Quantitative validation of NASA ARIA damage proxy maps for detection of ground displacement from surface fault rupture from the 2019 Ridgecrest earthquake sequence

Abstract

National Aeronautics and Space Administration Advanced Rapid Imaging Analysis (ARIA) Damage Proxy Maps (DPMs) are developed by the NASA Jet Propulsion Laboratory (NASA JPL) to identify potentially damaged areas based on interferometric coherence loss in Synthetic Aperture Radar (SAR) data. DPMs are typically based on data from Sentinel-1 satellites that orbit every 12 days, meaning that results can be provided within 1-2 weeks of an event. Although DPMs have been qualitatively validated as being able to detect surface effects of earthquakes, quantitative validations of their ability to differentiate damaged from undamaged areas and different types and levels of surface effects are lacking. We propose a framework for quantitative validation and apply it to surface fault rupture data from the 2019 Ridgecrest Earthquake sequence. The quantitative analyses take two forms: (1) the statistical distribution of a DPM index ( $I_{\mathrm{DPM}}$ ) for different fault displacement ranges through box and whisker plots, and (2) empirical fragility functions that relate $I_{\mathrm{DPM}}$ to probabilities of displacements exceeding certain thresholds. These relationships are developed for DPM1 (derived from one pre-event pair and one co-event pair of SAR images) and DPM2 (derived from multiple pre-event and co-event pairs of SAR images). We show that both DPM types perform similarly well for distinguishing between no surface displacement and some surface displacement. The predictive power of $I_{\mathrm{DPM}}$ metrics, as measured by a dispersion term in the fragility model, shows the best performance for low surface fault displacements and DPM2-based indices. Recall and precision performance metrics show favorable performance of fragility models for identifying locations of fault displacement but increasing rates of false positives as fault displacement increases (i.e. predictions of displacements exceeding a threshold that did not occur). While this validation study was performed for a single earthquake sequence involving a specific area of California, these results demonstrate both the capabilities and limitations of $I_{\mathrm{DPM}}$ as a rapid, post-event predictive tool for identifying locations and severity of ground displacements in environments similar to those in the Ridgecrest area (flat terrain, limited vegetation).

Introduction

This study examines relationships between National Aeronautics and Space Administration Advanced Rapid Imaging Analysis (ARIA) Damage Proxy Maps (DPMs) and different levels of surface displacement following a major earthquake event. Since being developed in the early 2000s, DPMs have been utilized to rapidly detect “changes” on the surface of the earth following natural and anthropogenic disasters (Yun et al., 2015). Our aim is to quantitatively link, for the first time, the numerical index that is the basis for the maps to actual ground displacements.

DPMs are formulated with Synthetic Aperture Radar (SAR) data. Interferometric analyses of SAR data (InSAR) can be used to develop interferograms, which measure radar wave phase change as a function of position on the earth. With information on wave frequency and propagation velocity, phase change between two times at a given location can be converted to displacements on the earth’s surface at that location, down to a minimum resolvable displacement of 1/32 of a SAR wavelength. In the case of the Copernicus Sentinel-1 satellites, which utilize C-band SAR waves, the implied minimum detectable displacement is about 0.175 cm based on a 5.6 cm wavelength (Burgmann et al., 2000; Massonnet and Feigl, 1998).

Prior work suggests a potential theoretical upper limit to the displacement levels that can be distinguished using InSAR data. When surface displacements in the line of sight (LoS) direction (i.e. from a satellite to the ground surface) exceed half of a SAR wavelength (for Sentinel-1 which utilizes C-band SAR waves, about 2.8 cm) over the length of a SAR pixel, coherence loss occurs and displacement changes can no longer be resolved (Peltzer et al., 1994). Coherence loss is a term used in InSAR literature that refers to displacement change not being resolvable. In addition to excessive phase shift due to large displacements, coherence loss can also result from re-orientation of scatterers within a specified area on the earth’s surface (i.e. pixel) between the acquisition times of the two SAR images. DPMs aggregate coherence loss across a specified area (collection or analysis window of adjacent pixels), which is converted using different algorithms to an index that ranges from -1.0 to 1.0 for DPM1 and 0.0 to 1.0 for DPM2, with higher values corresponding to more coherence loss in both instances. The DPM is a map of this index, converted to color, that provides a visual representation of “change” from before an event occurred to after it. It is reasonable to assume that DPMs would be able to identify areas where surface fault rupture exceeds half a SAR wavelength. However, it is harder (if not impossible) to measure scatterer reorientation. Due to these multiple potential causes of coherence loss, only one of which is related to ground displacement, formal validation is needed to see if displacement can be resolved from DPM indices.

DPMs have been qualitatively evaluated for their ability to detect landslides (Yun et al., 2015), building damage (Sextos et al., 2018; Yun et al., 2015), and surface fault rupture (Fielding et al., 2005). Prior to the present work, validations of DPMs were mostly performed using Boolean (and/or multi-class) labels for the ground truth data and the presence of colored DPM pixels (whose DPM index values are not explicitly stated) to test the ability of DPMs to identify damage. The only prior quantitative validation using actual measurements and DPM index values focused on their ability to detect structural damage and blast effects on building facades (Sadek et al., 2022). In this paper, we seek to quantitatively validate DPMs’ ability to detect different levels of surface displacement. We introduce a process to establish links between DPM index and (1) the binary distinction of whether displacement was or was not recorded at a given location and (2) the amount of displacement. We anticipate that such numerical procedures may be of general interest, even for other remote sensing technologies (Dell’Acqua and Gamba, 2012). We utilize a dataset gathered following the 2019 Ridgecrest Earthquake sequence consisting of surface fault rupture measurements along the major fault ruptures associated with both the M6.5 and M7.1 seismic events (Ponti et al., 2020 and DuRoss et al., 2020a,). This dataset includes ground displacement amounts and directions for 522 discrete locations.

Following this introduction, we describe the bases for different types of DPMs, present the data that is considered along with the analysis methods, and conclude with a presentation and discussion of the results.

