## Abstract

This paper utilizes a combination of theory and simulations to examine synthetic aperture imaging across a wide range of turbulence conditions. Extensive wave optics simulations are used to validate existing theory and to investigate the use of a common measurement technique. It demonstrates the applicability of earlier synthetic aperture laser radar (ladar) (SAL) research across a wide range of turbulence conditions, and examines the metric approaches and limitations for the imaging conditions normally seen in practical SAL systems. To examine the full impact of turbulence on SAL, the derivations, simulations, and analyses include three different resolution metrics as well as a commonly used contrast metric: the integrated sidelobe ratio. This paper demonstrates the integrated effects of turbulence on SAL imaging. Finally, suggestions are given for measuring the true resolving power of operational SAL systems.

© 2017 Optical Society of America

## 1. OVERVIEW AND BACKGROUND

There is a significant body of analytic and experimental work detailing the effect of atmospheric turbulence on standard imaging. Standard imaging is defined here as any system that collects intensity data with a fixed aperture, a power-law focal plane, and typical integration times of milliseconds or longer. Many metrics exist that describe the environment for standard imaging including coherence diameter (${r}_{0}$), isoplanatic angle (${\theta}_{0}$), Rytov variance, and others. These metrics allow estimation of conventional imaging performance in the presence of turbulence.

In synthetic aperture laser radar (ladar) (SAL), less work has been done to quantify the impact of turbulence on SAL images. To date, there are two papers that have presented a theoretical basis for SAL resolution in turbulence. The first paper, by Karr, made a careful study of the wave optics origins of the SAL signal [1]. In order to simplify the mathematics and generalize the results, many assumptions were made that might limit the applicability of the theory. Experiments or simulations are required to validate this work under a broad range of turbulence conditions.

In the second paper, by Lucke, initial numerical simulations were performed and compared to Karr’s theoretical analysis [2]. Lucke pointed out that Karr’s analysis did not include the contrast degradations in the impulse response function (IPR). Karr’s analysis was based on a modulation transfer function (MTF) derived resolution metric that did not adequately address contrast information. Lucke’s analysis qualitatively addressed the contrast issue, but did not present metrics to quantify the contrast degradations. Furthermore, Lucke’s simulations only examined a single realization of the atmosphere. The results provided some very useful insights; however, they did not capture the general trends that provide a more thorough depiction of the overall effectiveness of SAL imaging in the atmosphere.

This research was undertaken to examine and extend this previous work and to gain a better understanding of theoretical SAL performance in the atmosphere. Through a series of simulations, it was found that for long-term averages, the measured resolution was in good agreement with the theoretical SAL coherence length.

A common method of measuring the resolution of a SAL system has been to measure the width of the intensity peak of individual point targets in a SAL image and average the results. This research shows that this method can produce spurious results for longer apertures in the presence of atmospheric turbulence, and is not a reliable measure of the resolving power of a SAL system.

In Section 2 of this paper, the existing theory is reviewed to lay the groundwork for understanding the simulation results. Sections 3 through 5 follow the development of the existing theory in [1]. Section 3 presents the Fried resolution metric and modifications for its use in SAL [3]. Section 4 presents the wave structure function (WSF) and its use in understanding SAL resolution in a turbulent environment. Section 5 presents the wave optics simulation used to characterize SAL performance across a wide range of turbulence regimes. The results of the wave optics simulations are presented with calculations of the SAL WSF and are compared to theoretical values for both the SAL coherence diameter and for resolution. Section 6 shows results under likely imaging scenarios with respect to averaging. Finally, conclusions are drawn with a discussion of possible future work.

## 2. SAL SIGNAL INTRODUCTION

A review of the principles of SAL is presented to ensure common use of terminology, to introduce the coordinate system that will be used in this paper, as well as to introduce the mechanisms for evaluating the perturbations in the SAL phase history data (PHD).

