Detecting Ships in the New Zealand Exclusive Economic Zone: Requirements for a Dedicated SmallSat SAR Mission

The performance tradeoffs of a smallsat synthetic aperture radar (SAR) system for maritime surveillance in the coastal waters of New Zealand are investigated. The lower costs of smallsat platforms allow for a constellation of SAR satellites that can be launched from New Zealand using an existing local launch service provider. The minimum SAR image quality necessary for a smallsat system to achieve a desired detection performance is determined using existing <inline-formula><tex-math notation="LaTeX">$X$</tex-math></inline-formula>-band satellite data. The image quality is specified in terms of noise-equivalent sigma zero (NESZ) and resolution. It was found that for a resolution cell of 4 m<inline-formula><tex-math notation="LaTeX">$^2$</tex-math></inline-formula> a system NESZ of <inline-formula><tex-math notation="LaTeX">$-1.7$</tex-math></inline-formula> dB is sufficient to detect small fishing vessels with a probability of detection of 0.5, while maintaining the probability of false alarm below <inline-formula><tex-math notation="LaTeX">$10^{-10}$</tex-math></inline-formula>. These requirements are translated into a preliminary SAR system design.

. Islands of New Zealand and surrounding ocean, including the EEZ outlined in red (adapted from [6]).
The ships of interest are typically 30 m in length and 7 m in width [7]. This size corresponds to a small overseas trawler. Ships smaller than these dimensions may occur in the EEZ, but will most likely not be an IUU vessel of interest. This is because most illegal fishing activities in New Zealand's EEZ are conducted by foreign ships with a port of call located several thousand kilometres away from New Zealand. Ships smaller than 30 × 7 m are unlikely to make such a journey.
Detection is currently achieved via patrol ships and sporadic flights by the New Zealand Defense Forces [8], however, this method only permits the identification of a small number of illegal ships, and the cost to identify a single ship is significant. This solution is not suitable for monitoring the entire EEZ, as the majority of illegal ships are not detected. Accordingly, spaceborne remote sensing, such as SAR, is under consideration.
SAR is an attractive solution, as it is capable of achieving regular coverage independent of weather conditions. Due to the small footprint size on the ground of typical SAR systems, This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ and the need for near real-time maritime surveillance, a group (or constellation) of small SAR satellites is required. Each satellite needs an SAR system optimized for detecting ships on the ocean surface. Preliminary design principles for small SAR systems have recently been proposed [9]. This work extends these ideas by proposing a cost-effective SAR mission with small satellites that benefits from a significant reduction of the power demands for a specific application scenario.
The aim of the ship detection SAR system described in this article is to improve on the current situation, in which almost all illegal ships are missed. Accordingly, constraints on the probability of detection P d for a dedicated small-satellite SAR mission can be relaxed. In this article, a minimum P d of 0.5 is deemed acceptable. Furthermore, a conservative ship detection algorithm is used as the general problem of ship detection is outside the scope of this article, as is the task of designing a ship detection system capable of high-fidelity detection in a wide variety of circumstances. For the probability of false alarm (PFA), the assumption is made that not more than two false alarms per whole coverage of New Zealand's EEZ are tolerable. This is to minimize the risk of unnecessarily deploying expensive resources, such as spotting aircraft or intercepting naval vessels. It is shown that this limit on the number of false alarms leads to a maximum permissible PFA of 1 × 10 −10 .
In particular, this work identifies the minimum image requirements for an SAR system in order to detect typical fishing vessels with specified values for the PFA and probability of detection. Monte Carlo (MC) simulations are used to determine the probability of detection P d for different image resolutions and different NESZ values. The PFA is determined numerically by assuming that pixels falsely exceeding a constant false alarm rate (CFAR) threshold are binomially distributed. Ambiguities are not considered explicitly, as ship targets are considered localized scatterers. Therefore, ambiguities can be easily removed in postprocessing, as they are defocussed and the locations are known [10]. In principle, the ambiguities could even be used to confirm the detection results by identifying patterns corresponding to the system's expected ambiuity locations.
SAR systems on larger satellites have been widely used for maritime surveillance; for example, both ERS-1 and Seasat [11]. Images from the Canadian Radarsat-1 have also been subject to various detection studies [12]- [14]. Similarly, its successor Radarsat-2 is used for ship monitoring as part of the Polar Epsilon project [15]. Another well-known SAR mission is TerraSAR-X [16]. While these satellites have been used for ship detection, this is not their primary mission objective. In particular, the small number of satellites limits the revisit time for maritime monitoring applications. However, recently, commercial missions are beginning to employ formations of multiple small-satellite platforms [17], [18]. The use of small, lightweight, and cost-effective platforms enables, in turn, the launch of more platforms for a given budget and paves the way for new mission design paradigms that are better suited to address emerging challenges such as the quasicontinuous monitoring of spatially limited areas.
The outline of this article is as follows. Section II illustrates how ship targets are modeled in the simulation process, and describes the detection algorithm. Simplifying assumptions are also explained. Section III shows how simulations are used to derive requirements for the SAR image quality necessary to detect ships for a specified probability of detection and PFA. Section IV shows how these requirements can be translated into a simple system design. Finally, Section V presents the conclusions of this work.