Damage proxy maps

NASA-JPL produces two types of DPMs (DPM1 and DPM2). As noted in the Introduction, for each a DPM index is computed from SAR data for a given pixel. The procedures used for these calculations are described below. These descriptions have been adapted from relevant literature, with certain details that are important for conceptual understanding more clearly stated and some mathematical details corrected (specifically, Eq. 4).

DPM1

The DPM1 index is based on the difference in interferometric coherence of two pairs of SAR images, one pre-event pair (i.e. a pair of SAR images acquired before the disruptive event) and one co-event pair (i.e. a pair of SAR images comprising a pre-event image and an image taken after the disruptive event). Interferometric coherence is a measure of the consistency in radar wave phase between two SAR images of the same area at different times. Interferometric coherence is measured for each pixel using the magnitude of complex coherence (Fielding et al., 2005; Rosen et al., 2000),

$$\gamma \left(i\omega \right)=\frac{〈g_1{g}_2^{*}〉}{\sqrt{〈{\left|g_1\right|}^2〉〈{\left|g_2\right|}^2〉}}$$

(1)

where g ₁ and g ₂ describe the complex-valued and frequency-dependent radar wave properties recorded during different passes of a satellite over a given location, each of which can be described by,

$$g_i=a_i{e}^{i{\phi }_i}$$

(2)

where index $i$ refers to a particular time or epoch (i.e. passage of satellite over the pixel), ${\phi }_i$ is the phase angle of the received SAR signal from that pixel, and $a_i$ is the signal amplitude. In Eq. (1), the 〈〉 operator represents integration over a spatial domain of pixels surrounding the subject pixel; $〈{\left|g_i\right|}^2〉$ are auto power spectral density functions, which are frequency-dependent and real-valued; $〈g_1{g}_2^{*}〉$ is the cross power spectral density function, which is frequency-dependent and complex-valued; and the symbol * indicates the complex conjugate. The interferometric coherence in Eq. (1) is generally referred to as coherency in signal processing literature (Ljung, 1987; Pandit, 1991), but we adopt the “interferometric coherence” terminology of Fielding et al. (2005). While in general the coherence in Eq. (1) and the received signal in Eq. (2) are frequency-dependent, in effect these quantities correspond to a frequency of 5 GHz with a narrow bandwidth of 50 MHz. As such, they effectively correspond to a single frequency and frequency-dependence is dropped subsequently in our notation.

The spatial integration represented by the 〈〉 operator computes power spectral densities (and coherence) across a window defined by a particular number of SAR pixels. For Sentinel-1, SAR pixels have dimensions of roughly 3 × 14 m and a typical window size is 16 × 4. As a result, integration occurs across a window roughly 48 m × 56 m in size. Some authors use the term “spatial correlation” for the interferometric coherence because the spatial integration is a measure of the spatial correlation of the phase values (Fielding et al., 2005).

The DPM1 index ( $I_{\mathrm{DPM}1}$ ) is based on the difference of interferometric coherence amplitudes for two pairs of SAR images, one pre-event pair and one co-event pair, as follows:

$$I_{\mathrm{DPM}1}=\left|{\gamma }_0\right|-\left|{\gamma }_c\right|$$

(3)

where $\left|{\gamma }_0\right|$ is the pre-event coherence magnitude and $\left|{\gamma }_c\right|$ is the co-event coherence magnitude. Following computation of $I_{\mathrm{DPM}1}$ , the indices are geocoded and resampled using a Digital Elevation Model (DEM) based on the 1 arc-second Shuttle Radar Topography Mission (SRTM) data (USGS EROS Center, 2018), which has pixel dimensions close to 30 by 30 m.

To understand the physical meaning of $I_{\mathrm{DPM}1}$ , it is useful to first recognize that the complex coherence is high (close to unity) when the objects from which SAR waves reflect (referred to as scatterers) within the spatial window have not moved significantly relative to each other between epochs. In the case of the pre-event pair, scatterer movements ideally should be small, thus producing high values of $\left|{\gamma }_0\right|$ (near unity). Following a disturbing event producing scatterer movements, coherence loss of the co-event pair would occur, reducing $\left|{\gamma }_c\right|$ . As a result, such events should produce $I_{\mathrm{DPM}1}$ values close to unity whereas events with minimal scatterer movements would produce no appreciable change in coherence and $I_{\mathrm{DPM}1}$ values close to zero.

The DPM1 index can be ineffective for cases with low pre-event coherence because $\left|{\gamma }_0\right|$ and $\left|{\gamma }_c\right|$ may have similarly low values. $I_{\mathrm{DPM}1}$ would be negative if $\left|{\gamma }_0\right|$  <  $\left|{\gamma }_c\right|$ (its theoretical range is -1.0 to 1.0). Conditions leading to low values of $\left|{\gamma }_0\right|$ include changes in scatterer geometry, orientation, and dielectric properties due to weather, wind, and/or anthropogenic activity (e.g. motions of cars in a parking lot or vegetation oscillating during windy days). Three dimensional scatterers, like vegetation and buildings, are more sensitive to the changes caused by these phenomena than comparatively flat scatterers. Such concerns motivated the development of DPM2. However, in the case of the Ridgecrest earthquake, which occurred in a flat, desert environment, DPM1 would be expected to perform well.

DPM2

DPM2 uses the amplitude of complex coherence as given by Eq. (1). However, while DPM1 uses two pairs of SAR images (i.e. a pre-event pair and a co-event pair), DPM2s uses many pairs of SAR images (i.e. multiple pre-event pairs and at least one co-event pair). By using many pre-event images to characterize coherence, a more robust baseline is established for temporal coherence loss difference calculations. Zebker and Villasenor (1992) and Jung et al. (2016) describe a number of factors that influence coherence, including non-temporal factors and temporal factors.

