To obtain the data for each scenario comprising RASPNet, we make use of the RFView^® modeling and simulation tool. This physics-based, RF modeling and simulation platform integrates a global database of terrain and land cover for RF ray tracing simulations. RFView^® has been compared with measured datasets from VHF to K_a band (Gogineni et al., 2022). We further validate RFView^® versus the MCARM measured dataset (Babu et al., 1996) in Section 2.4.

The scenarios we consider simulate an airborne radar surveying various regions across the United States. For simplicity, the radar was assumed as static during the simulation. The complete list of radar platform parameters is provided in Table 4 of the Supplementary Information.

We position the radar platform at a preset, scenario-specific, latitude and longitude location, utilizing the parameters from Table 4. The radar platform aims $20$ km to the southwest and operates in spotlight mode. The clutter return is beamformed for each size $(\frac{48}{L}\times 5)$ receiver sub-array, condensing the array to size $(L\times 1)$. Formally, let $S=\{1,2,\ldots,M\}$ denote the set of scenario indices, where $M$ is the total number of scenarios within the dataset. We denote $Z^{i}\in\mathbb{C}^{\Lambda LG}$ as the random variable describing the clutter for scenario $i\in S$. Now, consider $z_{j}^{i}\sim Z^{i}$, which denotes the clutter return for the $j$th independent realization for scenario $i$. To obtain the clutter dataset for scenario $i$, we obtain $K$ i.i.d. realizations to form $\{z^{i}_{j},\,\forall j\in D\}$, where $D=\{1,2,\ldots,K\}$.

RASPNet comprises $M=100$ hand-picked scenarios from the contiguous United States, where the clutter dataset for each scenario, $\{z^{i}_{j},\,\forall j\in D\}$ with $i\in S$, is obtained via the procedure described in Section 2.1. The selection criteria for these scenarios prioritize geographical diversity, focusing on varied landforms and topographies that present distinct challenges for radar systems. The geographic coverage of RASPNet scenarios across the contiguous United States is depicted in Figure 1, wherein the indices, $i\in S$, are ordered from the westernmost to easternmost scenario by default.

To evaluate whether RASPNet encompasses a realistic and extensive feature set, we empirically verify the exemplar nature of our $M$ hand-picked scenarios in Section 2.3. Additionally contained within these $M$ scenarios are $30$ land-to-water transition scenarios, which capture the complexities associated with implementing adaptive radar processing algorithms in coastal regions. The complete tabulation of scenarios comprising RASPNet, along with short descriptions, is provided in Section G of the Supplementary Information.

To determine whether our $100$ hand-picked scenarios are exemplar, we compare their geographic coverage with that of a collection of $100$ scenarios obtained through the Partitioning Around Medoids (PAM) algorithm (Kaufman & Rousseeuw, 1990), from $500$ total scenarios uniformly selected at random across the contiguous United States. We denote $\smash{\,\overline{\!{S}}=\{1,2,\ldots,\,\overline{\!{M}}\}}$ as the set of indices pertaining to the $\smash{\,\overline{\!{M}}=500}$ scenarios, and denote $\smash{S^{*}\subset\,\overline{\!{S}}}$ as the set of indices yielded by PAM, where $\left|S^{*}\right|=M^{*}=100$.

To determine $S^{*}$ using the PAM algorithm, we employ the energy statistic ($\mathcal{E}$-statistic) as our distance metric (Székely & Rizzo, 2013). The $\mathcal{E}$-statistic quantifies the statistical distance between the distributions of random variables by aggregating pairwise sample distances, capturing differences in their distributional characteristics. Rooted in potential energy principles, the $\mathcal{E}$-statistic facilitates robust comparisons of complex, multivariate datasets beyond location or scale. The formal definition of the $\mathcal{E}$-statistic for our analysis is provided in Definition 2.1.

We now outline the PAM algorithm for our analysis. We first construct the $\smash{\,\overline{\!{M}}\times\,\overline{\!{M}}}$ $\mathcal{E}$-statistic matrix, $E$, for $\smash{\,\overline{\!{M}}=500}$, where each element, $E_{i,k}$, is precomputed using $\mathcal{E}(Z^{i},Z^{k})$ for every pair $\smash{i,k\in\,\overline{\!{S}}}$. This precomputation step facilitates efficient distance lookups during the PAM algorithm’s execution. Subsequently, over $T$ steps, the PAM algorithm iteratively refines a subset of $M^{*}=100$ representative scenarios, $S^{*}$, through a process of swapping current medoids with non-medoids to minimize the total cost, $C$, where $C$ is computed as the sum of distances from each scenario to its nearest medoid, based on the precomputed $\mathcal{E}$-statistic matrix, $E$. This procedure is summarized in Algorithm 1. The $500$ total scenarios and the $100$ representative scenarios from PAM are depicted in Section A of the Appendix.

