Physics-Informed Anomaly Detection of Terrain Material Change in Radar Imagery

PHYSICS-INFORMED ANOMAL Y DETECTION OF TERRAIN MA TERIAL CHANGE IN RAD AR IMA GER Y Abdel Hakiem Mohamed Abbas Mohamed Ahmed 1 , 2 , Beth J elfs 1 , Airlie Chapman 2 , Eric Schoof 3 , Christopher Gilliam 1 1 School of Engineering, Uni versity of Birmingham, UK 2 Department of Mechanical Engineering, Uni versity of Melbourne, Australia 3 Department of Electrical and Electronic Engineering, Uni versity of Melbourne, Australia Email: axm1614@student.bham.ac.uk, {b .jelfs, c.gilliam.1}@bham.ac.uk, {airlie.chapman, eschoof}@unimelb .edu.au ABSTRA CT In this paper we consider physics-informed detection of terrain mate- rial change in radar imagery (e.g., shifts in permittivity , roughness or moisture). W e propose a lightweight electromagnetic (EM) forward model to simulate bi-temporal single-look complex (SLC) images from labelled material maps. On these data, we derive physics- aware feature stacks that include interferometric coherence, and ev aluate unsupervised detectors: Reed-Xiaoli (RX)/Local-RX with robust scatter (T yler’ s M-estimator), Coherent Change Detection (CCD), and a compact con volutional auto-encoder . Monte Carlo experiments sweep dielectric/roughness/moisture changes, number of looks and clutter regimes (gamma vs K-family) at fixed proba- bility of false alarm. Results on synthetic b ut physically grounded scenes show that coherence and rob ust covariance markedly improv e anomaly detection of material changes; a simple score-le vel fusion achiev es the best F1 in heavy-tailed clutter . Index T erms — Change detection, RX detector , integral equa- tion model (IEM), terrain materials, anomaly detection. 1. INTR ODUCTION Detecting subtle, spatially localised changes in terrain materials from radar imagery underpins applications in infrastructure moni- toring, en vironmental surv eillance and security [1, 2, 3]. In coherent radar , long-standing strategies for detecting changes span intensity- based methods and coherent change detection (CCD), which ex- ploits interferometric coherence estimated between phase-registered image pairs with material or structural changes reducing the co- herence magnitude [4, 5, 6]. While state-of-the-art deep learning models have advanced bi-temporal change mapping, most works target pix el stacks statistically , with limited explicit ties to the EM material properties that actually derive radar backscatter and decor- relation [7, 8]. The radar backscatter obtained from terrain depends on a num- ber of factors, such as the comple x permitti vity and surface rough- ness parameters of the terrain, as well as the radar geometry (inci- dence angle). W idely used rough-surface models such as the Inte- gral Equation Model (IEM/AIEM) provide v alidated relationships between these parameters and the radar backscatter [2, 9], whereas, popular anomaly detectors like the Reed-Xiaoli (RX) score assume a (often Gaussian) background distribution and flag departures from this via Mahalanobis distance [10]. In radar imagery , heavy-tailed clutter is common, so a fixed global threshold does not preserve a stable false alarm rate across heterogenous backgrounds; using ro- bust covariance estimators (e.g., T yler’ s M-estimator) mak es detec- tors more CF AR-like, i.e., the decision threshold adapts to local clut- ter statistics so the false alarm probability remains approximately constant across the scene [3, 11]. Despite the dif ferent approaches to anomaly detection in radar imagery , there remains a lack of reproducible, controlled studies that link material-lev el changes (e.g., moisture-driven permitti v- ity or roughness changes) to expected changes in backscatter and coherence, and the relativ e performance of RX/robust-RX, CCD and modern unsupervised models. Material change (e.g., dry → wet soil, asphalt ageing, v egetation remov al) alters permittivity and roughness, shifting Fresnel/rough-surface reflecti vity and speckle statistics; simultaneously , coherence decreases in changed regions. Thus, coherence-aware features and robust background modelling are well-matched not only to radiometric change b ut also to material change [2, 3, 5]. In this work, we address the problem of terrain material change by proposing a