = 0.88).It can be concluded that the plane wave equivalent model generates very marginal errors for both the far-field and the close to near-field scenarios.The errors are generally larger when the target is farther away from the base line, which can be explained by the condition in (13).

IV. SHADOW PROFILE RETRIEVAL USING PLANE WAVE EQUIVALENT MODEL A. OPTIMIZATION METHOD
Here we briefly describe the optimization method for the model illustrated in Fig. 3. Different from [1] and [2], FSSR here is a function of the x and z coordinates of the target, rather than the coordinates of the receiver.As in [1] and [2], we divide the target shadow into N strips with equal width and each individual strip p is regarded as a rectangle (Fig. 6  for illustration).From ( 22), we then define Considering all the strips, the integration in (24) can be approximated as where a ′ p and b ′ p are the upper and lower limits of strip p, respectively, and f a ′ p, b ′ p , x is given by and G p (z) is given by in which c ′ p and d ′ p respectively mark the z ′ coordinates of the right and left limits of strip p.Note that in this retrieval method, the location of the target is assumed known and the width of the strips are predefined, so c ′ p , d ′ p and z are considered known.In (26) and (27), erfi (x) is the imaginary error function [15]. Let and 134990 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Equations ( 25) and (26) then become and Equation ( 22) then becomes Suppose that there are M observations of the FSSR, each is denoted by ε q for the target at (x q , 0, z q ), q = 1, • • • , M , solving (a p, b p ) can then be formulated as a nonlinear leastsquare problem: where By solving (33), the optimal values for (a p, b p ) can be calculated and thus the shadow profile of the target is retrieved.Note that a p and b p are respectively the x coordinates of the upper and lower limits of strip p.Hence, it is convenient to graph the retrieved shape on the x − z plane.

B. RETRIEVAL WITH ONE RECEIVER
We use the forward scatter radar setup illustrated in Fig. 1 to numerically examine the retrieval of target shadow profile via the plane wave equivalent model.The distances of D 1 and D 2 are fixed at 3 × 10 5 and 10 5 , respectively, while targets of various sizes and shapes are tested to cover both the far-field and the close to near-field scenarios.Since for any 3D shapes only the shadow silhouette plays a part in the FSSR, in our analysis we assume that all targets are 2D and located on a plane (x ′ − z ′ plane in Figs. 1 and 3).The targets all move on the plane perpendicular to the y-axis and in the direction of the z-axis.The true FSSR is calculated using (2), ( 3) and ( 8).
The shadow profile retrieval is achieved by solving the optimization problem (34), where R is calculated using (1) from D 1 and D 2 (R is 7.5 × 10 4 in this case).We also assume that a sufficient number of FSSR samples are available without any errors.As shown in our previous work [3], the minimum value of the FSSR can provide a good estimation of the size of the target.For instance, a minimum value of 0.8 in the FSSR samples corresponds to a circular target with an approximate diameter of 100 for R = 7.5 × 10 4 .We quadruple this dimension and uses z = −200 to z = 200 on the x − z plane as the target retrieval range.The number of strips covering this range is then set to 40 (strip width is 10) so all c ′ p and d ′ p are known.The nonlinear least-square function ('lsqnonlin' in MATLAB) with the trust-region-reflective algorithm [16] is used to solve the optimization problem.Once the estimated limits (â p , bp ) for the strips are ready, we mark the location of each strip on the x−z plane only if âp > bp .The strip is considered nonexistent if âp < bp .
The first target we use is a circle with a diameter of 100 (Fig. 7(a)).It moves along the line of x = 0, so the projection of the z-axis on the x ′ − z ′ plane overlaps with the z ′ -axis (see Fig. 7(a)).The true values of the FSSR when the target is moving are shown by the black curve in Fig. 7(b).We take a finite number of samples of the FSSR to retrieve the shadow profile of the target.In this case, 90 samples (shown by the red circles in Fig. 7(b)) with a fixed increment in the z-axis are selected.
The final retrieved target shadow profile is shown in Fig. 7(c), which shows a high level of agreement with the true target shape.We also plot the sampled FSSR (ε), the  FSSR of the first receiver is denoted by the black curve with 80 samples marked by the black circles in Fig. 10(a).For the second receiver, the true FSSR is given by the red curve with 80 samples marked by the red crosses.The retrieval outcome is shown in Fig. 10(b).Compared to Fig. 8(c), it can be seen that the additional receiver not only corrects the position of the target but also improves the accuracy of the shadow profile retrieval.
Two more targets are used to verify the performance of the retrieval with two receivers.The first target is a rectangle with  12(c).From Figs. 11 and 12, we can see that the retrieved shadow profiles using the FSR system with two receivers display a high level of agreement with the true shadow profiles.