A SAL measures the evolution of the phase from a target or group of targets from a translating aperture, as shown in Fig. 1. It measures the range (${R}_{n}$) to the $n$th target at $({x}_{n},{z}_{n})$ from the aperture at each position $x(t)=vt$ along the synthetic aperture (SA), where $v$ is the velocity of the aircraft.

The return from each pulse, defined as fast-time, is typically short enough (usually $<100\text{\hspace{0.17em}}\mathrm{\mu s}$) to be unaffected by the atmosphere intrapulse, but may have a piston offset and temporally fixed, spatially distributed phase and amplitude patterns imposed on the whole pulse. Atmospheric turbulence will have a significant impact from pulse to pulse, defined as slow-time, and will directly impact the cross-range, or along-track, resolution. This research will focus on the slow-time perturbations where the equations of interest are those for cross-range compression.

In the absence of turbulence, the measured phase of a single pulse is dependent on the range to the target, $R(t)$, given by

where $z$ is the range direction perpendicular to the line of flight, $x$ is the cross-range distance measured from the SA center, the platform is moving in the $x$ direction at a velocity $v$ and the $n$th target is located at $({x}_{n},{z}_{n})$ [3].If the $n$th target is assumed to be at ${x}_{n}=0$ with unity amplitude, the unperturbed time-dependent return signal, ${s}_{r,0}(t)$, is written

When the atmosphere is included, the signal, ${s}_{r}(t)$, will include amplitude and phase fluctuations due to atmospheric turbulence expressed as

where $\chi $ and $\varphi $ are log-amplitude and phase perturbations as a function of SA time or slow-time.For simplicity, all targets will be assumed to be at the same range, ${z}_{0}$. A reference signal made up of an ideal point target at scene center $({x}_{0},{z}_{0})$

Note that for every target position ${x}_{n}$ the signal in Eq. (5) has a linear phase ramp. The linear phase ramp will, when Fourier transformed to the image coordinates, result in a shift in position, but does not otherwise affect the image. This research utilizes isolated point targets, so the shifts are removed for ease of analysis. Removing the linear phase is equivalent to putting each target at $({x}_{0},{z}_{0})$, and Eq. (5) is reduced to just the phase and log-amplitude perturbations.

This signal is converted from the time domain to the azimuth spatial frequency domain via the transformation

where $\overrightarrow{x}=x(t)$ evaluated over $t\in [-{t}_{SA}/2,{t}_{SA}/2]$ and ${t}_{SA}$ is the time for the aircraft to traverse the SA of length $L$, and ${R}_{c}$ is the range from SA center to scene center. While the data havebeen collected in time or, equivalently, spatial coordinates, it can also be seen via Eq. (6) that the data represent the 1D field in the pupil plane. The resulting SA pupil data, or cross-range PHD, are## 3. FRIED’S RESOLUTION METRIC AND MODIFICATIONS FOR SAL

Passive imaging techniques are typically long-term processes with respect to atmospheric turbulence. Given the random nature of turbulence, standard imaging systems often measure average, or long-term, images. While it is possible to actively illuminate the scene using short-pulse lasers to acquire a short-term image, most turbulence theory has been developed for long-term images.

In standard electro-optic/infrared (EO/IR) imaging, there are many definitions of resolution. The diffraction limit of $1.22\lambda R/D$ and full width at half-maximum (FWHM), or 3 dB width, are common metrics. The diffraction-limited resolution is the optimal theoretical performance, but it is not achieved in a system with aberrations. The FWHM, while easy to compute from measured data, presents challenges when deriving an analytic expression that includes aberrations such as turbulence.

One measure of resolution that is amenable to an analytic solution in a system with significant phase errors that has been used previously is

In a passive image, the coherent ensemble of pupil functions is combined over the integration time of the focal plane. A SAL image, however, is generally formed from a single, or at most a few, realizations of the SAL pupil function. In the past, it was common to measure the 3 dB width of each corner cube in a scene consisting of many such targets and then average their widths in order to arrive at an average resolution. As mentioned, this approach works well for measured data but does not lend itself to development of a mathematical formula.