physics-informed anomaly detection frame work. W e use a compact, IEM-inspired forward model to map labelled mate- rial maps to bi-temporal single look complex (SLC) images. Per- pixel parameters for permittivity , rms height, correlation length and incidence angle determine mean backscatter; multiplicativ e speckle and controlled cross-epoch correlation generate SLC pairs that emu- late decorrelation for genuine material/structural change [2, 9]. From these images, a ph ysics-aware feature stack is estimated in local win- dows combining log -intensities, simple texture, incidence angle, and interferometric coherence ˆ γ [5]. W e then perform a comparative study of unsupervised detectors to highlight regimes where ph ysics- aware features and robust cov ariance are decisi ve. W e consider both global and local RX with a robust scatter estimator for heavy-tailed clutter [3, 10, 11]; CCD via decorrelation [6]; and a lightweight con- volutional auto-encoder trained on unchanged tiles [7, 12]. The remainder of this paper is structured as follo ws: Sec- tion 2 details the proposed forward model and Section 3 describes the features and detectors. In Section 4 we outline the datasets used and the Monte-Carlo protocol which cov ers changes in per- mittivity , rms height, number of looks and Signal to Noise Ratio (SNR). Section 5 presents results and ablation data reporting Re- ceiv er Operating Characteristic-Area Under the Curve (R OC-A UC), A verage Precision (AP) and F1 score at low Probability of False Alarm (PF A). Section 6 concludes the paper , highlighting that on © IEEE 2026 Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in an y current or future media. synthetic, but physically grounded data, coherence-aware methods decisiv ely outperform traditional intensity-based detectors, with a simple score-lev el fusion achieving the highest performance. 2. PHYSICS-INFORMED FOR W ARD MODEL In this section, we describe our SLC image formation framew ork using a physics-informed forward model. In our model we assume a monostatic, side-looking geometry with a kno wn radar pose. A base digital elevation model (DEM) provides local surface normal n ( r ) (with r the geospatial location in the DEM/frame); the local incidence angle is θ ( r ) = arccos( ˆ k · n ) , where ˆ k is a unit vector denoting the look direction. A labelled material map assigns, per pixel p , complex permittivity ε r,p ( f ) , where f is the radar operating frequency and roughness parameters: rms height σ p and correlation length l c,p . These parameters control microw ave backscatter accord- ing to rough-surface EM models such as IEM/AIEM [2, 9, 13, 14]. The exact IEM/AIEM backscatter requires spectral integrals and regime checks [9]. Therefore to allo w large-scale Monte-Carlo stud- ies, we use a lightweight surrogate that preserves the main depen- dencies [15, 16]: σ 0 p ( θ ) ≈ F  ε r,p , θ  G  θ , σ p , l c p , f  , (1) where F is a Fresnel term and G is a roughness attenuation inspired by IEM/AIEM. For a single polarisation (e.g., v ertical co-planar, VV), we take: F ( ε r , θ ) =      ε r cos θ − p ε r − sin 2 θ ε r cos θ + p ε r − sin 2 θ      2 , (2) and a smooth attenuation such as G ( θ , σ , l c , f ) = exp  − (2 k σ cos θ ) 2  ϕ ( l c ) , k = 2 πf /c, (3) with ϕ ( l c ) a bounded, monotone factor capturing correlation-length effects. Although (1) is not a substitute for AIEM, it tracks the trend that higher ℜ{ ε r } (or moisture) and appropriate roughness alter σ 0 systematically [2, 9]. Where fidelity is crucial, we sw ap G for the closed-form IEM terms within a restricted validity re gime [13, 14]. 2.1. SLC image formation Under fully dev eloped speckle, a SLC pixel is modelled as S ( t ) p = q σ 0 , ( t ) p η ( t ) p e j ϕ ( t ) p , (4) where η is circular complex Gaussian (unit mean intensity) and ϕ in- cludes deterministic phase from geometry . L -look intensities follow a Gamma distribution; heterogeneous terrain often exhibits heavy- tailed, compound-Gaussian (e.g., K ) statistics, impacting CF AR be- haviour [3]. Between two epochs ( t 1 , t 2 ) , we synthesise correlated speckle via: W 2 = γ W 1 + p 1 − | γ | 2 W ⊥ , W 1 , W ⊥ ∼ C N (0 , 1) , (5) using a spatially v arying coherence parameter γ ( r ) and where C N is the complex normal distribution. In unchanged