Precursors for synthetic aperture radar

In this paper we study waveform design for synthetic-aperture radar (SAR) imaging through dispersive media. Under the assumptions of scalar wave propagation through a causal dielectric medium and single-scattering from an isotropic point scatterer, we use asymptotic analysis to derive the asymptotic approximation to the scattered electric field. From this asymptotic approximation, we define a sensing precursor that we propose for the transmit waveform for SAR imaging through dispersive material. We compare our sensing precursor with previously defined optimal waveforms (Varslot et al 2011 SIAM J. Appl. Math. 71 1780–800) in terms of both propagation and scattering capabilities, as well as imaging performance. With the filtered back-projection imaging algorithm that we use, we find that for high levels of signal-to-noise ratio (SNR), the optimal waveforms contain higher frequencies and thus produce better images. For low levels of SNR, the transmitted optimal waveforms and the sensing precursors are similar, thus giving comparable images.


Introduction
When an electromagnetic pulse travels through a dispersive material, each frequency of the transmitted pulse changes in both amplitude and phase, and each frequency at its own rate. * Author to whom any correspondence should be addressed.
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As a consequence, broadband pulses propagating in dispersive material experience significant amplitude distortion and changes in pulse velocity. Asymptotic analysis of the exact integral representation of the propagated field, which utilizes the full causal dispersion relation of the dispersive material, provides a complete far-field description of the propagated pulse. In 1989, Oughstun and Sherman [22] utilized asymptotic methods to show that in a dielectric (nonconducting) material the low-frequency component of the propagated field, the so-called Brillouin precursor, has a peak amplitude that decays algebraically with propagation distance (as the inverse square root of propagation distance), whereas other pulse components decay exponentially. More recently, Oughstun has shown that the peak amplitude of the propagated field actually decays as a frequency-chirped Lambert-Beer's law [20]. In 2005, Oughstun [19] proposed a Brillouin pulse as the pulse that experiences the least attenuation due to material absorption. It has been hypothesized that the slow decay rate of the Brillouin pulse can be used to advantage in radar detection and imaging applications [12,19].
The question of whether or not the Brillouin precursor is the ideal waveform for synthetic aperture radar (SAR) imaging through dispersive material was partially addressed by Varslot, Morales, and Cheney in a pair of papers appearing in 2010 and 2011 [26,27]. These authors used a filtered back-projection algorithm to derive an optimal filter and its associated optimal waveform. The optimization is based upon minimizing the mean square error (MSE) of the L 2norm between the ideal image and the reconstructed image. The optimal waveform is numerically derived for each noise level by solving a fixed-point equation for the absolute value of the optimal pulse spectrum and then applying a minimum-phase algorithm to obtain the optimal waveform. The authors conclude that the optimal minimum-phase waveforms have a 'transmit spectrum that is concentrated around the frequencies which are conducive to the generation of precursors' [27], but no conclusive statement could be made.
In this paper, we provide an asymptotic expansion of the electric field component of the impulse response due to scattering by an isotropic point source in a frequency-dependent dispersive (and lossy) material. We subsequently define a sensing precursor based on this asymptotic expansion. The sensing precursor is analogous to Oughstun's Brillouin pulse [19]. We compare the propagation and imaging performance of this sensing precursor to that of the optimal waveform derived by Varslot et al [27]. Our results show that, in general, the optimal waveform of Varslot et al [27] is not a precursor waveform. For high levels of signal-to-noise ratio (SNR), the optimal waveform is comprised of frequencies higher than those that comprise the sensing precursor. For low levels of SNR, the optimal waveform is similar to that of a sensing precursor.
In section 2 of this paper, we provide a brief history and derivation of the Brillouin precursor and Brillouin pulse in dispersive material. In section 3, we formulate the imaging problem and give a description of the optimal minimum-phase results of [26,27]. Section 4 derives the sensing precursor for an isotropic point scatterer. In section 5, numerical studies provide comparisons between the propagation characteristics and imaging performance of the two waveforms. In addition, we provide a discussion of resolution for imaging in a dispersive media and compare the two waveforms in that respect. Finally, conclusions are give in section 6.