Temporal decorrelation is defined as the decorrelation caused by abrupt (sudden) events, which may have small or large impacts (e.g. person walking by or earthquake, respectively) or by gradual changes such as the growth of vegetation (Jung et al., 2018; Zebker and Villasenor, 1992). Temporal decorrelation is a measure of abrupt or gradual changes of the positions of scatterers and/or radar targets on the ground. Temporal decorrelation is influenced by three contributing factors:

1. Temporally-correlated coherence ( $\left|{\gamma }_{0,\mathrm{tc}}\right|$ when used for pre-event pairs and $\left|{\gamma }_{c,\mathrm{tc}}\right|$ when used for one or more co-event pairs) due to gradual changes over time;

2. Temporally-uncorrelated coherence (similarly $\left|{\gamma }_{0,\mathrm{tu}}\right|$ or $\left|{\gamma }_{c,\mathrm{tu}}\right|$ ) due to abrupt changes;

3. Other decorrelation factors (e.g. geometric, thermal, volumetric; Fielding et al., 2005).

Sources in factor #3 are negligible in interferometric SAR applications such as those we are describing in this paper (i.e. Jung et al., 2018). As such, temporal decorrelation can be solely described by the temporally-correlated and temporally-uncorrelated terms.

The temporally-correlated coherence has been observed to decay exponentially as the time window used for the pre-event analyses lengthens, which can be referred to as an interval length effect (Jung et al., 2018). This coherence loss is generally thought to result from a gradual accumulation of changes in vegetation or other surface materials. To account for this length effect, DPM2 uses an exponential temporal decorrelation model (i.e. Eq. (15) in Jung et al., 2018). This model is used to de-trend the exponential decaying term of temporal decorrelation. The remaining portion of the temporal decorrelation, after removing the model for the interval length effect, is $\left|{\gamma }_{0,\mathrm{tu}}\right|$ . This is the coherence component attributed to abrupt changes for pre-event pairs, multiple observations of which can be used to define the probability density function (PDF) of $\left|{\gamma }_{0,\mathrm{tu}}\right|$ , which is denoted as $f_{\gamma }\left(\left|{\gamma }_{0,\mathrm{tu}}\right|\right)$ . To evaluate temporal decorrelation from a specific abrupt event, coherence is computed from co-event pairs that are adjusted for the length effect by subtracting the decorrelation model to estimate the co-event temporally-uncorrelated coherence $\left|{\gamma }_{c,\mathrm{tu}}\right|$ .

The DPM2 index, $I_{\mathrm{DPM}2}$ , is computed for a pixel as a “probability of damage” conditioned on the $\left|{\gamma }_{c,\mathrm{tu}}\right|$ value for the pixel as (adapted from Jung et al., 2018 to reflect the notation presented here),

$$I_{\mathrm{DPM}2}=P\left(\left|{\gamma }_{c,\mathrm{tu}}\right|\right)=1-\underset{0}{\overset{\left|{\gamma }_{c,\mathrm{tu}}\right|}{\int }}{f}_{\gamma }\left(\left|{\gamma }_{0,\mathrm{tu}}\right|\right)d\gamma$$

(4)

To conceptualize the meaning of Eq. (4), consider the case of $\left|{\gamma }_{c,\mathrm{tu}}\right|$  = 1.0. In this case, the integration covers the full range of coherence magnitudes and the integral provides the cumulative distribution ordinate of the $\left|{\gamma }_{0,tu}\right|$ distribution sampled at its upper limit, which is necessarily close to unity. Accordingly, $I_{\mathrm{DPM}2}$ would be close to zero. Next consider a case of low $\left|{\gamma }_{c,\mathrm{tu}}\right|$ (e.g. 0.2). If the pre-event coherence likelihoods (i.e. PDF ordinates) are low over the range of [0, 0.2], the integral would be close to zero and $I_{\mathrm{DPM}2}$ would be close to unity. In this manner, $I_{\mathrm{DPM}2}$ distinguishes cases where the abrupt event produces coherence loss that is distinguishable from the non-temporal condition.

Although DPM2 is considered to provide a methodological improvement to DPM1, the differences might be anticipated to be modest for the Ridgecrest case because of its flat, desert environment. The DPM2 method involves a much larger amount of data processing than the DPM1 method.

Ground movement data and DPMs

Field reconnaissance data

The surface disruption event considered in this paper is from an earthquake sequence near Ridgecrest, California on July 4 and 5, 2019. The sequence began on July 4 with a M6.5 earthquake on the left-lateral, northeast-striking Salt Wells Valley fault. That event was followed 34 hours later by a July 5 M7.1 earthquake on the right-lateral, northwest-striking Paxton Ranch fault. These earthquakes caused significant ground failure, including landslides (Jibson, 2020), widespread liquefaction (Zimmaro et al., 2020 and Brandenberg et al., 2020), and surface fault rupture (Ponti et al., 2020 and DuRoss et al., 2020a).

Following the earthquake sequence, teams led by the USGS gathered 522 surface fault rupture measurements along the approximately 18 km and 52 km of fault rupture from the M6.5 and M7.1 events, respectively (Ponti et al., 2020 and DuRoss et al., 2020a). These measurements all included at least one or a combination of horizontal, vertical, and offset crack width fault displacements (Figure 1). More details regarding the reconnaissance effort and the way measurements were taken can be found in DuRoss et al. (2020a).

Figure 1.

Fault measurement locations and “true zero displacement” points from 2019 Ridgecrest earthquake sequence. The moment tensors were generated using the Python library, ObsPy (Beyreuther et al., 2010), using moment tensor data from the USGS event page.

The dataset from DuRoss et al. (2020b) is essentially complete for locations of ground cracks. However, it is not complete for displacement vectors (e.g. there are areas where surface fault rupture features were observed, without associated vector measurements). It also contains few zero displacement data points (26 points). To achieve balance in the dataset between data points with and without measured displacements, 260 “true zero displacement” points were added to the dataset based on observations from on-ground inspections and helicopter overflights. The selected true zero data points are all located close to each other in a relatively small area. Before selecting these data points, we pre-selected a larger number of candidate areas. However, areas other than the selected one presented ground failure features (e.g. landslides, liquefaction, ground cracks, and/or surface fault rupture features) or were outside reconnaissance boundaries. The selected true zero area is the only one for which we are sure that no ground failure occurred at these locations (i.e. at these locations post-earthquake observations confirmed that ground failure did not occur; Figure 1 of Zimmaro et al., 2020). The number of “true zero” points added was selected to balance displacements (i.e. horizontal, vertical, and crack width) by percentiles into five equally size bins (i.e. null, >0-25th percentile, 25-50th percentile, 50-75th percentile, and 75-100th percentile).