We observe that the geographic coverage of the $100$ hand-picked scenarios in RASPNet closely mirrors the coverage of the $100$ representative scenarios determined through PAM. A large concentration of these scenarios are in the Western United States — this region is characterized by a wider variety of terrain types, which introduces greater variability in the distribution of geographic features, leading to a greater concentration of representative scenarios. We also note that the $100$ hand-picked scenarios in RASPNet are more dispersed across the contiguous United States by construction, particularly because they encompass identifiable geographical landmarks representing a broad spectrum of environmental conditions. A greater emphasis was also placed on incorporating land-to-water transition zones in the $100$ hand-picked scenarios, acknowledging these as inherently more varied scenarios.

To evaluate the realistic nature of the RFView^® simulator, we compare it with the MCARM measured dataset, focusing on Flight #5 Acquisition 575 (Babu et al., 1996). The MCARM radar parameters are tabulated in Section B of the Appendix. Let $\smash{Z^{i}\in\mathbb{C}^{\tilde{\Lambda}\tilde{L}\tilde{G}}}$, for $\smash{i\in\hat{S}=\{\text{MCM, RF, BE}\}}$, denote the clutter random variables for the MCARM, RFView^®, and bald-earth (BE) models, wherein the bald-earth model is a more classical clutter model without terrain heights. For the RFView^® and BE models, we generate the clutter datasets $\smash{\{z^{\text{RF}}_{j},\,\forall j\in D\}}$ and $\smash{\{z^{\text{BE}}_{j},\,\forall j\in D\}}$, which we then reshape to obtain $\smash{\mathbf{Z}^{\text{RF}},\mathbf{Z}^{\text{BE}}\in\mathbb{C}^{\tilde{\Lambda}\tilde{L}\tilde{G}\times K}}$. The MCARM Flight #5 Acquisition 575 provides a single clutter realization, $\smash{z^{\text{MCM}}_{1}}$, which we reshape to form $\smash{\mathbf{Z}^{\text{MCM}}\in\mathbb{C}^{\tilde{\Lambda}\tilde{L}\tilde{G}\times 1}}$.

We produce range-Doppler plots for both the RFView^® simulated data and the MCARM measured data by performing Doppler processing via the Fast Fourier Transform (FFT), coherently integrating across antenna elements. The range-Doppler plots for the MCARM, RFView^®, and BE models are shown in Figure 2. We clearly observe that the RFView-generated data exhibits a higher degree of similarity to the MCARM measured data than the classical BE model.

To validate the similarity between the RFView^® simulated data and the MCARM measured data, we perform a modified version of the likelihood-based test from (Johnson & Abramovich, 2010). Let $\smash{\mathbf{Z}^{\text{MCM}}[l,g]\in\mathbb{C}^{\tilde{L}\times 1}}$, $\smash{\mathbf{Z}^{\text{RF}}[l,g]}$, and $\smash{\mathbf{Z}^{\text{BE}}[l,g]\in\mathbb{C}^{\tilde{L}\times K}}$ form the MCARM, RFView^®, and bald-earth clutter realizations, for the $l$-th Doppler bin and $g$-th range bin. The null hypothesis, $H_{0}$, asserts that $\mathbf{Z}^{\text{MCM}}[l,g]$ is from the same distribution as $\mathbf{Z}^{\text{RF}}[l,g]$ — we repeat this procedure for $\mathbf{Z}^{\text{BE}}[l,g]$). Next, $\forall l,g$, we compare the likelihood of the MCARM clutter realization to the empirical distribution of likelihoods from the RFView^® clutter realizations. We let $\mathbf{\Sigma}_{\text{RF}}[l,g]$ denote the sample covariance matrix for the RFView case, where the likelihood of the MCARM data given $\mathbf{\Sigma}_{\text{RF}}[l,g]$, and the likelihood of the $j^{\text{th}}$ RFView clutter realization given $\mathbf{\Sigma}_{\text{RF}}[l,g]$, are given by:

To determine if the real-world data is consistent with the simulated data distribution, we compare $\mathcal{L}_{\text{MCM}}[l,g]$ to the empirical distribution of $\{\mathcal{L}_{\text{RF},j}[l,g],\ \forall j\in D\}$. We form the $95\%$ confidence interval from the percentiles of $\mathcal{L}_{\text{RF},j}[l,g]$. If $\mathcal{L}_{\text{MCM}}[l,g]$ falls within this confidence interval, we fail to reject $H_{0}$. We compute the total fraction of how often the MCARM likelihood falls in the confidence interval over all Doppler and range bins, denoted $\rho\in[0,1]$:

In the RFView^® case ($\mathbf{Z}^{\text{RF}}[l,g]$), we obtain $\rho=0.1969$ for the real part of $\mathcal{L}_{\text{MCM}}[l,g]$, $\rho=0.436$ for the imaginary part of $\mathcal{L}_{\text{MCM}}[l,g]$. In the bald-earth case ($\mathbf{Z}^{\text{BE}}[l,g]$), we obtain $\rho=0$ for both the real and imaginary parts of $\mathcal{L}_{\text{MCM}}[l,g]$. This study shows that while RFView^® does not offer a complete match with the real-world data, it is more realistic than the bald-earth model. Furthermore, this study suggests that for real-world applications, it is feasible to train networks on RASPNet scenarios, and fine tune them using real-world samples. We explore this further in Section 4.3.

To prepare RASPNet for the target localization and transfer learning tasks, we consider sub-regions from the scenarios, $\smash{i\in\tilde{S}}$, which we call radar processing regions. The minimum and maximum range of each radar processing region from the airborne radar platform and the azimuth angle bounds of each radar processing region are provided in Table 3 (see Appendix Section C.1), and the range bin indices are $g\in\{75,76,\ldots,95\}\subset\{1,2,\ldots,G\}$, whereby $G^{\prime}=21$. We run a set of $N^{\prime}=1000$ independent experiments, where in each experiment, a point target is placed in range and azimuth, uniformly at random inside each radar processing region. We select the radar cross section (RCS) of each target independently at random from the uniform distribution, $\mathcal{U}[15,25]$ (dBsm), and we generate $K^{\prime}=100$ independent random realizations of the radar return for each target placement. We consider stationary targets in this example, with $\Lambda^{\prime}=1$, wherein there is no Doppler processing. For compatibility with our generated target data, we divide the $K=10{,}000$ clutter realizations provided by RASPNet for scenario $\smash{i\in\tilde{S}}$ into $N^{*}=100$ separate clutter datasets with $K^{\prime}=100$ realizations, where $K=N^{*}K^{\prime}$. As $\Lambda^{\prime}=1$, we only consider the first pulse from the clutter realizations. For the CVNN target localization benchmark task, we solely consider the first $5$ of the $100$ realizations, wherein $K^{\prime}=5$. Using this construction, we now consider the signal model outlined in Section C.2 of the Appendix.

For both non-CVNN target localization and transfer learning tasks, we leverage datasets of heatmap matrices (Venkatasubramanian et al., 2023), which comprise test statistics obtained via the signal model from Appendix Section C.2, where $\phi=0$. To produce each heatmap matrix, we sweep the radar steering vector, $\mathbf{\tilde{a}}(\theta,0,0)$, across azimuth with step size $\Delta\theta=0.4^{\circ}$, from $\theta_{\text{min}}$ to $\theta_{\text{max}}$, producing an NAMF azimuth spectrum at each range bin. Arranging these spectra across the $G^{\prime}$ range bins gives a size $G^{\prime}\times(\lfloor\frac{\theta_{\text{max}}-\theta_{\text{min}}}{\Delta\theta}\rfloor+1)$ matrix of NAMF test statistics. The values for $\theta_{\text{min}}$ and $\theta_{\text{max}}$ are in Table 3 for scenarios $\smash{i\in\tilde{S}}$. The cumulative dataset contains $N=N^{\prime}N^{*}=100{,}000$ heatmap matrices, formed using all combinations of $\mathbf{X}^{i}[g]$ and $(\mathbf{Z}^{i}[g]+\mathbf{A}^{i}[g])$. For the CVNN target localization task, we consider $3$-dimensional feature vectors, which are formed from the radar return data matrices described in Section C.2 of the Appendix, where $\phi=0$. The cumulative dataset comprises the first $25{,}000$ feature vectors of the $100{,}000$ total combinations.