Wave propagation in dispersive media
Consider a homogeneous, isotropic, locally linear, dispersive material with relative dielectric permittivity ϵ R (ω) that occupies all of space. Let E(0, t) denote the temporal behavior of a linearly-polarized electromagnetic pulse on the plane z = 0 that is propagating in the positive z direction. The electric field component on any plane z > 0 is given exactly by the integral representation E(z, t) = 1 2πˆCẼ (0, ω)e ik(ω)z−iωt dω, (2.1) where C is a Bromwich contour in the upper half plane whose real part is greater than the abscissa of absolute convergence for the input field E(0, t),Ẽ(0, ω) is the Fourier transform of E(0, t), and k(ω) = (ω/c) ϵ R (ω) is the complex wave number [21]. Here, c denotes the speed of light in vacuum. Asymptotic analysis of the integral representation of the propagated electric field component (2.1) for a step-modulated sinusoid began in 1914 with Sommerfeld [23] and Brillouin [5,6] who used the then recently-developed method of steepest descent. Their results were later improved upon by Oughstun and Sherman [22] who used better approximations of the saddle point locations and uniform asymptotic methods [10,15,25]. The complete uniform asymptotic expansion of the propagated step-modulated sinusoid was given by Cartwright and Oughstun [7]. A significant result that appeared from the work of Oughstun and Sherman [22] is the fact that the peak amplitude point of the Brillouin precursor decays algebraically with propagation distance, as z −1/2 , whereas the remainder of the field decays exponentially. In 2015, Oughstun showed that the peak amplitude of the Brillouin precursor actually follows a frequency-dependent Lambert-Beer law [20].
In order to provide a basic understanding of these results and to provide context for our new results, we briefly review the asymptotic analysis of the integral representation equation (2.1) for a step-modulated sinusoid E(0, t) = sin(ω c t) of fixed carrier frequency ω R traveling through a Debye-type dielectric with relative dielectric permittivity Here, ϵ ∞ is the high frequency limit of the permittivity, ϵ s is the static, low-frequency permittivity, and τ is the relaxation time of the polarized molecules. We choose to concentrate our work on a Debye-type dielectric as this model was used by Fung and Ulaby [14] to describe the permittivity of foliage and it was the dispersive model used in the calculations by Varslot et al [26,27]. The Debye model is also a common model of the permittivity of dry sand [13]. Nevertheless, the analysis presented here may be extended to other causal models of material permittivity such as Lorentz [7], Rocard-Powles [19], and Drude [8] models. Figure 1 shows the real and imaginary parts of the index of refraction n(ω) = n r (ω) + i n i (ω) = ϵ R (ω) for a Debye-type dielectric with material parameters ϵ ∞ = 1.45, ϵ s ≈ 2.43, and τ = 8 ns. We will continue to use these parameters throughout this paper, as these values depict the dense vegetation model given by Fung and Ulaby [14]. We define the absorption band of the material to be those frequencies at which n i (ω) > 0.1. The lower and upper bounds of the absorption band are denoted by ω L and ω U , respectively, in figure 1. Throughout this paper we will relate propagation distances to an absorption depth given by z = c/(ωn i (ω)) ≈ 13.7 m when ω = 1/τ , at which the signal amplitude has decreased by a factor of 1/e. The value ω = 1/τ is also denoted in figure 1.
With this model of permittivity, an initial pulse given by a step-modulated sinusoid E(0, t) = sin(ω c t) whose spectrum isẼ(0, ω) , we then define a dimensionless frequency ωτ = Ω and a large parameter for a Debye-type dielectric with material parameters ϵ∞ = 1.45, ϵs ≈ 2.43, and τ = 8 ns. The lower and upper edges of the absorption band are denoted by ω L and ω U , respectively. Absorption depths in this paper are defined at the value ω = 1/τ . so that equation (2.1) becomes (see [21] for a thorough treatment) The complex phase function ϕ is defined as with θ = ct/z a space-time variable (as originally done by Brillouin [5,6]). The complex phase function (equation (2.5)) has two branch points located at in the complex Ω-plane. A branch cut is created between these two branch points and analysis is restricted to the branch upon which the real part of ϕ is negative, as appropriate for lossy material. For values of θ < √ ϵ ∞ , convergence of the integral requires enclosure of the contour in upper half plane with the result that For values of θ > √ ϵ ∞ , convergence of the integral requires the contour to be enclosed in the lower half plane, where the branch cut lies. Rather than perform integration around the branch cut, saddle point methods are used to provide an asymptotic description of the field for large λ (i.e. for large propagation distances z). The complex phase function for the Debye model has one relevant (accessible) first-order saddle point Ω sp (θ) that moves in the complex Ω-plane as θ increases from √ ϵ ∞ . The saddle point Ω sp (θ) is initially located along the positive imaginary axis when θ = √ ϵ ∞ , moves downward along the imaginary axis with increasing θ, crosses the origin when θ = θ 0 , and continues toward the branch point at Ω = −i as θ → ∞. Expansion of the complex phase function about this first-order saddle point Ω sp (θ) gives, to first order, the low-frequency contribution to equation (2.1) due to the saddle point Ω sp (θ): as λ → ∞. Figure 2 is a plot of the Brillouin precursor equation (2.8) at a distance of z ≈ 54 m (roughly four absorption depths) into a Debye-type material with the same material parameters used for figure 1, which are the values Fung and Ulaby [14] use for dense vegetation with fractional leaf volume v l = 0.10 and fractional water volume v w = 0.20. The integrand of (2.4) has a simple pole located at Ω = Ω c . As the saddle point Ω sp (θ) moves down the imaginary axis, there will be a space-time point at which the deformed contour will encounter the simple pole, which results in the signal contribution [7]. For simplicity, we omit the analysis of the signal contribution; the interested reader is referred to [7]. A rectangular-modulated sinusoid is the difference between two step-modulated sinusoids of the same carrier frequency delayed in time. Consequently, its asymptotic approximation has the additional feature of a trailing-edge Brillouin precursor that is delayed in time and inverted. A picture of the complete field-the sum of the Brillouin precursor and the signal contribution-for a rectangular-modulated sinusoid of ten oscillations at a carrier frequency of ω c = 1 × 10 8 rad s −1 is given in figure 3.
The algebraic decay of the peak amplitude point of the Brillouin precursor follows directly from the asymptotic approximation (2.8): at the space-time point θ 0 = √ ϵ s at which the saddle point Ω sp (θ 0 ) crosses the origin, ϕ(Ω sp (θ 0 ); θ 0 ) = 0. Substitution of this value into equation (2.8) yields Historically, research on pulse propagation in dispersive material using asymptotic methods has focused on ultra-wideband pulses. If the spectrum of the input pulse contains a significant amount of frequencies below, within, and above the absorption band of the material, the amplitude of the Brillouin precursor (low-frequency contribution) will dominate the field only after a certain propagation distance, after the other frequency components have been sufficiently attenuated. This has led some to believe that a significant amount of energy loss necessarily precedes the appearance of the Brillouin precursor [2]. However, in 2005, Oughstun [19] defined a Brillouin pulse for a Rocard-Powles-Debye dielectric as where θ T = θ − cT/z d , T is the time delay between the leading and trailing-edge Brillouin precursors, and z d = [Im k(ω c )] −1 is one absorption depth at the carrier frequency ω c . This Brillouin pulse is essentially the Brillouin precursor of the propagated field of a square-modulated sinusoid and, as such, loses less energy than the square-modulated sinusoid upon propagation and achieves 'near optimal material penetration' [19,20]. It is this pulse that is hypothesized to be advantageous for radar detection and imaging through dispersive material [12,19,20].