The displacement data can be represented in different ways for the comparisons to DPM indices. We consider the along-strike horizontal-component, vertical-component, crack width, and the square-root-sum-of squares (SRSS) of the horizontal-components, vertical-components, and crack width offsets. When a displacement measurements type was missing at a given location (typically in the vertical or crack width fields), they were taken as zero. Table 1 summarizes the numbers of displacement data points that were considered in the analyses presented subsequently.

Table 1.

Displacement type data sets.

Displacement type	Measurement locations	“True zero” points	Size of total dataset
Horizontal	419	131	550
Vertical	312	78	390
Crack Width	202	51	253
SRSS	522	260	782

SRSS: square-root-sum-of squares.

DPMs

After the earthquake sequence, on 11 July 2019, a DPM1 was released that mapped the heavily affected portion of California from these events, including Searles Lake. The DPM1 index calculation used a 16 × 4 window (DPM1-16 × 4) of SAR pixels and was geocoded and resampled to have the same resolution as the SRTM DEM (1-arcsecond or approximately 30 m pixel spacing). The map was created using ascending track SAR data from the Copernicus Sentinel-1 satellites, operated by the European Space Agency (See Table A2). Interferometric processing was performed with the ISCE2 package (Rosen et al., 2012). A first version of the map was made publicly available through the ARIA event page (https://aria-share.jpl.nasa.gov/20190704-0705-Searles_Valley_CA_EQs/DPM/). The DPM1 initially available on the ARIA event page was formatted in a manner that favors readability for event responders. That map was prepared by making DPM indexes transparent below a pre-defined threshold and color-coded when index values were above this threshold. In our analyses we used “raw” DPM maps. In these maps, each pixel in the calculation domain is assigned a DPM index (i.e. they are all visible and none of them is transparent). For this study, four raw DPMs were generated by NASA-JPL: a raw DPM1 version of that initially published on the ARIA event page (DPM1-16 × 4), a DPM1 utilizing the same ascending track Sentinel-1 data and a smaller calculation window of 8 × 2 (DPM1-8 × 2), and two DPM2s derived from ascending and descending satellite tracks (denoted DPM2-TA and DPM2-TD, respectively) that both used the regular 16 × 4 pixel windows (See Table A2). As a result, including DPM1-16 × 4, four DPMs were considered in this study (available at the JPL Open Repository; https://doi.org/10.48577/jpl.0LBWZG). This facilitated comparison between DPM1 and DPM2, between different DPM1 window sizes, and between DPM2s from ascending and descending tracks.

The reason different flight track directions are considered for DPM2s is that it can affect DPM accuracy in areas with topographic features. In the present case, where the fault ruptured in flat areas, we do not expect to see significant differences between the alternate DPM2s.

Data analysis and results

Using the Geographic Information System (GIS) tool QGIS, 60 m diameter buffers were generated surrounding all 782 locations with either surface fault displacements or zero displacements (Figure 1). This buffer diameter was chosen because it is roughly equivalent to the width of the default, 16 × 4 SAR pixel spatial integration window (56 m). For each DPM type, the mean DPM index within the 60 m buffer around each measurement point was computed for the analyses presented in this section. This choice was made because we consider the mean index to be representative of the index values within each buffer.

For each displacement type, the different DPMs were evaluated with respect to the following criteria:

• Criterion 1: Distinguish no surface fault rupture (0 cm) from some surface fault rupture (> 0 cm);

• Criterion 2: For locations with measured displacement, distinguish moderate (0 to 50th percentile) from severe (50th to 100th percentile) surface fault displacements;

• Criterion 3: For locations with measured displacement, distinguish among surface fault displacement percentile bins (0 to 25th, 25th to 50th, 50th to 75th, and 75th to 100th percentiles).

The analyses performed to evaluate DPMs’ ability to distinguish these displacement amounts were evaluated using box and whisker and fragility analyses, as described next.

Box and Whisker analyses

The first comparisons between DPM indices and the measured surface rupture displacements are presented through box and whisker plots (i.e. graphs showing groups of numerical data where a box represents the 25^th and 75^th percentiles, whiskers represent the 95^th and 5^th percentiles, and a line inside the box represents the median value of that bin). Through these figures, performance metrics were quantified which give a means of directly quantifying the performance of the various DPMs at detecting different offset levels.

Box and whisker analyses were conducted on every displacement type and DPM and repeated to address each of the three criteria listed above. The available data were separated into bins depending on the criterion being investigated, as follows:

1. For Criterion 1, two bins were considered, one consisting of buffers where the displacement was greater than 0 cm and the other where the offset was either measured or inferred to be zero.

2. For Criterion 2, only locations with non-zero displacements were considered and two bins with different displacement percentile ranges were used (> 0-50th, 50-100th).

3. For Criterion 3, again only non-zero displacement locations were considered, and four displacement percentile bins were used (> 0-25th, 25-50th, 50-75th, and 75-100th).

The displacements corresponding to different percentiles are given in Table 2 for each displacement type.

Table 2.

Displacement percentiles.

Percentile	Horizontal (cm)	Vertical (cm)	Crack Width (cm)	SRSS (cm)
0	0	0	0	0
25	9	6	5	13
50	35	17	14	36
75	100	35	36	105
100	520	374	500	520

SRSS: square-root-sum-of squares.

Figure 2 shows a histogram of the 522 measured displacements (using SRSS), including a bin showing the number of observations for zero displacements and the bounds of the percentile-based bins.

Figure 2.

Histogram for SRSS fault measurement locations and “true zero displacement” points from 2019 Ridgecrest earthquake sequence.