V. CONCLUSION
In this paper, we present an analysis of the forward scatter shadow ratio using a spherical wave model and the Kirchhoff diffraction formula.Aiming to overcome the limitations of previous studies, we build an FSR system model in which the transmitter is regarded as a point source and the target is moving across the transmitter-receiver baseline.A comparison between the new point source model and the existing model based on the Fresnel formula is given, and a plane wave equivalent model for the new model is derived.Using numerical analysis, it is shown that the plane wave equivalent model generates very marginal errors for both the far-field and the close to near-field scenarios.
The optimization method for target shadow profile retrieval is discussed within the framework of the new model.In this approach, the target shadow profile, represented as a finite set of rectangular strips, is retrieved using discrete samples of the FSSR.The plane wave equivalent model is applied to enable the utilization of the nonlinear least-squares algorithm to optimize both the upper and lower limits of the strips.Different targets with various sizes and shapes are used to evaluate the shadow profile retrieval capability of the proposed method under both far-field and close to near-field scenarios.It is suggested that incorporating two receivers, as opposed to just one, aids in resolving the ambiguity associated with target localization and enhances the precision of target shadow profile retrieval.

FIGURE 1 .
FIGURE 1.An illustration of an idealized forward scatter radar system.

FIGURE 2 .
FIGURE 2. The numerically calculated FSSRs for a plane perpendicular to the baseline for: (a) a 180 by 120 rectangular target on a plane 200,000 from the transmitter, (b) a circular target of diameter 100 on a plane 300,000 from the transmitter.The receiver is 400,000 from the transmitter for both cases.

FIGURE 3 .
FIGURE 3.An illustration of the forward scatter model where the Fresnel diffraction formula is used.

FIGURE 4 .
FIGURE 4. The plane wave equivalent models for the FSR systems in Fig.2and the numerically calculated FSSRs for a plane perpendicular to the direction of propagation.

FIGURE 5 .
FIGURE 5.The difference between the FSSR calculated in the point source model by the Kirchhoff diffraction formula and that in the plane wave equivalent model by the Fresnel diffraction formula.(a) The target is a circle with a diameter of 100.(b) A circle with a diameter of 200.(c) An isosceles triangle with base 100 and height 150.(d) An isosceles triangle with base 140 and height 210.

FIGURE 6 .
FIGURE 6.The target shadow is divided into strips, one of which is by the shaded area.Each strip is regarded as a rectangle.

FIGURE 7 .
FIGURE 7. (a) Dimensions and location of the circular target.(b) The true FSSR is shown by the black curve, and the 90 FSSR samples when the target is moving in the z direction are marked by the red circles.(c) The retrieved shadow profile through the plane wave equivalent model.(d) The sampled FSSR (black circles), the recalculated FSSR (black crosses) using the retrieved shape and their difference (red squares).

FIGURE 10 .
FIGURE 10.(a) The true FSSRs for the first and second receivers are shown by the black and red curves, respectively.The samples used for retrieval are marked by the black circles (the first receiver) and the red crosses (the second receiver).(b) The retrieved shadow profile using two receivers.

FIGURE 11 .
FIGURE 11.(a) Dimensions and location of the rectangular target.(b) The true FSSRs for the first and second receivers are shown by the black and red curves, respectively.The samples used for retrieval are marked by the black circles (the first receiver) and the red crosses (the second receiver).(c) The retrieved shadow profile using two receivers.

FIGURE 12 .
FIGURE 12. (a) Dimensions and location of the irregular target.(b) The true FSSRs for the first and second receivers are shown by the black and red curves, respectively.The samples used for retrieval are marked by the black circles (the first receiver) and the red crosses (the second receiver).(c) The retrieved shadow profile using two receivers.

TABLE 1 .
Normalized values of distances and dimensions.