areas | γ | ≈ 1 (af- ter coregistration), whilst material/structural changes reduce | γ | [5]. The sample coherence ov er a window W is: ˆ γ = P p ∈W S ( t 1 ) p S ( t 2 ) p ∗ r P p ∈W    S ( t 1 ) p    2 r P p ∈W    S ( t 2 ) p    2 , (6) with bias/variance properties detailed in [5]. For a CCD baseline, we also consider the maximum-lik elihood change statistic of [6], which offers a CF AR-like behaviour with respect to clutter-to-noise ratio. Material change is injected by modifying { ε r , σ, l c } within se- lected regions (e.g., dry → wet soil, asphalt → gravel). The simulator outputs paired SLCs, intensities, and auxiliary maps (incidence, slope). Limitations (discussed in Section 6) include the single- frequency , single-pol default and omission of volume scattering and layover; nonetheless, the setup suffices to study how material perturbations propagate to intensity/coherence statistics [2, 9]. 3. FEA TURES AND DETECTORS W e now detail the features and detectors we will use to study the anomaly detection problem. 3.1. Physics-awar e feature stack T o obtain a representati ve feature stack, given SLCs  S t 1 , S t 2  we form: I 1 =   S t − 1   2 , I 2 =   S t 2   2 , R log = log I 2 + ϵ I 1 + ϵ , texture : { µ, σ 2 } ov er N × N , θ (incidence) , | ˆ γ | from (6) . (7) These features jointly encode radiometry , local heterogeneity , ge- ometry and coherence, all of which respond to ∆ ε r and roughness changes [2, 5]. 3.2. Detectors In all of our analysis we standardise indi vidual scores from each of the detectors and compute a fused score by a weighted sum or a lo- gistic regressor fitted on a small validation subset. Operating points are selected by fixing a global PF A (e.g., 10 − 3 ) using background pixels. Isolated f alse alarms are remov ed via morphological open- ing/closing. Reporting follows standard detection practice: R OC- A UC, AP and F1 at fixed PF A [3]. RX and Local-RX anomaly detection Stacking features x ∈ R d , the RX score is the Mahalanobis distance: D 2 ( x ) = ( x − µ ) ⊤ Σ − 1 ( x − µ ) , (8) with ( µ, Σ) estimated from a background set; Local-RX uses a slid- ing neighbourhood to adapt to non-stationary en vironments [10]. Under a Gaussian background and known Σ , D 2 ∼ χ 2 D , enabling CF AR thresholding; in practice, estimation error and non-Gaussian clutter require robustification. Robust scatter for heavy-tailed clutter Radar clutter is fre- quently heavy-tailed (Gamma / K − distrib uted intensities), violating Gaussian assumptions [3]. W e therefore replace the sample co vari- ance in (8) by T yler’ s M-estimator [11], which is distribution-free within the complex elliptically symmetric family: Σ ← d n n X i =1 ( x i − µ )( x i − µ ) ⊤ ( x i − µ ) ⊤ Σ − 1 ( x i − µ ) , with tr(Σ) = d, (9) where d is the number of features in x and tr( · ) is the trace operator . In our experiments, we use a regularised form: 30 fix ed-point iter- ations with 5% ridge shrinkage tow ards a scaled identity for small- sample conditioning. This choice is standard for heavy-tailed radar clutter and helps maintain CF AR-like behaviour . 2 Fig. 1 . Representative trial - feature maps. Left → right: log- I 1 , log- I 2 , coherence magnitude | ˆ γ | , ground truth (GT). Fig. 2 . Representative trial - detector score maps. Left → right: RXrob, Local-RX, CCD ( 1 − | ˆ γ | ). T able 1 . Global parameters used in the simulations. Parameter Range Complex SNR 12 - 28 dB Compound texture shape ν ∈ [0 . 3 , 1 . 2] Residual co-registration jitter σ xy ∈ [0 . 08 , 0 . 25] px Residual phase jitter σ ϕ ∈ [0 . 10 , 0 . 30] rad V egetation/volume fraction 0 . 08 - 0 . 18 Unchanged coherence magnitude | γ bg | ∈ [0 . 90 , 0 . 95] Changed coherence magnitude | γ chg | ∈ [0 . 45 , 0 . 65] Looks L ∈ { 2 , . . . , 8 } Coherent Change Detection (CCD) CCD treats change as decor- relation. The simplest score is: CCD( p ) = 1 − | ˆ γ ( p ) | , (10) thresholded to control PF A. W e also report the maximum-likelihood CCD statistics of [6], which exploits finite-look statistics and ex- hibits CF AR properties in speckle. Unsupervised Auto-Encoder (AE) A compact conv olutional AE operates on feature tiles (e.g., 32 × 32 × d ). Trained only on un- changed tiles (from t 1 or masked stable regions), it minimises recon- struction loss ∥ ˆ X − X ∥ 2 2 and yields an anomaly score e = ∥ ˆ X − X ∥ at test time. Surveys document strong change detection performance from Siamese/AE families, while noting domain shift and limited physical interpretability [8, 12, 17]. 