Figures 3 and 4 present box and whisker plots for SRSS displacements evaluated using Criteria 1 and 2; four plots are shown in each case for the four considered DPM types. Outliers are represented with black circles on the ends of each whisker. Figure 3 shows that there is a significant gap between the $I_{\mathrm{DPM}}$ values for no-displacement and displacement bins for each DPM type, with the no-displacement indices being consistently lower. While the $I_{\mathrm{DPM}1}$ distributions (Figure 3a and b) are consistently lower than $I_{\mathrm{DPM}2}$ distributions (Figure 3c and d) for the same bin types, the separation between bins for a given DPM type are consistently large (about 0.3-0.4). Figure 4 shows similarly formatted plots related to Criterion 2 (i.e. are the DPMs able to distinguish between smaller (0-36 cm) and larger (36-520) cm displacement levels?). In this case, the distributions have appreciable overlap but the median $I_{\mathrm{DPM}}$ values are consistently lower for the lower displacement levels. These outcomes suggest that DPMs are better suited to distinguish between zero and some displacement than between moderate and severe displacements. This result is somewhat expected as it is a direct consequence of the Sentinel-1 satellite wavelength. Hence, this outcome is satellite-specific and can be generalized only for satellites with similar band and wavelength characteristics.

Figure 3.

Box and whisker plots for SRSS displacements comparing “no displacement” vs “some displacement”: (a) DPM1-16 × 4, (b) DPM1-8 × 2, (c) DPM2-TA, and (d) DPM2-TD.

Figure 4.

Box and whisker plots for SRSS displacements comparing “moderate displacement” vs “large displacement”: (a) DPM1-16 × 4, (b) DPM1-8 × 2, (c) DPM2-TA, and (d) DPM2-TD.

To evaluate the distinctions between data distributions in a quantitative manner, we use an F-test (Snedecor and Cochran, 1989). This test compares the statistical performance of sub-models (e.g. the two distributions for a given DPM type in either of Figures 3 or 4) with that of a full model (i.e. the combination of the two distributions). The F-test assumes that the bins of data are normally distributed and have similar variances, both of which are satisfied for all of the data bins generated for this analysis. For a given distribution, the median $I_{\mathrm{DPM}1}$ value is subtracted from the $I_{\mathrm{DPM}1}$ value for each buffer and the square of the differences are summed, forming the residual sum of squares (RSS). RSS values for the two bins are denoted $\mathrm{RS}{S}_1$ and $\mathrm{RS}{S}_2$ , respectively, whereas the value for the combined distribution is $\mathrm{RS}{S}_f$ . The one-way analysis of variance F-statistic, $F_1$ , is then computed as (adapted from Snedecor and Cochran, 1989):

$$F_1=\frac{\mathrm{RS}{S}_f-(\mathrm{RS}{S}_1+\mathrm{RS}{S}_2)}{(\mathrm{RS}{S}_1+\mathrm{RS}{S}_2)/(N_f-2)}$$

(5)

where N _f is the number of buffers in the dataset. This F-statistic can be compared to the F distribution to evaluate a significance level (p) for the test. This was done utilizing the “f-one way” function from the Python library SciPy. Small values of F ₁ and large values of p (> 0.05) suggest that the models are not distinct. In Figures 3 and 4, the $F_1$ statistics and p-values are listed below each plot. For both sets of plots, the distributions for each satellite type are statistically distinct because p-values are far below the 0.05 threshold, but the levels of distinction are greater for Criterion 1 than Criterion 2.

Figure 5 presents box and whisker plots for the SRSS displacements evaluated using Criterion 3 (i.e. to check DPMs performances for various displacement bins). Median $I_{\mathrm{DPM}}$ values generally increase as displacements increase, with the exception of the third-to-fourth bins where saturation occurs (i.e. the increases are negligible or small decreases occur). In this case, p-values based on F-statistics were computed between successive bins. The results indicate distinction between bins at low displacement levels, but reduced distinction (or non-distinction) at higher levels. Moreover, higher levels of distinction are achieved for $I_{\mathrm{DPM}1}$ than for $I_{\mathrm{DPM}2}$ , which is likely a result of $I_{\mathrm{DPM}2}$ values being higher and having less range over which to change between bins.

Figure 5.

Box and whisker plots for SRSS displacements for different displacement levels: (a) DPM1-16 × 4, (b) DPM1-8 × 2, (c) DPM2-TA, and (d) DPM2-TD.

Although not shown here for brevity, similar results are obtained for horizontal displacements. The DPM indices were generally less effective at distinguishing vertical displacements and offsets. This is likely a consequence of fault rupture having been strike-slip with predominantly horizontal displacement. Box and whisker plots for each displacement metric and corresponding F-test results are provided in O’Donnell et al. (2024).

Overall, the results suggest that both DPM2s are marginally better than both DPM1s at distinguishing between areas with no surface fault rupture and some surface fault rupture, although each DPM type is effective from a statistical perspective. However, both DPM1s are marginally better than the DPM2s at distinguishing between moderate- and large surface fault rupture displacements. These results suggest that in a flat, desert environment with little vegetation, DPM1s and DPM2s are both effective but may be best used in tandem.

Fragility analysis

Additional comparisons between DPM indices and measured surface rupture displacements are presented using fragility functions, which are curves relating the probability of a certain damage threshold being exceeded to a demand parameter or a proxy (Baker, 2015; Porter et al., 2007).

For each DPM and displacement type, fragility functions were derived conditioned on $I_{\mathrm{DPM}}$ values. This was done by dividing the buffers for each DPM and displacement type into 20 bins based on percentiles of $I_{\mathrm{DPM}}$ . For each bin $j$ , the number of buffers with a displacement ( $\mathrm{\Delta }$ ) exceeding a threshold value ${\mathrm{\Delta }}_i$ was evaluated and normalized by the total number of buffers in the bin. This gives the probability of failure ( $P_j$ ) associated with median $I_{\mathrm{DPM}}$ value of that bin, which can be expressed as follows (adapted from Kwak et al., 2016),

$$P_j(\mathrm{\Delta }>{\mathrm{\Delta }}_i|I_{\mathrm{DPM}}=i_{\mathrm{dpm},j})=\frac{N{D}_j}{N_j}$$

(6)

where $i_{\mathrm{dpm},j}$ is the median DPM index for bin j, $N{D}_j$ is the number of buffers with $\mathrm{\Delta }>{\mathrm{\Delta }}_i$ for bin $j$ , and $N_j$  = total number of buffers for bin $j$ .