SAR imaging through dispersive media
In 2010 and 2011, Varslot et al [26,27] adapted the statistical framework and filtered backprojection imaging algorithm developed in [17,18,28,29] for traditional SAR imaging to SAR imaging through dispersive material. They derive an optimal filter and optimal waveform based on minimizing the MSE between the ideal image and the reconstructed image. They conclude that for high noise levels, the waveform 'looks like a few oscillations of a sinusoid' and has a band-limited spectrum concentrated around frequencies that are 'conducive to the generation of precursors' [27]. However, they were unable to determine if their optimal waveform was equivalent to a precursor. As it is one purpose of this paper to determine if the optimal pulse of [27] is the Brillouin pulse given in [19], we now formulate the imaging problem and summarize the results given in [26,27].
We adopt the problem formulation of [26,27] and assume that the sensor and scatterers are embedded in a homogeneous, isotropic, locally linear dispersive material with a known dielectric permittivity. We choose to retain this (physically impossible) formulation so that we may compare results with those in [27] and delay the more difficult problem involving a half space until this problem is well understood. Here, we summarize the assumptions, formulation, and results of [27] closely following their exposition; the reader is referred to both [26,27] for a complete account.
In a homogeneous, isotropic, locally linear material in which the magnetic permeability µ 0 is that of free space, each component of the electric field satisfies whereε is the (time domain) dielectric permittivity and j s is the source term. Writing both the total electric field component and the total source term as a sum of incident and scattered contributions, E tot = E in + E sc and j s = j in + j sc , the scalar wave equation separates into where j in models the transmitting antenna and j sc models the scattering object's response to the incident field. The object's response will depend on geometric and material properties of the object, which is represented by the reflectivity function v(x, t), used with the assumption of stationary targets comprised of linear materials. For an isotropic point-like antenna moving along the path γ(s) parameterized by slow time s ∈ (s min , s max ), the incident source is modeled by . With this incident source, the solution for the incident field equation (3.2) may be given in terms of the Green's function as where P(ω) is the spectrum of the transmit pulse p(t).
It is assumed that scattering occurs at a known surface (3.5) and the target dispersion is known. Therefore the target reflectivity function may be written as With use of the Born (single-scattering) approximation it then follows from equation (3.3) that each electric field component of the scattered field satisfies If the same antenna is used for both transmit and receive, and if the antenna position remains fixed from transmit to receive (the start-stop approximation), the solution of the scattered field may be written in terms of the Green's function as Here,T is a modified target that accounts for both the target reflectivity and non-planar surface area, y = (y 1 , y 2 ) is a two dimensional vector position, and |r s,y | = |Ψ(y) − γ(s)| is the distance between the antenna and the target.
It is assumed that the received signal is corrupted by additive white noise η(s, t). The image is formed by applying a filter Q to the noisy data and backprojecting using only the real part of the refractive index in order to account for phase delay, where the function A is defined as In order to determine the optimal filter a Stolt change of variables is performed in equation (3.9) where The actual image in equation (3.9) is then compared with the ideal image where defines the data collection manifold. This image is considered ideal because if the data collection manifold was the entire R 2 plane then I Ωz (z) = T(z) exactly. In reality we are limited by our bandwidth and the flight path. The filter Q in equation (3.9) is then chosen to minimize the error between the ideal image in equation (3.12) and the actual image in equation (3.9). The error between these two images is the MSE, i.e. the expected value of the L 2 -norm of the point-wise error between the reconstruction I(z) and the ideal image I Ωz (z). The expected value is included since the error is a stochastic quantity due to the random noise η(s, t). In addition, it is assumed that each target is a realization of a random field, therefore the filter Q is designed to perform well for a family of candidate reflectivities. In particular, it is assumed that T is wide-sense stationary with zero-mean and known covariance function. The noise is also assumed to be a zero-mean second-order random field. T and η are assumed to be statistically independent. The noise η is assumed to be stationary in t and statistically uncorrelated in s. We note that the MSE is known to be made up of the variance of the point-wise error as well as the square of the L 2 norm of the bias. In this case, where it is assumed that the target has zero mean, and since the model and reconstruction algorithm are both linear, the bias term is equal to zero. Therefore minimizing the MSE is equivalent to minimizing the variance. An optimal filter Q opt is derived via this minimization using a variational derivative of the variance of the point-wise error between the ideal and reconstructed image with the result .
(3.14) where σ ηT (ω, s) = S η (ω, s)/S T (ω, s, z) is the noise-to-target ratio (with S η and S T the spectral density functions of the noise and target random fields, respectively) and J is the Beylkin determinant [1] from the Stolt change of variables (3.11).
With this optimal filter, the second paper of Varslot et al [27] uses the method of Lagrange multipliers in order to minimize the MSE of the reconstruction error subject to the constraint that the total transmit energy along the flight path is fixed. This leads to the fixed-point equation to solve for the magnitude of the spectrum |P| 2 = |P(ω)| 2 of the optimal pulse. Here, S T = S T (ω, s, z) is the target spectral density function. A Hilbert transform method is then used to find a minimum-phase waveform corresponding to |P(ω)|; this is the optimal waveform p opt (t).
It is mentioned in section 5 of [27], that a Gaussian filter is applied to the magnitude of the spectrum so that it is nonzero along the imaginary axis. We should note here that in order to obtain results similar to those in [27], we found |P| required considerable processing, similar to what we found in [24]: 'the coefficients of low frequency components are reduced to a negligible size, all spectral components less than −50 dB below the maximum value of |P| are set to a minimum threshold value, and then a Gaussian filter is applied to the this modified spectrum.' The parameters of the Gaussian filter are dependent upon the carrier frequency ω c and the dispersive properties of the material; the parameters are chosen such that the main lobe of P(ω, s) remains (relatively) intact and side lobes are (relatively) suppressed. As an example of such filtering, consider the spectrum |P(ω, s)| resulting from the fixed-point equation for the Debye model dielectric with parameters for dense foliage and SNR = −40 dB. Frequencies below 29 MHz and frequencies whose amplitude is more than 50 dB below the maximum of |P(ω, s)| are set to a minimum nonzero value before a Gaussian filter is applied. The absolute value of the spectrum resulting from the fixed-point algorithm |P| (solid red), and the absolute value of the spectrum after filtering the result of the fixed-point algorithm |P filter | (dashed blue), are shown in figure 4.