Equation (8) can be modified to follow a similar averaging process by using the average MTF, $\u27e8|\mathcal{O}|\u27e9$, instead of the average OTF. This is akin to measuring the resolution, $\mathcal{R}$, of a single realization of the atmosphere for a single point target, and then averaging the results over many realizations and many point targets, or a large number of anisoplanatic targets in the same scene. This resolution, denoted ${\mathcal{R}}_{I}$, is then the incoherent average of the individual MTFs:

As the length of the SA grows, the incoherent average and 3 dB resolutions continue to increase beyond the coherent resolution. However, when the baseline is much larger than the atmospheric coherence diameter, these measures become dominated by noise and converge to the speckle limit, i.e., the resolution of the SA baseline with random phase noise.

As noted in Eq. (10), the OTF is the normalized autocorrelation of the SAL phase history $H({\overrightarrow{K}}_{a})$. Because there is a linear relationship between spatial frequency, ${\overrightarrow{K}}_{a}$, and SA coordinates, $\overrightarrow{x}$, the OTF can be rewritten using Eq. (6):

## 4. ADDITIONAL METRICS

#### A. WSF for Standard Imaging

Another common device for examining the effect of turbulence on imaging is the WSF, $\mathcal{D}(\mathrm{\Delta}\overrightarrow{x})$, and its components, the phase and log-amplitude structure functions, ${\mathcal{D}}_{\varphi}$ and ${\mathcal{D}}_{\chi}$, respectively. The WSF is the variance of the phase and log-amplitude differences as a function of separation in the pupil. It is related to the statistical autocorrelation of the pupil function. Equations for the phase and log-amplitude structure functions derived from first principles exist for standard imaging, which can predict imaging performance in a given set of turbulence conditions [4,6]. The WSF is

The standard WSF, for short-term, near-field imaging conditions, is related to the normalized autocorrelation, $\mathrm{\Gamma}(\mathrm{\Delta}\overrightarrow{x})$, by

#### B. Role of the WSF in SA Imaging

Previous research concluded that the SAL WSF will follow the same form as the WSF for standard imaging given in Eq. (14) [1]. For SAL, the normalized autocorrelation, also known as the OTF, has the same form as Eq. (15), where ${\mathrm{\Gamma}}_{ap}$ is the normalized autocorrelation of the SAL pupil function, ${H}_{0}(\mathrm{\Delta}\overrightarrow{x})$, and the WSF will be ${\mathcal{D}}_{SA}$. The $\u27e8{\overline{\mathrm{\Delta}a}}^{2}\u27e9$ in the exponential of Eq. (16) removes the slope variance from the WSF so that the result corresponds to the autocorrelation of the tilt-removed PHD, ${\mathrm{\Gamma}}_{p}$. The tilt is removed, since it has no effect on the image resolution or contrast. The far-field is defined as $L\ll {(R\lambda )}^{1/2}$, where $R$ is the path length, and $L$ is the SA length [4]. The scenarios, which will be considered in the modeling and results section, have an imaging range of 50 km, wavelength of 1.5 μm, and SA lengths ranging from 0.17 to 8 m. To be in the far-field for this configuration, $L\ll 0.2739$; therefore, except for the first few SA lengths in this simulation, the system will operate in the near-field.

Using Equation (15), the coherent resolution from Eq. (8) is rewritten as

Earlier research posited that the SAL WSF is four times the standard WSF [1]. This new SAL WSF, ${\mathcal{D}}_{SA}(\mathrm{\Delta}\overrightarrow{x})$, is written

#### C. Contrast Metric

Another metric needed to evaluate SAL imaging performance in turbulence is the integrated sidelobe ratio, (ISLR). The 3 dB width and the resolution ($\mathcal{R}$) metrics only capture the size of the main lobe, while ISLR gives a measure of the amount of energy in the sidelobes, i.e., the contrast in the image. Lower ISLRs indicate better image quality. The ISLR is given by

## 5. MODEL-BASED ANALYSIS

To determine the form and impact of the SAL WSF, a wave optics propagation model was developed and run under eight atmospheric conditions. In each case, the atmosphere was modeled using a Hufnagel-Valley ${C}_{n}^{2}$ profile with a constant multiplier, average winds set to 21 m/s and a ground ${C}_{n}^{2}$ set to the value in Table 1.