The blue bars in Figure 6 show the variation of $P_j$ with $I_{\mathrm{DPM}2}$ for the case of SRSS displacement types, DPM2-TA, and a threshold displacement of ${\mathrm{\Delta }}_i=0$ . The results show that the probability of exceeding zero displacement increases from nearly null to nearly unity over the range of $I_{\mathrm{DPM}2}$ ≈ 0.5 to 0.7. We fit the data in the bar chart using a log normal cumulative distribution function, selected based on visual inspection, which produces the smooth curve shown in Figure 6. The fitted fragility function is described by two parameters, which are the natural log mean ( ${\mu }_{\mathrm{lnI}}=-0.48$ , which provides a median of 0.62 in arithmetic units) and natural log standard deviation ( ${\sigma }_{\mathrm{lnI}}$  = 0.09). A low standard deviation indicates strong predictive power of the independent variable ( $I_{\mathrm{DPM}2}$ in this case).

Figure 6.

Bar chart showing probabilities of exceeding null displacement for buffers with variable levels of $I_{\mathrm{DPM}2}$ and curve fit using log normal cumulative distribution function. SRSS displacement type and DPM2-TA.

Figure 7 shows a similarly derived fragility function for a threshold displacement of ${\mathrm{\Delta }}_i=$ 36 cm (50^th percentile of SRSS measurements). In this case the probabilities do not approach unity and the spread in the dispersion curve is wider, with ${\sigma }_{\mathrm{lnI}}$  = 0.30.

Figure 7.

Bar chart showing probabilities of exceeding 36 cm displacement for buffers with variable levels of $I_{\mathrm{DPM}2}$ and curve fit using log normal cumulative distribution function. SRSS displacement type and DPM2-TA.

Fragility analyses were repeated for each displacement type and DPM type and for various displacement levels corresponding to different percentiles from Table 2. Table A1 (in the electronic supplement to this paper) reports means and standard deviations of log-normal CDF fragility functions for each analyzed displacement metric and DPM type. As shown in Figure 8, DPM indices demonstrate stronger predictive power (lower ${\sigma }_{\mathrm{lnI}}$ ) for smaller threshold displacement values, DPM2 vs DPM1, and in the case of DPM1 the larger window size (16 × 4). The results in Table A1 also show reduced ${\sigma }_{\mathrm{lnI}}$ for the horizontal and SRSS displacement types vs vertical displacement or crack width.

Figure 8.

Dependence of ${\sigma }_{\mathrm{lnI}}$ on SRSS displacement threshold and DPM type.

Optimal threshold analysis

An important outcome of the fragility analyses presented in the previous section is the level of DPM index that effectively differentiates buffers at which a displacement threshold is and is not exceeded. An intuitive choice for this threshold is the index value ( $I_{\mathrm{DPM}1}$ or $I_{\mathrm{DPM}2}$ ) corresponding to $P_j$  = 0.5. In this section, we conduct alternative, and arguably more formal, analyses to identify the optimal threshold.

We take the optimal threshold as the DPM index that differentiates buffers with displacements below and above the displacement threshold. To illustrate the process for optimal threshold identification, consider for example a displacement threshold of zero and $I_{\mathrm{DPM}2}$ , for which the optimal threshold is denoted $I_{\mathrm{DPM}2}^{\mathrm{th}0}$ . Figure 9 shows histograms of $I_{\mathrm{DPM}2}$ for buffers with zero displacement (blue) and non-zero displacement (orange). The cumulative relative likelihood of bins from the zero displacement histogram with $I_{\mathrm{DPM}2}<I_{\mathrm{DPM}2}^{\mathrm{th}0}$ is N while the cumulative relative likelihood of bins from the non-zero histogram with $I_{\mathrm{DPM}2}>I_{\mathrm{DPM}2}^{\mathrm{th}0}$ is F. Following Mohammed et al. (2024), a confidence index can be computed as,

$$I_{\mathrm{conf}}=\frac{N+F}{2}$$

(7)

Figure 9.

Histograms of DPM indices for buffers with displacements below and above a displacement threshold value that are used to identify an optimal DPM index threshold. Data shown are for a zero-displacement threshold and $I_{\mathrm{DPM}2}$ (TA).

The value of $I_{\mathrm{DPM}2}^{\mathrm{th}0}$ that maximizes $I_{\mathrm{conf}}$ is taken as the optimal threshold, and is marked in Figure 6 by the red line. This value is 0.63 for the data shown in Figure 9, which can be compared to the $I_{\mathrm{DPM}2}$ value where $P_j$  = 0.5 from the fragility curve, which (per Table A1 in the electronic supplement to this paper) is $\exp (-0.48)$  = 0.62. Hence, the two values are nearly equivalent in this case. The values have greater variability for data sets with larger standard deviations. The optimal threshold procedure demonstrated here is effective regardless of the plateau level of probability in fragility plots, which can be <0.5 for large threshold displacements (e.g. Figure 7).

Application of fragility curves for predictions of fault rupture displacement probabilities

The fragility curves in Figures 6 and 7 can be used to predict the probability of exceeding null or 36 cm displacement levels conditional on $I_{\mathrm{DPM}2}$ , which allows DPMs to be converted to maps or other diagrams indicating the probability of exceeding these displacement levels. Figure 10 shows one such map for the areas affected by surface rupture from the Ridgecrest Earthquake sequence. The figure shows measured on-fault SRSS displacement values overlain on DPM pixels colored to indicate probabilities of $P_k(\mathrm{\Delta }>0|I_{\mathrm{DPM}2})$ as given by the fragility model (where $k$ is a pixel index). Where $P_k$  < 0.5, the pixels are transparent. The map in Figure 10 has some limitations, in that the scale is too small to see details and it is visually confusing because most of the DPM-related coloration is unrelated to fault rupture (although it may indicate other effects, such as landslides).

Figure 10.