Sensing precursor
We now return our attention to the integral representation of the scattered field in order to derive an expression for a sensing precursor. With the assumption that the target T is an isotropic point scatter independent of frequency and located at position y, we consider the ω integration in equation (3.8) and define This scattering integral has the same form as the integral for the propagating field given in equation (2.1) and may be evaluated using asymptotic methods. As in section 2, we consider a Debye-type dielectric with permittivity given in equation (2.2), a dimensionless frequency Ω = ωτ , and an impulse response with P(ω) = 1. We express the scattering integral as where the complex phase function ϕ is that given in equation (2.5) and the space-time parameter θ is defined as Here, the factor of 2 accounts for travel to and from the scatterer. An application of Jordan's lemma shows that As in the propagation integral (2.4), there is one accessible saddle point Ω sp (θ) located along the positive imaginary axis when θ = √ ϵ ∞ , which then moves down the imaginary axis with increasing θ, crosses the origin when θ = θ 0 , and approaches the branch cut at −i as θ → ∞. However, unlike in integral (2.4), the scattering integral (4.2) has a factor of Ω 2 in the amplitude function. Thus, the stationary point Ω sp (θ 0 ) = 0 coincides with an amplitude critical point Ω 2 P(Ω/τ ) = 0 when θ = θ 0 . In this case, a uniform asymptotic expansion is needed in order to provide an asymptotic approximation that is uniformly valid for all values of θ > √ ϵ ∞ .
We follow the method of Bleistein [3] and introduce a change of variable appropriate for an isolated first-order saddle point In order for to remain finite, we require v = −α when Ω = Ω sp . We fix the origin by setting v = 0 when Ω = 0 so that β = ϕ(0; θ) = 0. Under this change of variable, the amplitude function is written as where H 0 (v, θ) has removable singularities at v = 0 and v = −α. Substitution of equations (4.6) and (4.8) into the integral representation of the scattered field equation (4.2), along with a deformation of the contour through the valleys of the saddle point ω sp (θ), yields Evaluation of this integral gives the first-order, low-frequency contribution to the scattered field When θ = θ 0 , the saddle point Ω sp (θ 0 ) and the complex phase function ϕ(Ω sp (θ 0 ); θ 0 ) are both zero so that equation (4.10) reduces to That is, the peak amplitude of the low-frequency contribution to the scattered field decays as |r s,y | −3/2 , as |r s,y | → ∞. This decay rate agrees with the results of Bleszynski et al [4], which were derived by approximating the slope of the spectrum and the transfer function in a sparse discrete media. We emphasize here that the difference in decay rate of the peak amplitude of the propagated pulse equation (2.9) and that of the scattered pulse equation (4.12) is due to the factor of Ω 2 that appears in the spectrum of the scattered field (equation (4.2)) but not the spectrum of the propagated field (equation (2.4)). The variations in the factor of Ω 2 are significant for wideband pulses, such as those presented in this work. We now define a sensing precursor pulse following the idea of Oughstun's Brillouin pulse [19] given in equation (2.10). We align the pulse so that θ = √ ϵ ∞ corresponds to time (4.13) The value 2d > 0 is the distance at which the low-frequency component of the scattered field is evaluated. Figure 5 shows sensing precursor waveforms (equation (4.13)) for values d = 10, 25, 50, 75 and 100 m (left) and their corresponding spectra (right) for a Debye-type dielectric with material parameters ϵ ∞ = 1.45, ϵ s ≈ 2.43, and τ = 8 ns. These distances correspond to absorption depths (determined when ω = 1/τ ) of approximately 0.7, 1.8, 3.6, 5.4 and 7.2, respectively. We note that the bandwidth of each waveform decreases as d increases due to the absorption of higher frequencies. Here, the energy of each waveform has been normalized to a fixed value, that given by a rectangular-modulated sinusoid of eight cycles at a carrier frequency of 0.1 GHz (see section 5 below). In our numerical simulations we use a sampling frequency of 10 GHz and 2 15 sample points.

Numerical simulations
We now consider numerical comparisons of three types waveforms for SAR application in a Debye-type material: a rectangular-modulated sinusoid, the optimal waveforms of Varslot et al [27], and the sensing precursor. Again, the material parameters are ϵ ∞ = 1.45, ϵ s = 2.43, and τ = 8 ns, appropriate for dense foliage [14]. In these comparisons, the energy of all initial (transmit) waveforms are normalized to that of a rectangular-modulated sinusoid comprised of eight cycles at a carrier frequency of 0.1 GHz (chosen so that its initial frequency content overlaps the region of anomalous dispersion). In our numerical calculations, we use a sampling frequency of 10 GHz and 2 15 sample points.
In order to simulate data with noise we first set our desired SNR value. Note we consider SNR levels between −40 dB and 40 dB to showcase both high and low levels of noise. SNR is commonly given by an expression of the form SNR = 20 log 10 A signal where A signal is some measure of the signal amplitude and E|η| 2 is the expected value of the random noise. We follow [26] and use the maximum value of the scattered rectangular-modulated pulse (equation (3.8)) as our 'signal' and use the fact that E|η| 2 = S η in the case of white Gaussian noise (which has a constant spectral density equal to the variance of the noise). Thus, for a desired value of SNR, S η is found to be the following constant value To add noise to our data we add S η times a complex Gaussian vector with zero mean and unit complex variance to our scattered field data. We also note that the optimal filter and waveforms of Varslot et al [27] are given in terms of the noise-to-target ratio σ ηT from equations (3.14) and (3.15). In [26] it is shown that σ ηT is directly related to the SNR, in particular, Therefore when calculating the optimal filter and optimal waveforms of Varslot et al [27] we use the above relationship (5.4) and the desired SNR level to calculate σ ηT . We follow the work of [27] and reference the SNR in order to distinguish the various optimal waveforms but remind the reader that the SNR value is used to calculate the optimal waveform of Varslot et al only; SNR is not used in the determination of the rectangular-modulated sine wave or the sensing precursor. Also note that in section 5.2 we consider received waveforms after scattering but before noise has been added; we do not explicitly include noise in the data until we consider target reconstruction in section 5.3.