4. SIMULA TIONS 4.1. Simulation setup W e run N =200 Monte Carlo trials on 256 × 256 scenes. Each trial samples global nuisance parameters uniformly within physically credible ranges to emulate heterogeneous terrain and acquisition imperfections, see T able 1. The compound texture shape allo ws interpolation between heavy-tailed K -family and multilook-Gamma clutter and unchanged coherence has an additional decorrelation factor 0 . 75 - 0 . 90 to represent natural variation. Incidence varies smoothly around 35 ◦ to reflect a side-looking geometry . These factors follo w established clutter/CF AR consideration in SAR tar get detection under non-Gaussian statistics [3]. Material change is injected by perturbing per-pixel ( ε r , σ, l c ) within a compact region (e.g., square/rectangle/ellipse). Mean backscatter σ 0 ( θ ) is generated by an IEM-inspired surrogate that preserves the Fresnel and roughness/geometry dependencies; where appropriate, closed-form IEM/AIEM components are used within their validity regimes [2, 9, 13, 14]. This ties simulated radiometry to dielectric/roughness changes, such as dry → wet soil or asphalt ageing [2]. Paired SLCs  S ( t 1 ) , S ( t 2 )  are formed with multiplicativ e speckle and optional compound texture. Cross-epoch speckle is correlated using (5) to provide spatially varying coherence fields; sample coherence ˆ γ is estimated in a 7 × 7 boxcar , following clas- sical bias/variance analysis [5]. In addition to 1 −| ˆ γ | , we include the maximum-likelihood CCD statistics of [6] as a likelihood-based comparator . 4.2. Detectors, training and fusion W e ev aluate: global RX [10]; robust-RX using T yler’ s M -estimator with light shrinkage for heavy-tailed background [11, 18, 19]; Local- RX with a dual ring (outer/guard windows 21 / 9) ; CCD ( 1 −| ˆ γ | ) [6]; and a compact con volutional autoencoder (AE) trained on un- changed tiles (patch 16 50 epochs), reflecting unsupervised/deep change-detection practice [7, 8, 12]. Scores are z -normalised and fused either equally or with two/three-way weights learned on a 10% calibration subset per trial. For a representative qualitativ e figure, we select the single trial with the highest visibility (coherence-dominated separation) and render feature/score maps with the ground truth contour , together with ROC/PR curves. The looks-dependence of coherence variance explains the strong ef fect of L on visual separability [5]. 5. RESUL TS AND DISCUSSION The results for a representati ve trial are sho wn in Figs. 1, 2, and 3. Figure 1 illustrates the log-intensity at t 1 and t 2 , the coherence mag- 3 T able 2 . Aggregate detection performance (mean ± 95% CI ov er N =200 trials). Method R OC-A UC AP (PR-A UC F1 RX (global) 0 . 775 [0 . 761 , 0 . 788] — —- RXrob (T yler) 0 . 775 [0 . 760 , 0 . 788] 0 . 157 [0 . 139 , 0 . 180] 0 . 086 [0 . 071 , 0 . 106] Local-RX 0 . 489 [0 . 486 , 0 . 493] 0 . 019 [0 . 018 , 0 . 020] 0 . 001 [0 . 001 , 0 . 002] CCD 0 . 901 [0 . 891 , 0 . 910] 0 . 401 [0 . 371 , 0 . 432] 0 . 304 [0 . 276 , 0 . 332] FUSE (equal 2-way) 0 . 891 [0 . 880 , 0 . 899] — — FUSEw (learned 2-way) 0 . 901 [0 . 891 , 0 . 910] 0 . 399 [0 . 368 , 0 . 429] 0 . 302 [0 . 275 , 0 . 332] FUSE3w (learned 3-way) 0 . 901 [0 . 891 , 0 . 910] 0 . 390 [0 . 359 , 0 . 420] 0 . 289 [0 . 263 , 0 . 317] AE (compact) 0 . 645 [0 . 633 , 0 . 657] 0 . 054 [0 . 048 , 0 . 062] 0 . 014 [0 . 012 , 0 . 018] Fig. 3 . Representativ e trial. ROC computed on the same trials as Figs. 1 2 nitude and the ground-truth mask. Figure 2 shows the corresponding score maps for RXrob, Local-RX, and CCD. Finally , Fig. 3 sho ws the R OC results for the trial. These results highlight that the change region is more identifiable in detectors that rely on interferometric coherence. It can be seen that the changed re gion is more prominent in the interferometric coherence | ˆ γ | in Fig. 1 compared to the log- intensity images and this directly translates to the output of the de- tectors; the region is easily seen in the CCD output in Fig 2 whereas it is very subdued in the Local-RX output. This observation