Map showing locations and amounts of SRSS displacements overlain on pixels colored to indicate probabilities of exceeding null displacement conditioned on $I_{\mathrm{DPM}2}$ as given by DPM2-TA.

Model performance is commonly evaluated using the performance metrics “Recall” and “Precision” that are defined as follows:

$$\mathrm{Recall}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}}$$

(8)

$$\mathrm{Precision}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}}$$

(9)

where TP = true positive, FN = false negative, and FP = false positive. The meaning of recall is the fraction of positive incidents (in this case, measured displacements that exceed the threshold) that are correctly predicted by the $I_{\mathrm{DPM}2}$ -based model. An individual prediction from the fragility model is identified as meeting a threshold displacement criterion based on the probability level (i.e. if the probability exceeds a threshold, the limiting displacement is predicted to be exceeded). Using a threshold probability of 0.5, the recall for the $P_k(\mathrm{\Delta }>0|I_{\mathrm{DPM}2})$ model is 0.98, which indicates that very few displacement measurements are not captured by the model. The meaning of precision is the fraction of positive incidents to predictions of positive incidents, including incorrect predictions (i.e. FP cases). The precision of the $P_k(\mathrm{\Delta }>0|I_{\mathrm{DPM}2})$ model is also 0.98, which indicates a low rate of FP cases. While this large value might be surprising based on visual inspection of Figure 10, only data within buffers are considered, for which there are few FP cases.

The threshold probability used with the fragility model is subjective. In the case of the $P_k(\mathrm{\Delta }>0|I_{\mathrm{DPM}2})$ model, a 50% threshold is mid-way within the data range for the fragility curve (Figure 6), which matches the threshold value in Figure 9. Sensitivity analyses using different threshold probability levels indicate that optimal performance (i.e. maximized recall and precision) is achieved near the median level. On the other hand, the data used to develop the $P_k(\mathrm{\Delta }>36\mathrm{cm}|I_{\mathrm{DPM}2})$ model ranges from null to 60% probability (Figure 7). Testing of alternate probability thresholds showed that recall and precision were relatively low at a 50% threshold, whereas better results were obtained using a threshold ( $I_{\mathrm{DPM}2}^{\mathrm{th}36}$ ) developed from the optimal threshold analyses presented in the previous section. That analysis produced a threshold level of 0.73 which translates to a probability of 27% (Figure 7).

Using the 27% probability level with the $P_k(\mathrm{\Delta }>36\mathrm{cm}|I_{\mathrm{DPM}2})$ model, the recall remains high at 0.96 but the precision is relatively low (compared to the $P_k(\mathrm{\Delta }>0|I_{\mathrm{DPM}2})$ model) at 0.53. This indicates that as ${\mathrm{\Delta }}_i$ is increased, the model retains the ability to identify positive incidents but it produces a larger rate of FP cases.

Figure 11a shows the performance of the $P_k(\mathrm{\Delta }>0|I_{\mathrm{DPM}2})$ model based on the optimal threshold analysis. A correct prediction is either a SRSS displacement larger than the threshold that has a large $I_{\mathrm{DPM}2}$ -based probability (TP, shown in green) or a null displacement that has a small $I_{\mathrm{DPM}2}$ -based probability (TN, not depicted due to few such data points). Similarly, FP cases (shown in yellow) and FN cases (shown in black) indicate incorrect predictions. Figure 11b provides similarly formatted results demonstrating the performance of the P_k(Δ > 36 cm|I_DPM2) model.

Figure 11.

Map of $I_{\mathrm{DPM}2}$ -based probability indicating areas where fault SRSS displacements are expected to be > 0 (red) and zero (white), overlain by buffers color-coded to indicate prediction type (TP, FN, FP) for displacement thresholds of: (a) 0 cm and (b) 36 cm. The thresholds used for mapping are the optimal values ( $I_{\mathrm{DPM}2}^{\mathrm{th}0}\mathrm{and}{I}_{\mathrm{DPM}2}^{\mathrm{th}36}$ ).

To more clearly show the spatial relationships between observed displacements and predicted exceedance probabilities, Figure 12 shows these quantities as a function of along-strike position for the (b) the Paxton Ranch fault (M7.1 event) and (d) Salt Wells Valley fault (M6.5 event). Also depicted are the predicted exceedance probabilities independent of location for the “true zero-displacement” points (c). Above the abscissa we show points at the positions of measured SRSS displacements along the fault. Below the abscissa we show $I_{\mathrm{DPM}2}$ -based displacement exceedance probabilities taken as the average for the pixels at the along-strike position for the null displacement threshold. These plots demonstrate that generally, DPM indices indicate a high probability that null displacement is being exceeded at all locations with greater than null displacement, regardless of the size of the measured displacement. However, the few locations with lower probabilities (at or around 50%) generally correspond to measured SRSS displacements below 50 cm. This is further illustrated by the limited range of exceedance probabilities associated with the “true zero-displacement” points, which are mainly between 0% and 60%.

Figure 12.

(a) SRSS fault measurement locations with epicenters for Paxton Ranch fault M7.1 event (red) and Salt Wells Valley fault M6.5 event (blue) and SRSS displacement and $I_{\mathrm{DPM}2}$ -based probability of exceeding null measured displacement plotted against distance along fault for (b) Paxton Ranch fault and (c) “true zero-displacement” points (graphed independent of position) and (d) Salt Wells Valley fault.

In Figure 13, we extend the performance metrics discussed above for different DPM types and a range of ${\mathrm{\Delta }}_i$ thresholds from 0-120  cm using ${\mathrm{\Delta }}_i-$ dependent $I_{\mathrm{DPM}2}$ thresholds derived from optimal threshold analyses. Based on Figure 13, all DPMs perform similarly at all displacement thresholds. The sensitivity to displacement threshold is variable, being negligible for recall but significant for precision. Although not shown here for brevity, if the threshold probability level were fixed at a single value (e.g. 50%), recall would decrease modestly with increasing ${\mathrm{\Delta }}_i$ (the reduction is smaller than for precisions). These results indicate that DPMs are very effective at distinguishing between areas with no fault rupture and areas with some fault rupture. However, for all DPMs, the drop in precision with increasing ${\mathrm{\Delta }}_i$ indicates appreciable over-estimation of large-displacement areas (FP cases).