Transmit
First, we compare transmit waveforms. Because we adopt the physical parameters from Varslot et al of a circular flight path with radius 100 m at an altitude of 10 m (as discussed in section 5.2), we choose a sensing precursor with d = 100 m (approximately seven absorption depths when ω = 1/τ ) as the waveform with which to compare. The reasoning here is that the sensing precursor with d = 100 m will consist of those frequencies that remain in the scattered field; any higher frequencies in the transmit waveform will be attenuated after propagation and scattering and thus, not contribute to imaging. The left of figure 6 shows the transmit waveforms for an eight-cycle rectangular-modulated sinusoid of carrier frequency 0.1 GHz (dash-dotted black), optimal waveforms of Varslot et al one for SNR = +40 dB (solid blue) and one for SNR = −40 dB (dashed blue), and the sensing precursor (equation (4.13)) with d = 100 m (solid red). The corresponding spectra of these transmit waveforms are shown on the right in figure 6. The sensing precursor with d = 100 m has a spectrum concentrated below the absorption band of the material in anticipation that those frequencies in the absorption band will be highly attenuated during propagation, and thus not significant for imaging through the lossy material. The spectrum of the optimal waveform with SNR = −40 has a similar spectrum to that of the sensing precursor. In contrast, the optimal waveform with SNR = 40 dB consists of a significant amount of frequency between 0.05 GHz and 0.4 GHz.
We are now in a position to address the question of whether or not the optimal pulse of Varslot et al is the Brillouin pulse of Oughstun-in general, it is not. The Brillouin pulse, equation (2.10), is the difference between two exponentials with the same peak amplitude and width (determined by the material parameters) but separated in time. The optimal pulses cannot be formed by the difference of two such Gaussian pulses.
Neither is the optimal pulse equivalent to one sensing precursor with a fixed value of d (4.13). For scattering in dispersive material, the sensing precursor is the analog of the Brillouin pulse in dispersive wave propagation problems. Both the Brillouin pulse and the sensing precursor are the low frequency response of the propagated or scattered field, respectively, obtained from the uniform asymptotic expansion of the integral representation of the propagated or scattered field. The spectra of these pulses consist mainly of those frequencies that lie below the absorption band of the material. In contrast, the optimal pulse is the result of using the optimal filter to balance the effects of dispersion and noise. With low noise levels, the optimal pulse spectrum spans a wide range of frequencies, below and within the absorption band of the material, as evident in figure 6. With higher levels of noise, the optimal pulse spectrum is concentrated in the region below the absorption band of the material and the waveform begins to look like the sensing precursor, again evident in figure 6. We do note that there may be values of d for which the sensing precursor is similar to an optimal pulse of Varslot et al for a given value of SNR. This idea will be addressed further in section 5.3.

Receive
Next, we compare the scattered waveforms and their spectra after propagation through the dispersive material, scattering off a point scatterer, and propagating back to the receiver. We consider the same flight path given in [26,27]: a (full) circular path centered at (0, 0) with radius 100 m at an altitude of 10 m, all within the dispersive material. The target scene consists of two isotropic point scatterers located at (0, −3, 0) and (0, 3, 0), also within the dispersive material. Recall that in [26] there is an assumption that the target is represented by a random field and hence knowledge of the target spectral density S T is needed in the calculation of the optimal filter equation (3.14). If any a priori information about the target(s) is known, such as location, scattering strength, or spatial correlation between different scatterers, this could be incorporated into the assumed S T . For our purposes, as was done in [26], we assume a white target spectrum, i.e. S T is constant.
The  very similar to the sensing precursor. The scattered rectangular-modulated sinusoid is comprised of leading-and trailing-edge sensing precursors [compare to figure 3]. The amplitudes of the sensing precursor and the optimal pulse with SNR = −40 dB dominates the other two waveforms because of their low frequency content, although all amplitudes are small (on the order of 10 −11 ) due to the material absorption. The large attenuation of frequencies near and above 0.1 GHz has caused the spectra of the optimal waveform with SNR = 40 dB and the rectangular-modulated sinusoid to lose a significant amount of its spectral content above 0.1 GHz [see figure 6 for comparison]. However, both the optimal waveform for SNR = 40 and the rectangular-modulated sinusoid retain some frequency content above 0.1 GHz and therefore can be considered as having a larger bandwidth than the sensing precursor.

Target reconstruction
We introduce noise before comparing reconstructions of the target scene. To model receiver noise, white Gaussian noise with constant spectral density equation (5.3) is added to the scattered field at each antenna position. The noisy received spectra is multiplied by the optimal filter Q(ω, s, z), equation (3.14), and backprojected using the real part of the refractive index, as given in equation (3.9).
Reconstructions of the target scene using the three transmit waveforms, the rectangularmodulated sinusoid, the optimal waveform of Varslot et al and the sensing precursor with d = 100 m, are given in columns one, two, and three of figure 8, respectively. The noise level decreases from top to bottom with SNR = 40, 0, −40. We observe that the optimal waveform provides the best reconstruction for SNR = 40 and 0 and that the rectangular sinusoid also provides better reconstructions than the sensing precursor with d = 100 when SNR = 40 and 0. For SNR = −40, all three waveforms provide poor images. These results are not surprising given the initial bandwidths of the transmitted fields and the optimal filter used to make these reconstructions. It is assumed that we have a closed-form expression for the imaginary part of the index of refraction n i (ω) that accounts for the attenuation of the signal. The optimal filter equation (3.14) attempts to reverse this attenuation, thereby restoring much of the initial spectrum which is then used in the backprojection, although noise levels affect how well this works. Thus, although the large amplitudes of the sensing precursor in figure 7 are enticing, one must remember that the filtered back-projection algorithm used here does not solely utilize the received fields; it first applies the filter Q opt which tries to overcome the effects of both attenuation and noise. We discuss this in detail in section 5.4.
Given that target resolution will not solely depend upon the received field but also the filter Q opt , it is an interesting question as to whether or not there exists a value of d that optimizes the image resolution of the sensing precursor for a given filter Q. It is a question that is outside the scope of this work but one we plan to address in the future. Our purpose here is to compare the image reconstructions of the optimal pulse of Varslot et al with those from a sensing precursor, an analytic waveform based on the asymptotic solution to the scattered field. However, as an illustration of the dependence of target reconstruction on the value of d, we now compare reconstructions of the target scene using five sensing precursors with d = 10, 25, 50, 75, 100 m, shown in rows one through five of figure 9, respectively. The noise level increases from left to right with SNR = 40, 0, −40. We observe that all sensing precursor waveforms provide relatively poor target reconstructions for SNR = −40 dB. The sensing precursors with d = 50, 75, 100 are not differentiating the two scatterers at the SNR = 0 dB level. The reconstructions for SNR = 40 dB are better for sensing precursors with lower values of d. Again, these results are expected given the role of Q opt and the larger bandwidth of the transmit pulses for small d (see the right figure in figure 5).