Having obtained the datasets of heatmap matrices for scenarios $\smash{i\in\tilde{S}}$, we benchmark the non-CVNN target localization performance of two existing algorithms — we consider the classical peak-cell midpoint algorithm outlined in (Venkatasubramanian et al., 2022), and compare it with the regression CNN architecture introduced in (Venkatasubramanian et al., 2023). We clarify that other classical algorithms exist for target localization, including gradient-based local search methods; these algorithms can also be benchmarked using RASPNet. The considered regression CNN architecture is further described in Section D.1 of the Appendix.

We now produce the training and test datasets for each scenario, $\smash{i\in\tilde{S}}$. We partition each dataset of $100{,}000$ heatmap matrices such that the first $N_{train}=0.9N$ examples form the training dataset, and the remaining $N_{test}=0.1N$ examples form the test dataset. Let $(\mathbf{\bar{\text{$r$}}}\vphantom{r}_{t}^{i},\mathbf{\bar{\text{$\theta$}}}\vphantom{\theta}_{t}^{i})$ denote the range and azimuth of the midpoint of the peak heatmap matrix cell, and let $\smash{(\mathbf{\tilde{\text{$r$}}}\vphantom{r}_{t}^{i},\mathbf{\tilde{\text{$\theta$}}}\vphantom{\theta}_{t}^{i})}$ denote the azimuth and velocity values predicted by the regression CNN, for example $t$ from our dataset for scenario $i$. While our heatmap matrices follow the polar coordinate system, we can transform their ground truth target locations into Cartesian coordinates to report the localization error in meters, whereby $\smash{(\mathbf{\bar{\text{$r$}}}\vphantom{r}_{t}^{i},\mathbf{\bar{\text{$\theta$}}}\vphantom{\theta}_{t}^{i})\rightarrow(\mathbf{\bar{\text{$x$}}}\vphantom{x}_{t}^{i},\mathbf{\bar{\text{$y$}}}\vphantom{y}_{t}^{i})}$ and $\smash{(\mathbf{\tilde{\text{$r$}}}\vphantom{r}_{t}^{i},\mathbf{\tilde{\text{$\theta$}}}\vphantom{\theta}_{t}^{i})\rightarrow(\mathbf{\tilde{\text{$x$}}}\vphantom{x}_{t}^{i},\mathbf{\tilde{\text{$y$}}}\vphantom{y}_{t}^{i})}$. Let $(r_{t}^{*i},\theta_{t}^{*i})\rightarrow(x_{t}^{*i},y_{t}^{*i})$ be the ground truth target location and velocity for example $t$ from our dataset. For vector inputs $\mathbf{s}_{t}^{*i},\mathbf{\hat{\text{$\mathbf{s}$}}}\vphantom{s}_{t}^{i}$, the localization error, $Err(\mathbf{s}_{t}^{*i},\mathbf{\hat{\text{$\mathbf{s}$}}}\vphantom{s}_{t}^{i})$, over the $N_{test}$ test examples, is defined as:

Regarding our case, $(\mathbf{s}\vphantom{s}_{t}^{*i},\mathbf{\hat{\text{$\mathbf{s}$}}}\vphantom{s}_{t}^{i})=([x_{t}^{*i},y_{t}^{*i}],[\mathbf{\hat{\text{$x$}}}_{t},\mathbf{\hat{\text{$y$}}}_{t}])$, with $(\mathbf{\hat{\text{$x$}}}\vphantom{x}_{t}^{i},\mathbf{\hat{\text{$y$}}}\vphantom{y}_{t}^{i})=(\mathbf{\bar{\text{$x$}}}\vphantom{x}_{t}^{i},\mathbf{\bar{\text{$y$}}}\vphantom{y}_{t}^{i})$ for the peak cell midpoint $[Err_{\text{NAMF}}]$, and $(\mathbf{\hat{\text{$x$}}}\vphantom{x}_{t}^{i},\mathbf{\hat{\text{$y$}}}\vphantom{y}_{t}^{i})=(\mathbf{\tilde{\text{$x$}}}\vphantom{x}_{t}^{i},\mathbf{\tilde{\text{$y$}}}\vphantom{y}_{t}^{i})$ for the regression CNN $[Err_{\text{CNN}}]$. For each scenario, $\smash{i\in\tilde{S}}$, we train the regression CNN until convergence on the training dataset using MSE as our loss function, and we leverage the Adam optimizer (Kingma & Ba, 2014) with a learning rate of $\alpha=1\text{e-}3$. The localization errors on the test dataset are summarized for the peak cell midpoint and the regression CNN in Table 1. We see that as the scenarios become more challenging — measured using the $\mathcal{E}$-statistic with respect to the baseline scenario — the localization performance of both methods worsen.