Figure 13.

Performance metrics vs displacement threshold, ${\mathrm{\Delta }}_i$ , for four DPM types for (a) Recall and (b) Precision, in both cases based on ${\mathrm{\Delta }}_i-$ dependent $I_{\mathrm{DPM}2}$ thresholds as derived in previous section.

The performance metrics considered to this point have been based on fragility curves derived for the SRSS displacement type. To investigate sensitivity to displacement type, we average metrics across all DPMs (due to the similarity of performance metrics across DPMs; Figure 13), and plot in Figure 14 the resulting average performance metric against displacement threshold. The performance metrics derived for displacements in the line of sight (LoS) direction associated with the ascending Sentinel-1 track are of special interest because SAR wave phase change, and hence $I_{\mathrm{DPM}}$ values, should be most precisely resolved in the LoS direction. However, this resolution is expected to drop off beyond 2.8 cm, equal to half of a C-band wavelength, which is the theorized displacement above which coherence loss across one pixel is anticipated (Peltzer et al., 1994). Somewhat unexpectedly, the plots indicate that there is no noticeable drop off in LoS performance metrics at 2.8 cm and the performance metrics are either similar to (recall) or lower than (precision) the metrics for other displacement types. This may indicate that larger fault displacements are associated with greater amounts of other surface disruptions adjacent to the fault.

Figure 14.

Average performance metrics vs displacement threshold, ${\mathrm{\Delta }}_i$ , for five displacement types for (a) Recall and (b) Precision. Results apply for probabilities at optimal thresholds.

Conclusion

We quantify the performance of four types of SAR-based Damage Proxy Maps (DPMs) using surface fault rupture displacement data collected following the 2019 Ridgecrest earthquake sequence. DPM results for a given location (pixel) are represented using DPM indices ( $I_{\mathrm{DPM}}$ ) derived from pre- and co-event SAR wave coherence differences derived from Sentinel-1 imagery (5.6 cm radar wavelength, C-band). Field performance within the pixel is represented using various types of ground displacement, including horizontal, vertical, crack width, SRSS, and line-of-sight (LoS). The aim of this research is to establish quantifiable relationships between $I_{\mathrm{DPM}}$ and ground displacements.

Four DPM types were considered: two DPM1 types based on different window sizes (representing different resolution levels) and two DPM2s based on different satellite orbital motions (i.e. ascending and descending). Box and whisker plots indicating $I_{\mathrm{DPM}}$ statistics for bins of observations with different displacement amounts show that no displacement and >0 cm displacement cases are very well distinguished by all DPMs. However, the separation of $I_{\mathrm{DPM}}$ distributions for bins with different amounts of non-zero-displacement was relatively small and better resolved by both DPM1s than by the two DPM2s. This appears to have been caused by $I_{\mathrm{DPM}2}$ values being relatively high (closer to unity) and therefore operating across a narrower numerical range, limiting their ability to distinguish between areas with varying displacement levels. The numerical range for $I_{\mathrm{DPM}1}$ values was broader and produced greater sensitivity to variable displacement amounts.

We examined relationships between I_DPM values and probabilities of exceeding various threshold displacements, which were developed empirically and then fit using log-normal cumulative distributions functions. The standard deviation terms in those functions indicate the predictive power of the $I_{\mathrm{DPM}}$ used as the independent variable. We found the best performance (lowest standard deviations) for low threshold displacements, horizontal and SRSS displacement metrics, and DPM2-based indices. Application of the fragility relations to predict displacements within the Ridgecrest study area demonstrates high recall (cases with ground displacement are likely to be correctly identified) and variable levels of precision (the fraction of actual ground displacement locations relative to those identified by the model as having ground displacement). Precision was found to be high for a small displacement threshold (about 0.97 for ${\mathrm{\Delta }}_i$  = 0) and to drop as ${\mathrm{\Delta }}_i$ increases. This indicates a stronger ability to distinguish no displacement from some displacement than to distinguish different amounts of displacement among a population where some displacement is known to have occurred.

Although our analyses are based on an individual earthquake sequence, we are able to provide quantitative insights into the applicability and use of DPMs. This research demonstrates both the value and the limitations of DPM indices for identifying areas of ground deformation. While DPM2 has some clear advantages, the narrow range of its index ( $I_{\mathrm{DPM}2}$ ) limits its effectiveness for some applications. This could potentially be addressed with refinements of its formulation. As stated earlier, the present work applies for the conditions present in the Ridgecrest study area, which consists mainly of level ground with little vegetative cover. Different outcomes are possible for different field conditions, terrain, pre-disaster levels of disturbance and vegetation, and different surface disruption types (e.g. structural damage, landslides, etc.). Our results use the C-band (5.6 cm) radar of Sentinel-1. The longer radar wavelength of the upcoming NASA-ISRO SAR (NISAR) mission (24 cm; L-band) should expand the applicability of this method to more vegetated environments. However, several of our results are expected to be generally applicable, including: (1) $I_{\mathrm{DPM}2}$ saturation at high values, and (2) the narrow range of variation of $I_{\mathrm{DPM}2}$ values.

Data and resources

The surface fault rupture data used in this paper is available at DuRoss et al. (2020b). Moment tensor data for the 2019 Ridgecrest earthquake sequence are from the USGS event page, available at: https://earthquake.usgs.gov/earthquakes/eventpage/ci38457511/executive (last accessed on 10 March 2025). The DEM used in this study is from the 1-arc sec data from the USGS EROS Center (2018). Preliminary DPMs are available at the National Aeronautics and Space Administration (NASA) Advanced Rapid Imaging and Analysis (ARIA) Jet Propulsion Laboratory (JPL) event page at https://aria-share.jpl.nasa.gov/20190704-0705-Searles_Valley_CA_EQs/DPM/. The DPMs used in this analysis are available from the JPL Open Repository (DOI: 10.48577/jpl.0LBWZG).