Resolution and noise tradeoff
The images here and in [26] have rather poor resolution. This is surprising given that we assume prior knowledge of the index of refraction n(ω) and are thus able to presumably filter out the Here, we clearly see (with equation (3.10)) the imaginary portion of the refractive index n i (ω) is entirely filtered out after Q is multiplied by the received data. In this case, following the resolution analysis in [9] but assuming n i (ω) ≈ 0, we find that the cross-range resolution is approximately the same as in the non-dispersive case whereas the range resolution is A derivation of this is given in appendix. Thus, the range resolution in dispersive material differs from the non-dispersive case by the n r (ω) in the denominator, which may improve or worsen resolution depending on its value. We will confirm this result by plotting the image of a single target (which is equivalent to the point-spread function (PSF) when the data is noiseless) in the noiseless data case (equation (3.9) with η = 0). For simplicity, we plot results using only the sinusoidal waveform; we see similar results for the precursor and optimal waveforms. We use the same flight path, frequencies, and waveforms as used above but now assume a single point target is located at (−0.3889, 3.1111, 0). Note this location is directly on our imaging grid. While this is typically seen as an 'inverse crime' [11], we choose to use a location on the grid to demonstrate the image in the best scenario possible. We will discuss the error introduced by having targets off grid in section 5.6.
First, simulated data, i.e. the scattered field from equation (3.8) with no added noise, is shown in figure 10, both with (left) and without (right) the effect of absorption. That is, on the right, we simulated data assuming n i (ω) = 0. We see the effects of the absorption clearly in the left plot, when n i (ω) is included in the simulation. We will use these plots later for comparison with filtered data in order to discuss the effects of the optimal filter.
Next, we consider the image of a single target in the case of noiseless data (equation (3.9) with η = 0) using the standard back-projection filter defined in equation (5.5) (which does not include a noise term). In figure 11 we plot the image using the assumption that there is no attenuation (n i (ω) = 0) in both the data simulation as well as the filter (left); having n i (ω) accounted for in the data but not in the filter (middle); and having n i (ω) accounted for in both the data and the filter (right). (Here, not accounting for n i (ω) in the filter means the absorption Figure 11. Image for a single point target assuming noiseless data using the standard back-projection filter equation (5.5) and a sine waveform. Left: the effect of n i (ω) has not been included in either the data or the filter. Middle: the effect of n i (ω) is included in the data but not the filter. Right: the effect of n i (ω) is included in both the data and the filter. factor e −2ωn i (ω)|rs,y|/c = 1 in equation (5.5)). We see in the right plot that the optimal filter in the noiseless case does an excellent job of filtering out the effects of n i (ω) with no qualitative change in the resolution of the image.
The question then remains as to why we see such poor resolution in the images above, namely in figure 8. The answer lies in the fact that we must address noise when we design the back-projection filter. With data corrupted by noise, the use of the standard back-projection filter equation (5.5) multiplies the noise by the reciprocal of the absorption, thereby amplifying the noise (recall that the optimal noiseless filter is given by equation (5.5) along with the definition of A in equation (3.10)). That is, when Q multiplies the noise term, which does not contain a factor of AP, it will multiply the noise by a factor of e 2ωn i (ω)|rs,y|/c . The amplification of noise by the standard back-projection filter is seen clearly in figure 12. Thus, a filter designed to handle noise, like the optimal filter given in [26,27] and presented in equation (3.14), is a necessity.
We now assume there is noise present in the measured data, η ̸ = 0. We return to the same imaging expression equation (3.9) but we now use the full optimal back-projection filter, which in practice is given by If we perform the Stolt change of variables (see equation (3.11)) we obtain It is not a straightforward task to rewrite the kernel of the PSF in terms of ξ. Nor is it obvious that we can factor K as in done in the noiseless case (as shown in appendix). We therefore must rely solely on numerical studies to determine the changes in resolution when noise is present. As discussed in section 3 and in [26,27], the optimal filter equation (3.14) was originally designed to minimize the MSE between the reconstructed image and the ideal image. In these works it is assumed that the target is stochastic and has zero mean, in which case the bias of the image is zero and minimizing the MSE is equivalent to minimizing the variance. In practice, however, the target is deterministic and does not have zero mean. It is not tractable to calculate the resulting filter that minimizes the MSE as it requires one to solve a complex integral equation [29]. Therefore, we are forced to minimize simply the variance of the reconstructed image and, by making this choice, we neglect to put any restraint on the bias of the reconstructed image. The bias piece of the MSE of the back-projected image is directly related to the resolution of the image. As was shown in [16], constraining the image to be unbiased results in the same constraint as choosing the back-projection filter such that the PSF is as close as possible to the delta function in the microlocal sense [9]. This constraint is precisely how one creates a PSF that is approximately the product of two sinc functions and leads to the standard (no attenuation) SAR resolution results [9]. By ignoring the bias piece of the MSE, resolution performance may be traded for the benefit of variance reduction and hence noise mitigation.
We first investigate the effects of the optimal filter by looking at the data after filtering but prior to backprojection, i.e. we have multiplied the scattered field by Q but have not yet multiplied by the exponential or integrated as given in the image equation (3.9). Figure 13 shows a plot of the filtered data in the case of a sine waveform and SNR = 40 dB. The top left shows the result of using the optimal filter on noisy data. We see the target data faintly in the middle of the image, surrounded by noise. The top right shows the result of using the optimal filter but with absorption being neglected in the data simulation (i.e. we assume n i (ω) = 0 in the data simulation). In this case, the filter is able to filter out noise quite well. The bottom left shows noiseless data filtered with the optimal filter in equation (5.7). Here, the data still has some effects of absorption like that seen in figure 10. Finally, for comparison, the bottom right shows noiseless data filtered with the standard (noiseless) filter equation (5.5). Here, the effects of n i (ω) have been effectively filtered out. These plots show that the optimal filter does not filter the effects of absorption as well as it filters noise.
Similar effects are seen in figure 14, which shows the images of the single target for these same scenarios. The top left shows the image obtained by applying the optimal filter to noisy data that includes absorption. The top right shows the image obtained by applying the optimal filter to noisy data but with no absorption in the data and image. In the bottom left we see the image formed using the optimal filter on data that includes absorption but not noise, and the bottom right shows the image formed using the standard (noiseless) filter on data with absorption but no noise. We see again that the optimal filter can filter out noise well when Figure 13. Plots of filtered data from a single point target with a transmit sinusoidal waveform and SNR = 40 dB. Using the optimal filter on data that includes noise and absorption (top left); optimal filter on data with noise but no absorption (top right); optimal filter on data that includes absorption but no noise (bottom left); standard (noiseless) filter on data that includes absorption but no noise (bottom right). Note the color is displayed on a log scale. the effects of absorption are left out. However, if noise is removed from the data, the optimal filter does not succeed in filtering out all the effects of absorption. While the image obtained using the optimal filter is an improvement of the image seen in figure 11 in which the effects of absorption are not filtered out at all, it is still showing a significant decay in resolution over the standard (noiseless) filter. We leave as future work a more detailed resolution analysis to determine exactly how much the resolution is degraded due to this noise/resolution tradeoff, as well as a study of other imaging techniques to improve resolution without a reduction in noise mitigation.