The model simulates a monostatic SAL with a real aperture diameter, $D$, of 15 cm. It propagates a laser beam from an aircraft to the ground through 20 equal-strength Fourier-series phase screens [10]. The beam is then reflected off a point target, propagated through the same phase screens back to the aperture, and the resulting field is integrated over the receiving aperture, $D$. This process is then repeated at each slow-time aperture position to build up a SAL phase history (in azimuth only).

Platform motion effects are included by moving the platform appropriately between pulses. No wind was used in these simulations, so the phase screens remain stationary.

Using the discrete representation of the ${C}_{n}^{2}$ profile, model outputs were compared to theoretical values of beam wander, beam size, scintillation index, and scale size. All were found to be in excellent agreement.

A classification scheme for SAL performance has been presented that has three distinct “regions,” based on Fried’s coherence diameter [11]. In Region I, ${r}_{0}>L$, where $L$ is the SA length. Here the SA sees a fairly uniform atmospheric “lens.” In Region II, $D\le {r}_{0}\le L$, the coherence diameter is larger than the real aperture, $D$, but smaller than the SA. Region III is characterized by a coherence diameter that is smaller than the real aperture, ${r}_{0}\le D$. As the sensing conditions change from Region I to Region III, the phase effects range from being uniform over the SA (Region I) to being uniform over individual apertures but varying over the SA (Region II), to being nonuniform over a single real aperture (Region III). This research used a physical aperture of 15 cm and included the ${r}_{0}$ values shown in Table 1. The maximum baseline for each case was set to $L/{\tilde{r}}_{0}=33$; therefore, the simulated SAL data covered from Region I to the lower end of Region II. Region III was not addressed in this research.

For each pulse in the SAL simulation, field data was used at the end of propagation to calculate the phase structure function. The simulation included more than 132,000 pulses over 162 independent realizations of the atmosphere across an 8 m baseline. The ensemble average phase structure function was calculated for the propagations. The measured values and theory are in excellent agreement. As an example, Fig. 2 shows one set of atmospheric conditions.

## 6. RESULTS UNDER REPRESENTATIVE IMAGING CONDITIONS

The nominal form of the structure function given by Eq. (19) was validated following the example of [1,4] and by simulating multiple SA lengths at multiple ${r}_{0}$ values and fitting the theoretical formula for resolution to the data.

Equation (17) was fit to data from the series of simulations with different atmospheric conditions given in Table 1. There are two free parameters in this equation: ${\tilde{r}}_{0}$ and $\alpha $. To find the best fit for ${\tilde{r}}_{0}$, an iterative approach was used varying $\alpha $ from Eq. (18) to find the best combination of $\alpha $ and ${\tilde{r}}_{0}$. One example of the output of this process is shown in Fig. 3, which plots normalized resolution as a function of normalized SA length. ${\mathcal{R}}_{1}$ is the resolution of an ideal aperture of length ${\tilde{r}}_{0}$. It is calculated using the resolution defined in Eq. (17) with $L$ set to ${\tilde{r}}_{0}$, and no turbulence, i.e., ${\mathcal{D}}_{SA}=0$ and ${\u27e8\overline{\mathrm{\Delta}a(L)}}^{2}\u27e9=0$. For this example, $\alpha =1$. The (green) dotted line shows the resolution that would be expected in the absence of turbulence. The (blue) dashed line shows the data from the simulation. The theoretical resolution for the coherent average OTF is shown in the (red) solid line. The SAL coherence diameter is indicated by the vertical and horizontal dashed lines.