A. Statistical Properties of Ship Targets
A TanDEM-X [19] dataset has been analyzed with the goal of determining the appropriate intensity distribution of ship targets. In particular, the normalized target backscatter coefficient σ ship is found, as this is necessary to determine the maximum NESZ sufficient for target detection. Fig. 2(a) shows the TanDEM-X image used for the analysis. 1 The scene was acquired close to the port of Singapore, and was chosen due to the variety of apparent target sizes. The resolution of the image is 3 ×3 m, the extent of the scene is 6.5 × 10.1 km, and the image has been calibrated radiometrically. For visualisation, the intensities in Fig. 2(a) have been scaled, such that the range of values is between 1 to 20 times the mean of the original image. The image contains 58 targets of varying size. In order to determine the intensity statistics for each target, a decision has to be made for every pixel as to which target (or clutter) it belongs. This has been obtained by applying a multistage threshold algorithm to the original SAR image. The resulting pixel map is shown in Fig. 2(b).
In order to determine the distribution that best describes ships in the TanDEM-X data, two distributions are fitted to each target image: the Γ-distribution, according to the Swerling target models [21], and, as an alternative, the log-normal distribution.
A common goodness-of-fit test is the Kolmogorov-Smirnov (KS) test [22]. The KS-test compares the empirical cumulative density function (ECDF) obtained from the data with the fitted cumulative density function (CDF). Fig. 3 shows the ECDF and fitted CDFs for both the Γ-distribution and the log-normal distribution for the target circled in Fig. 2(b), and is typical of targets in this image. It is observed in Fig. 3 that the log-normal distribution describes the target data more accurately than the Γ-distribution, as is indicated by the KS-test: for the log-normal distribution the test statistic attains a value of approximately 0.04, whereas for the Γ-distribution it is 0.64. On average, the KS-statistic distribution for the entire dataset is 0.05 for the log-normal distribution, and 0.63 for the Γ-distribution.
The probability density function (PDF) of the log-normal distribution is [22] f ln (I) = 1 The log-normal PDF is parameterized by the parameters α and β, which can be interpreted as the mean and standard deviation of the logarithm of the auxiliary data y = log e I, respectively, where I denotes the pixel intensity. Estimates for α and β for the 58 targets are shown in Fig. 4. In Fig. 4, it is observed that neither α nor β depend strongly on the target size N . Values for α and β can be used to compute σ ship for each target (2) Fig. 5 shows the normalized backscatter coefficient σ ship of the targets in Fig. 2 versus the target size in pixels. It is observed that smaller targets tend to be less reflective than larger ones. A target backscatter coefficient of σ ship = 2 dB is assumed using the data in Fig. 5, as this is a conservative estimate for targets smaller than N = 1000 pixels. This value is based on the observation that in Fig. 5 more than 85% of the targets have a normalized backscatter coefficient larger than 2 dB.