Effective bandwidth
As studied in [26], we can quantify the differences in resolution between waveforms by considering the effective bandwidth. The effective bandwidth is the bandwidth of the resulting signal after it has been filtered. In our case, as in [26], this bandwidth is the intersection of the bandwidth of the filtered signal, with the set of frequencies ξ obtained from the data collection manifold (as defined in [26]). Note that we suppress the dependence of W on s because it varies slowly with respect to s. We calculate the effective bandwidth for the different waveforms considered for the case when z = 0 by finding the width of the frequency band where W(ω, 0) exceeds −6 dB of its maximum. The result of these calculations are plotted in figure 15. We see in figure 15 that the optimal waveform and the sine waveform have comparable resolution, the precursor has less of an effective bandwidth. For comparison, the bottom curve shows the effective bandwidth of the sine waveform when the effects of absorption are not filtered out. This figure clearly shows that there is some improvement in the effective bandwidth when using the optimal filter compared to not filtering for absorption at all. However, as a comparison, we note that using the standard (noiseless) filter on noiseless data gives a bandwidth of approximately 2 × 10 9 Hz, which is an order of magnitude better than even the best case scenario seen in this plot. Therefore, again, we suspect the detrimental effects of absorption are not entirely mitigated by the optimal filter. We must also discuss the difference in appearance of the images in [27] with those presented here and in [26]. We appreciate the authors of [26,27] sharing their original numerical code used in producing their images, which enabled us to determine two changes that result in the much improved resolution and noise reduction presented in [27]. First, we note that in [27] the frequencies used in the simulations were not expressed as angular frequencies, used here and in [26], and so the bandwidth of the waveforms did not coincide with the portion of the refractive index where absorption is most significant. In addition, there was a mislabeling of figures. Most images in [27] were in fact for the case of rather sparse foliage (with v l = 0.001) instead of what was printed (v l = 0.04). There is less absorption in the sparse foliage which leads to significantly improved resolution over the dense foliage cases presented here. In addition, there was a slight modification made to the optimal filter in the computation that was not mentioned in the text of [27]. In the numerical code used to produce the images in [27], there was a regularization factor introduced in the second term of the denominator. This is a common tactic used when filtering noise. This factor, since it was larger than one, amplifies the noise mitigation effects of the optimal filter resulting in the significantly less noisy images seen in

Off-grid targets
Lastly, we discuss the effects of having a target location that does not align with the image grid used in reconstruction. Of course target locations are not known prior to imaging and so this is the case that happens most often in practice. We briefly discuss this issue because dispersion enhances the error due to having the target off grid when compared with the non-dispersive case of SAR imaging.
Recall our image is of the form where we have assumed we can substitute in the full form of Q, combine all exponential terms, and leave the remaining terms in E for simplicity. We consider the effect of a small difference between |r s,z | 2 and |r s,y | 2 , i.e. we assume |r s,y | 2 = r 2 and |r s,z | 2 = r 2 + ϵ where 0 < |ϵ| << 1.
In both exponents, the factors of |r s,z | − |r s,y | = √ r 2 + ϵ − √ r 2 can be approximated as Therefore, the exponential factors in equation ( Figure 16. PSF for a single point target located off grid using noiseless data and noiseless optimal filter with a sine waveform. Ideally, both exponents would be zero. In standard SAR, which does not consider propagation through a dispersive material, n r (ω) = 1 and n i (ω) = 0. With dispersion, the inclusion of n r (ω) in the first term may magnify or reduce the error due to the target being off grid, depending on its magnitude. In addition, and more significantly, the second factor is an exponential function with positive real exponent that will amplify the error, possibly significantly, depending on the values of ωn i (ω). The PSF (i.e. image of a single point target) for a target that is located off the imaging grid in shown in figure 16. The target is 0.2115 m from the nearest grid location. We see significant changes from the PSF for an on-grid target like that shown in figure 11. We do note, however, that in the case of additive noise, when one uses the optimal filter that takes noise into account (equation (5.7)), we no longer see such large differences between on and off grid target images. This is due to the additional term of σ ηT in the denominator of the filter, which is large enough to mask the error of the exponential term dependent on the imaginary part of n.