is to be expected as under stable geometry , material/structural change chiefly reduces interferometric coherence | ˆ γ | , which CCD tar gets directly , whereas modest radiometric contrast can be masked by hea vy-tailed clutter , and Local-RX’ s ring estimates can be contaminated in het- erogeneous neighbourhoods. T able 2 summarises the aggregate performance across the Monte Carlo simulations. Coherence-centric detectors (CCD and simple fusion) consistently lead in ROC-A UC and on the class imbalance- aware metric (AP and F1 at PF A = 10 − 3 ). In contrast, intensity-led baselines (RX/Local-RX) do not perform well, with robust scatter giving RX some stability b ut not closing the gap when decorrelation is the principal cue. These trends agree with the variance properties of the coherence estimator [5]. In more detail, CCD and coherence weighted fusion score best due to the injected material/structural changes mainly reducing co- herence magnitude | γ | (decorrelation) whilst radiometric contrast is comparativ ely modest and partly masked by compound Gaussian clutter . Robust cov ariance helps RX primarily through CF AR-like stabilisation rather than separability gains. This matches InSAR the- ory on decorrelation sources and coherence estimator beha viour [1, 4, 5] and the need for robust statistics in heavy-tailed radar back- grounds [3, 11, 18, 19]. Finally , the low performance of Local-RX is due to it assuming locally stationary background statistics. W ith veg- etation and spatially varying coherence, the dual ring configuration used in Local-RX often mixes populations (changed vs unchanged), contaminating the local mean/covariance and destabilising the com- putation of in verses for a small window , which explains the weak separation in T able 2. Similar sensitivity to window design and het- erogeneity are reported in anomaly detection studies [20, 21, 3]. On a verage our lightweight score fusion is equiv alent to CCD. In coherence-dominated re gimes, adding RX/AE contributes a lim- ited complementary signal and can actually add variance. Fusion becomes more beneficial when both radiometric and coherent cues carry information (e.g., larger ∆ σ 0 at higher equiv alent number of looks). Consistent with the bias/variance trade off of ˆ γ [5] and CCD practice [6]. In terms of sensiti vity to the simulation parameters the follo wing trends were observed: • Looks ( L ): higher L lowers the variance ˆ γ , improving CCD and fused metrics [5]. • T exture shape ( ν ): heavier tails degrade RX/local-RX more than CCD, motiv ating robust scatter [3, 11, 18, 19]. • Residual phase/co-registration: larger residuals depress background | γ | , narrowing separation and reducing CCD headroom, underscoring the need for precise co-registration [1, 4]. It should be noted that the simulator is single-polarisation and single-frequency by default and omits explicit v olume/layer and po- larimetric effects; nevertheless, it captures first-order links between ( ε, σ, l c ) and { σ 0 , | γ |} . Extending to PolSAR w ould enable Wishart- family change tests and omnibus statistics as likelihood compara- tors [22, 23, 24], and comparing recent SAR-specific deep change detectors would strengthen external v alidity [25, 26]. 6. CONCLUSION This paper presented a physics-informed frame work for detect- ing terrain material change in radar imagery , combining an IEM- inspired forward model with a coherence-aware feature stack and unsupervised detectors (RX/robust-RX, CCD, AE). In controlled Monte Carlo experiments spanning non-Gaussian clutter , residual co-registration/phase errors and looks sweep, coherence-centric methods dominated: CCD and a simple score level fusion achieved the highest PR-A UC and F1 at a fixed PF A ( 10 − 3 ). Robust scat- ter (T yler) stabilised RX in hea vy-tailed backgrounds, but RX lagged whenever decorrelation was the principal cue. These find- ings align with the physics backscatter and decorrelation [9, 2, 5], with likelihood-based CCD analysis [6], and with the role of robust statistics under heavy-tailed radar clutter [10, 11, 3]. 4 7. REFERENCES [1] P . A. Rosen et al. , “Synthetic aperture radar interferometry , ” Pr oc. IEEE , vol. 88, no. 3, pp. 333–382, 2000. [2] F . T . 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