B. Statistical Properties of Sea Clutter
It is important to know how the background pixels surrounding a target are distributed in order to apply a conventional CFAR ship detection algorithm [23]. Typically, radar returns from sea clutter are assumed to be K-distributed in the intensity domain [24]. However, this is computationally expensive to invert. In this article, a simpler approach based on experimental data is developed.
An analysis of TanDEM-X data reveals that the normalized backscatter coefficient of the sea surface is approximately −13dB. Comparing this value to the target backscatter coefficient yields a signal-to-clutter ratio of 15 dB. Therefore, the power of the clutter returns is significantly below the thermal noise level, such that the clutter can be neglected, and that only thermal noise determines the distribution of the background pixels. This assumption may be violated for some acquisitions (e.g., in stormy conditions). The loss of acquisitions during such harsh conditions is, however, tolerable in view of the much less expensive system.
Using the radar equation and system parameters that could be realised by a small-satellite SAR system with low transmit power, it was found that the NESZ will be on the order of −3dB (cf. Section IV), resulting in a clutter-to-noise ratio (CNR) of −10 dB. Assuming that the Iand Q-components of the noise follow a normal distribution, the intensity of the background pixels is negative exponentially distributed if the clutter is negligible [25].
The threshold T for a negative-exponentially distributed random variable (RV) can be computed by inverting its CDF, so that [22] where P FA, goal is the desired PFA-value, ξ is the rate parameter of negative exponential distribution. The decision to neglect clutter is supported by Fig. 6, which shows that the histogram of simulated SAR data with a CNR of −7 dB can be described by the PDF of the negative exponential distribution. The clutter is modeled as a Γ-distributed RV with a texture parameter equal to 1. The low texture parameter indicates a high heterogeneity [25]. The low value for the texture parameter results in spiky sea clutter, which complicates ship detection due to the generally higher number of clutter pixels exceeding the detection threshold (or the noise floor, for the case presented), leading to more false alarms [24]. Therefore, the results presented are also valid for higher texture parameters or more homogeneous clutter.
Computing the CFAR threshold using (3) is computationally much less expensive than numerically inverting the Kdistribution [26]. Therefore, neglecting the sea clutter significantly reduces the computational cost. It was found that the maximum CNR where this assumption remains valid is −7 dB. Similarly, a minimum NESZ of −6 dB is required when the normalised clutter backscatter is −13 dB.

C. Ship Detection in Simulated Data
The detection algorithm used for determining PFA and P d differs from conventional ship detection algorithms in that a distinction is made between detection at the pixel level and detection at the object level.
The detection algorithm consists of two steps; first a CFAR algorithm is used to identify pixels bright enough to exceed a certain threshold [23]. This threshold is selected to yield a desired PFA when applied to background pixels that do not contain a target.
The detection algorithm used for the system requirement analysis is not the focus of this article, and accordingly is less complex than conventional approaches. The detection algorithm only serves to determine if detection with a small-satellite SAR system is feasible.
In order to group the pixels exceeding the threshold T , a simple counting algorithm is employed: a window with dimensions adjusted to the smallest expected target size is convolved with the CFAR pixel map. For each location of the target window, a minimum number n min of pixels exceeding the threshold T have to be present for a target to be considered present.
In each iteration of the MC simulation, as many pixels as fit into a target detection box are simulated. The number N of simulated pixels depends on the target size of interest and the image resolution: where L ship and W ship are the length and width of the ship class of interest, and δ az and δ rg are the image resolution in azimuth and range, respectively. The pixel intensity values must consist of exponentially distributed noise intensities and log-normally distributed target intensities. The generation of these samples can be expressed as where u represents the pixels used in the simulation, t stands for the log-normally distributed target pixels in the intensity domain, φ is the uniformly distributed phase of the target, and n is complex Gaussian noise. In order to achieve an average noise intensity of 1, n ∼ CN (0, 1), where CN (a, b) indicates a complex Gaussian distribution with mean a and variance b.
In the case of no noise, n = 0, then u = t, which follows the log-normal distribution by definition. If no target is present, t = 0, then u = |n| 2 , which can be shown to follow the exponential distribution [25].
For all simulations, target dimensions of L ship × W ship = 30 × 7 m and an azimuth resolution of δ az = 2 m are assumed. The range resolution δ rg and the signal-to-noise ratio (SNR) are varied. To illustrate the simulated signal u, a range resolution of δ rg = 2m and an SNR of 6 dB are assumed. Using (4), N = 52.
In the MC simulation, the signal u is generated M times; the probability of detection P d is then approximated by dividing the number of targets exceeding the threshold n min by the total number of targets M . Thus, P d can be found for specific values of the CFAR-PFA (P FA, CFAR ) and the object threshold n min . In order to specify these values, the desired PFA on object level P FA, obj has to be determined. Once P FA, obj is fixed, the survival function of the binomial distribution can be inverted to determine the value of P FA,CFAR necessary to obtain the desired P FA, obj for varying values of n min An analytical method of inverting (6) has not been found; therefore, a numerical approximation has been implemented.