In order to arrive at a relationship between ${\tilde{r}}_{0}$ and ${r}_{0}$, 160 realizations were run for each of a broad range of atmospheric conditions. For each ${r}_{0}$, an ${\tilde{r}}_{0}$ value was calculated by fitting the measured resolution from the average OTF to the theoretical resolution, Eq. (17), and tilt slope variance, Eq. (18). These data are plotted in Fig. 4 along with the previous theory. The two methods produce different estimates of the SAL coherence length with the fit to the tilt slope variance, producing a value closer to the theory of Eq. (20).

Insight into the differences can be gained by looking at the SAL WSF. Figure 5 shows the measured SAL structure function alongside the structure function calculated with the ${\tilde{r}}_{0}$ measured from the resolution fit, Eq. (17) as shown in Fig. 3, as well as the theoretical value using Eq. (20). Despite having a good fit between the data and the resolution curve in Fig. 3, the measured WSF in Fig. 5 is much closer to the theory of Eq. (20), suggesting that the existing theory, given in Eq. (20), is correct for large ensembles.

In Fig. 5, the tail of the curve at small separations differs from theory in the same way as the measured WSF for standard propagation (Fig. 2) differed from its theory. These differences are likely due to sampling constraints of the simulation. Because of this potential artifact, both ${\tilde{r}}_{0}$ values are plausible. Further simulations are required to resolve the differences.

In SA radar, a typical method of measuring resolution is to measure the 3 dB width of each point target response in an image with many such targets and then find an average. This can be compared to the normalized versions of ${\mathcal{R}}_{C}$ and ${\mathcal{R}}_{I}$ using the following relation for 3 dB resolution, ${\mathcal{R}}_{3\mathrm{dB}}$:

The average MTF, $\u27e8|\mathcal{O}|\u27e9$, and average OTF, $\u27e8\mathcal{O}\u27e9$, are shown in Fig. 7, and the IPR from the average MTF is shown in Fig. 8. A long SA of 3.3 m is shown to emphasize the salient details. In Fig. 7, the average MTF has two main characteristics: a narrow central peak and a broad base. The average OTF has the same narrow central peak but without the broad base. The average OTF goes to zero outside of the cutoff, since the individual OTFs are complex, random, and cancel when summed. The average MTF is positive, so the fluctuations in the wings do not cancel but instead sum to the base as shown.

The origins of this behavior can be illuminated by considering a phase history of relatively uncorrelated noise. For that noise data, as the SA length increases, the width of the pedestal on the average MTF will increase, while the narrow correlated portion of the data will remain constant. Therefore, the resolution, ${\mathcal{R}}_{I}$, of Eq. (11) will integrate an increasing amount of noise with increasing SA length, as shown in Fig. 6 by the (black) dashed line.

Figure 8 shows the resultant IPR. The primary peak comes from the envelope of the average MTF, that is, the broad base. As the base of the MTF continues to increase, along with the SA length, the width of the primary peak of the IPR continues to decrease. This is the peak that would likely be measured to calculate the 3 dB width of a single point target. It is clearly spurious.

The achievable resolution, that is the ability to resolve two closely spaced targets, is given by the secondary peak. The width of this secondary peak is controlled by the width of the central peak in the average MTF that is governed by ${r}_{0}$; however, the width of the primary peak is proportional to the SAL baseline modulated by random phase noise. Unless many point targets are averaged in the image plane, or their OTFs are averaged in the pupil, the true resolution will not be observed in practice.

Figure 9 shows both the improving 3 dB width and the average ISLR. At $L=5{\tilde{r}}_{0}$, the approximate point at which the normalized resolution saturates, the ISLR is at $-9\text{\hspace{0.17em}}\mathrm{dB}$, a level that should produce acceptable image contrast. However, measuring image contrast beyond the resolution peak at $L=2{\tilde{r}}_{0}$ (Fig. 6) will produce increasingly suspect results.

## 7. CONCLUSIONS AND FUTURE WORK

The measured coherent resolution, ${\mathcal{R}}_{C}$, is based on many coherently summed realizations of the individual OTFs and agrees with the current theory. This expectation of the OTFs is positive and has little energy beyond the ${\tilde{r}}_{0}$ cutoff.