Apart from benchmarking the performance of algorithms, RASPNet can also be leveraged for transfer learning applications in realistic adaptive radar processing scenarios. We recall that the RFView^®-generated data exhibits some degree of statistical similarity to real-world data, as examined in Section 2.4. As such, suppose we have collected $K$ clutter realizations, $\{z_{j}^{\omega},\ \forall j\in D\}$ from some real-world scenario, $\omega$, where $z_{j}^{\omega}\sim Z^{\omega}$. Suppose we also have $100$ neural networks trained for some predictive adaptive radar processing task on each of the $100$ RASPNet scenarios, $i\in S$. Our aim is to determine which of the $100$ trained neural networks should be chosen to achieve the best possible performance for the same task on the real-world scenario, $\omega$. We can proceed by ranking the RASPNet scenarios, $i\in S$, by their similarity to scenario $b=\omega$, using Algorithm 2. We hypothesize that the scenario with the lowest $\mathcal{E}$-statistic, relative to the real-world scenario, is the optimal choice for selecting the corresponding trained neural network. To improve the performance of this trained neural network on the real-world scenario, we can use a small number of samples from the real-world scenario to fine-tune it for the task at hand.

As a proof-of-concept, we again consider the non-CVNN target localization task, where we train our regression CNN to localize targets in scenario $i=29$ (Bonneville Salt Flats, UT). However, we evaluate this trained network on each of the scenarios $\smash{i\in\tilde{S}\setminus\{29\}}$ to validate whether the scenario with the highest similarity to scenario $i=29$ from $\smash{\tilde{S}\setminus\{29\}}$ achieves the best localization performance. Paralleling Section 4.2, our training dataset comprises the first $0.9N$ of the $100{,}000$ heatmap matrices generated for scenario $i=29$. We consider four test datasets, each of which comprises the last $0.1N$ of the $100{,}000$ total heatmap matrices generated for scenario $\smash{i\in\tilde{S}\setminus\{29\}}$, and we consider four datasets for fine-tuning, each of which comprises the first $64$ of the $100{,}000$ total heatmap matrices generated for scenario $\smash{i\in\tilde{S}\setminus\{29\}}$. We train the regression CNN on the training dataset using MSE as our loss function, and we leverage the Adam optimizer (Kingma & Ba, 2014) with $\alpha=1\text{e-}3$. The localization accuracies on the test datasets, $[Err_{\text{CNN}}]_{\text{TL}}$, $\smash{\forall i\in\tilde{S}\setminus\{29\}}$, are depicted in Figure 5. As expected, the best performance is yielded by the neural network evaluated on the scenario $i=60$ (Central Kansas). This scenario has the lowest $\mathcal{E}$-statistic with respect to $i=29$ from $\smash{\tilde{S}\setminus\{29\}}$.

We now consider the case of fine-tuning the trained regression CNN using a small set of $64$ heatmap matrices from the scenarios $\smash{i\in\tilde{S}\setminus\{29\}}$. We freeze the convolutional and batch normalization layers of our trained regression CNN, halving the number of parameters. Using the $64$ heatmap matrices, the weights and biases of the remaining layers are fine-tuned using MSE Loss and the Adam optimizer with $\alpha=1\text{e-}3$. After fine-tuning the regression CNN for each scenario $\smash{i\in\tilde{S}\setminus\{29\}}$, we obtain the localization accuracies on the test datasets, $[Err_{\text{CNN}}]_{\text{FT}}$, $\smash{\forall i\in\tilde{S}\setminus\{29\}}$, depicted in Figure 5. We note that the improvements afforded by the regression CNN are largely recovered after fine-tuning.