Conclusions
In this paper, we formulate a sensing precursor waveform, the low frequency response of the scattered field at a distance 2d in a dispersive and lossy medium (see equation (4.13)). This sensing precursor is analogous to the Brillouin pulse for wave propagation but includes the effect of scattering. To use this sensing precursor as an initial waveform, the distance d needs to be chosen for the problem at hand. In the example provided here (a target scene with two isotropic point scatterers in a Debye-type dispersive space indicative of dense foliage), we chose the value d = 100 m to correspond with the distance from the antenna to the scatterers. We found that although the received signal of the sensing precursor with d = 100 m has a large amplitude, its limited low-frequency spectrum makes it unsuitable for target resolution when using the optimal filter Q opt with backprojection. The optimal pulse provides a better reconstruction of our target scene than both the sensing precursor with d = 100 m and the rectangular-modulated sinusoid of carrier frequency 0.1 GHz. One purpose of our paper was to address the question as to whether or not the optimal waveform derived by Varslot et al for SAR imaging through dispersive material is the Brillouin pulse proposed by Oughstun [19]. It is not. The optimal waveform utilizes the optimal filter Q opt whose purpose is to balance the effects of both dispersion and noise, thereby giving a wideband pulse when there are low levels of noise. Thus, the frequency content of the optimal pulse contains higher frequencies than those in the sensing precursor for high values of SNR. However, for low levels of SNR (high levels of noise), the optimal waveform is qualitatively similar to that of the sensing precursor.
The assumption of a known and exact index of refraction and the use of the optimal filter Q opt in the imaging algorithm means that waveforms with a wideband spectrum may create better images. We illustrated this in figure 9 by showing target reconstructions from sensing precursors with five different values d = 10, 25, 50, 75, 100 m. We saw that d = 10 m and d = 25 m provide better reconstructed images than the others, especially for low levels of noise. In future work, we would like to study two questions: (1) assuming a known complex index of refraction and the use of Q opt , is there is an optimal value of d that optimizes the target reconstruction of a sensing precursor? (2) If the complex index of refraction is not known a priori, is there a combination of Q and d that optimizes target reconstruction?
The significant amplitude and energy retention of the sensing precursor is enticing. Our work does not negate its potential use in imaging as we have not considered imaging techniques other than filtering with Q opt followed by backprojection, as discussed in [26,27]. Additionally, we have not addressed other problems with SAR imaging through dispersive material, such as target identification and backscatter, nor did we address algorithmic challenges for enhanced resolution.
Further, it was found that in using the optimal filter with any of the waveforms suggested here and in [27], there is a tradeoff between obtaining optimal resolution and noise mitigation. We find that the optimal filter is unable to fully filter out the effects of absorption, leading to poor resolution despite the fact that we assume perfect knowledge of the refractive index prior to imaging. This suggests the need for future resolution analysis as well as investigation of alternative filters, waveforms, or imaging techniques that do not cause resolution degradation in the pursuit of noise mitigation.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files). In the ideal scenario, the data collection manifold Ω z = R 2 , which implies the PSF is K(z, y) = δ(z − y) and hence I(z) = T(z). Obviously this will not happen in practice, but analysis of the data collection manifold and resulting PSF can determine how close an image is to ideal and also the resolution of the reconstructed image. We note resolution is a measure of how close two objects can be and still be seen as two separate objects in the image. If the PSF is the ideal delta function, the image is able to resolve arbitrarily close objects or targets. In nonideal scenarios, the width of the main-lobe of the PSF defines the resolution of the imaging operator.
With the assumption of a straight flight path, a flat earth, and finite bandwidth, the data collection manifold is a sector of an annulus. With the further assumption of a narrow antenna beam, the manifold can be approximated as a rectangle [9]. These assumptions allow the PSF to be factored as K(z, y) ≈ˆe i (z1−y1)ξ1 dξ 1ˆe i(z2−y2)ξ2 dξ 2 ; (A. 6) we analyze these factors separately to determine range and cross-range resolution, respectively. We use the convention that the interval [−b, b] in Fourier space corresponds to resolution 2π/b sinceˆb −b e i ρr dρ = 2 sin(br) r = 2bsinc(br). (A.7)

A.1. Cross-range resolution
To analyze the resolution in the cross-range direction, we consider a rectangular antenna of length L moving along a straight flight path γ(s) = (0, s, h) and two points on the earth whose coordinates differ only in the second coordinate. Therefore, the exponent in the PSF is (z − y) · ξ = (z 2 − y 2 )ξ 2 , which shows we need only consider the second coordinate of ξ when determining the data collection manifold. Near y = z, ξ ≈ (2ωn R (ω)/c) y2−γ2(s) R , where R = |y − γ(s)|. We now attempt to determine the ξ 2 values that lie in the data collection manifold, determined by the system bandwidth as well as the antenna beam pattern. The interval of s values for which y 2 is in the beam pattern is max|y 2 − s|, the width of the antenna footprint. To find the antenna footprint, we must consider the incident field radiated by the rectangular antenna, or dipole, we assumed earlier. First note, in general, using the far-field approximation, the incident field may be written as [9] E in (k, where the argument is real and it is straight forward to estimate the beam width for this antenna. Note the first zero of the antenna beam occurs when ωn r (ω)L sin(θ) 2c 0 = π, (A. 12) where x · e is written as sin(θ) and θ is the angle measured from the normal. With the approximation sin(θ) ≈ θ near zero, the first zero occurs at θ = 2π c Lωn r (ω) ≈ λ L (A. 13) where we used the fact that λ ≈ 2π c/(ωn r (ω)). The width of the mainlobe is twice that value, giving the beamwidth for this antenna to be 2λ/L. A simple arclength calculation gives the footprint as 2Rλ/L. Thus, max |ξ 2 | = ωn r (ω)/c 2λR L = 4π L , (A.14) which is precisely the standard resolution for SAR in a non-dispersive medium. We know that dispersion does degrade resolution however, and realize the assumption that n i ≈ 0 introduces error into our resolution estimate.

A.2. Range resolution
Assuming again that the data collection manifold is rectangular, we now aim to estimate the limits of the ξ 1 integral present in the approximate PSF equation (A.6). Near x = z, The maximum and minimum values of ξ 1 are estimated as 2 max(ωn r (ω)/c)z 1 /R and 2 min(ωn r (ω)/c)z 1 /R, respectively, where the slant range R is defined as R = z 2 1 + h 2 . With which, depending on the value of the refractive index, could increase or decrease resolution with respect to the dispersion free case. In the case of the Debye model, this formula predicts improved resolution, which is again counter intuitive. For the values used in our numerical simulations we find that in the case of no dispersion (i.e. k = ω/c) the resolution is predicted to be approximately 0.06 m while in the case of dispersion with the Debye model (i.e. k = ωn r (ω)/c) the resolution is slightly improved at 0.0498 m. One would expect that dispersion would decrease resolution and that is the case if we cannot filter out the effect of absorption due to the complex part of the refractive index, as seen in figure 11. Since we are assuming knowledge of the refractive index of the medium, we are able to include the exponential depending on the imaginary part of the refractive index in the denominator of the filter Q, in equation (5.5), and therefore we remove the effects of absorption due to the dispersive medium and achieve excellent resolution as predicted in the above analysis.