This research showed that while the average OTF method of calculating resolution has not been applied in many cases, it is capable of measuring the broad peak in the IPR associated with the coherence diameter that limits the resolution of a SAL. Simulations confirmed that the theoretical value for ${\tilde{r}}_{0}$ is likely correct, but that further research is required to resolve the differences in the two different measured ${\tilde{r}}_{0}$ values shown in Fig. 4. Results show that a commonly used SAL measure of the resolution, the average 3 dB width, does not adequately describe the resolving power of long SAs in turbulence. At values of $L/{\tilde{r}}_{0}$ greater than approximately 5, ${\mathcal{R}}_{3\text{\hspace{0.17em}}\mathrm{dB}}$ will tend to measure the width of the baseline $L$ with completely random phase. ${\mathcal{R}}_{3\text{\hspace{0.17em}}\mathrm{dB}}$ will exhibit increasing resolution but at significantly reduced main lobe peak levels.

The measured resolution of an isolated point does not coherently average multiple pupil realizations and will thus continue to increase beyond the ${\tilde{r}}_{0}$ cutoff, converging to the speckle limit, which is not a useful resolution.

Since ${\mathcal{R}}_{3\text{\hspace{0.17em}}\mathrm{dB}}$ measurements have been common in previous experimental SAL measurements, ${\mathcal{R}}_{I}$ was developed as a surrogate for ${\mathcal{R}}_{3\text{\hspace{0.17em}}\mathrm{dB}}$. ${\mathcal{R}}_{I}$ includes the same incoherent summation as ${\mathcal{R}}_{3\text{\hspace{0.17em}}\mathrm{dB}}$, but is measuring the bandwidth in spatial frequency similar to ${\mathcal{R}}_{C}$ rather than the width of the peak in image space as does ${\mathcal{R}}_{3\text{\hspace{0.17em}}\mathrm{dB}}$. This research showed that ${\mathcal{R}}_{I}$ will not produce accurate predictions of the SAL resolving power in turbulence because it will generally measure the primary peak, i.e., noise outside the ${\tilde{r}}_{0}$ cutoff, as shown in Fig. 8.

Isolated point targets see only a single realization of the atmosphere, and the measured resolution is consistent with ${\mathcal{R}}_{I}$ and ${\mathcal{R}}_{3\text{\hspace{0.17em}}\mathrm{dB}}$; that is, the measurements will continue to increase with longer SA lengths because the envelope of the MTF continues to grow. In order to measure the true resolving power of the SAL, OTFs from many isolated point targets measurements must be combined coherently or by summing in the image plane. This result will be consistent with ${\mathcal{R}}_{C}$. Cross-track (vertical) line targets will also produce the coherent sum of OTFs and exhibit resolution consistent with ${\mathcal{R}}_{C}$. Therefore, cross track line targets and/or multiple point targets should be included in laboratory and field measurements to facilitate accurate coherent measurements of SAL resolution.

More work is required to understand the specific mechanisms and dominant effects of turbulence that degrade SA imaging performance. In its nearly 70-year history, SA radar has made considerable progress in finding ways to overcome phase perturbations in collected data. Because of the significant differences in the sources of error, their magnitudes, and spatial coherence, much of that work is not directly applicable to SAL. Gaining a better understanding of the nature and source of the perturbations to the SAL phase history will make a considerable difference to those studying ways to mitigate the perturbations in signal processing. Future research will focus on decomposing atmospheric turbulence-induced SAL image degradations into individual mechanisms, quantifying and characterizing their impact on the SAL imaging metrics and identifying mitigation approaches.

## Funding

Air Force Research Laboratory (AFRL).

## Acknowledgment

This effort was supported in part by the United States Air Force Research Laboratory, Sensors Directorate, the University of Dayton, and by a grant of high-performance computer time from the DoD/AFRL Supercomputing Resource Center in Kehei, Maui, HI. The views expressed in this paper are those of the authors and do not reflect the official policy of the Air Force, Department of Defense, or the United States government.

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