III. IMAGE QUALITY REQUIREMENTS
The simulated dependence of the probability of detection P d on the NESZ for different values of the resolution area A res is shown in Fig. 7. The simulations were conducted for a desired PFA of P FA, goal = 1 × 10 −10 . In the MC simulations,  a resolution cell is either completely filled by a target or clutter. This leads to a discretization error for the P d -values, which is reduced using a Gaussian filter. For each individual data point in Fig. 7, the combination of n min and P FA, CFAR that yields the highest value for P d while keeping P FA, obj at the desired value of P FA, goal has been selected. Fig. 7 shows that to achieve a probability of detection of at least 0.5, an NESZ of more than −5.5 dB is sufficient if the resolution cell is 10 m 2 or less. Fig. 8 shows how the requirements for NESZ and A res change if the constraints on P FA, goal are tightened or relaxed. The desired value for P d has been set to 0.5 and with a decreasing P FA, goal the required NESZ also decreases as expected. However, the difference in required NESZ for different values of P FA, goal is only on the order of a few dB. This implies that changing the number of false alarms by one or two orders of magnitude does not significantly affect the system design in terms of resolution or NESZ.

IV. PRELIMINARY SYSTEM DESIGN
This section explores how the relationship between maximum NESZ and resolution cell area can be translated into an initial SAR-system design.  Table I  The NESZ is computed using the parameters in Table I where c is the speed of light, θ i is the angle of incidence, and r is the range from the satellite to ground. The range r and the satellite altitude H are connected by the relationship H = r cos θ. All other parameters are shown in Table I [10]. The desired object PFA is set to P FA, obj = 1 × 10 −10 . It can be shown that this value corresponds to approximately two false object detections per complete coverage of the area of interest (A EEZ ≈ 4 × 10 6 km 2 ): The number of target detection windows N win present in the entire area of interest A EEZ can be approximated by The required PFA to achieve N FA = 2 per whole coverage is then P FA, obj = N FA/N win ≈ 1 × 10 −10 . The parameters in Table I were chosen because they can reasonably be implemented on a small-satellite platform. In particular, the SAR transmit power is low compared to typical state-of-the-art SAR systems [19], due to the limited power available from small solar panels. The look-angle γ and the antenna width W a have been optimized to give a 3 dB beamwidth on the ground of at least 30 km.
The NESZ computed with (7) is plotted over the ground range R g in Fig. 9 for two different values of the bandwidth B r , corresponding to two different range resolutions. It should be noted that the NESZ requirements shown as dashed lines in Fig. 9 are adjusted for the appropriate range resolution. It is observed that for both bandwidths the system NESZ is better than required over a ground range of almost 30 km. As expected from (7), the NESZ falls off with decreasing B r . This means that the system becomes more sensitive to radar returns from weak scatterers, while simultaneously causing the range resolution to become coarser. The dashed lines in Fig. 9 show that the required NESZ also decreases when the bandwidth is reduced. This is in agreement with Fig. 8, which shows that for a coarser resolution a more sensitive system is required to detect a target with the same probability P d . For both bandwidths, Fig. 9 shows that the ground range over which the NESZ requirement is fulfilled is almost identical.
For B r = 100 MHz, the resulting NESZ curve is partially below the −6 dB threshold derived in Section II. Therefore, the NESZ values below the threshold may differ from what would be obtained if the sea clutter was taken into account in the simulations.

V. CONCLUSION
This article focuses on the design of a small-satellite SAR system for dedicated maritime surveillance in the New Zealand EEZ. A methodology was developed to assess the impact of ship detection requirement on the design of an SAR system. A simplified situation and model were used to demonstrate the process. Results apply only to this case, however, the methodology can be applied to other cases.
Specifically, TanDEM-X data were analyzed in order to extract statistical models of ship targets. It was found that the log-normal distribution is an appropriate fit for these targets. It was also shown that for a CNR below −7 dB, the sea clutter contribution to the radar signal can be neglected. This was shown by simulating heterogeneous sea clutter with additive Gaussian noise. With the normalized target backscatter coefficient found from the data, the minimum NESZ for which the assumption is valid is −6 dB.
To detect ships with dimensions 30 × 7 m, a curve has been computed that relates the maximum permissible NESZ to the resolution for a desired probability of detection of 0.5 and PFA of 1 × 10 −10 (which represents two false alarms over the entire EEZ). For example, with a resolution cell of 4 m 2 , the system NESZ must not exceed −1.7 dB, whereas at a resolution of 8 m 2 , the maximum NESZ is −5.4dB.
Future work will focus on a more detailed system design. Various methods of improving the system performance will be considered. For example, given that short revisit times are crucial for maritime surveillance, a concept of operations for a constellation of small satellites will be explored. Since 2019, he has also been a Lecturer with Ulm University, Ulm, Germany. He has authored or coauthored more than 80 research articles in peer-reviewed journals and international conference proceedings. He holds eight patents in the field of SAR. His research interests include the conception of innovative SAR modes for high-resolution wide-swath imaging and the development of low-cost SAR solutions for frequent and enhanced Earth monitoring. Dr