# Localized fault-zone dilatancy and surface inelasticity of the 2019 Ridgecrest earthquakes

William Barnhart, Ryan Gold, James Hollingsworth

► To cite this version:

William Barnhart, Ryan Gold, James Hollingsworth. Localized fault-zone dilatancy and surface inelasticity of the 2019 Ridgecrest earthquakes. Nature Geoscience, 2020, 13 (10), pp.699-704. 10.1038/s41561-020-0628-8. hal-03062390

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# Localized fault-zone dilatancy and surface inelasticity of the 2019 Ridgecrest earthquakes

William D. Barnhart ID 1✉, Ryan D. Gold ID 2 and James Hollingsworth ID 3

Earthquakes produce a spectrum of elastic and inelastic deformation processes that are reflected across various length and time scales. While elasticity has long dominated research assumptions in active tectonics, increasing interest has focused on the inelastic characteristics of earthquakes, particularly those of the surface fault rupture zone itself, and how they relate to ground rupture hazard and the mechanics of damage zones. Here we present detailed co-seismic surface-strain analysis of the 2019 Ridgecrest, California, earthquakes. We derive three-dimensional high-resolution surface displacements from satellite optical imagery, which we then invert for the co-seismic surface-strain tensors. We show that fault-zone dilation is pervasive throughout these earthquakes and that inelastic failure is present but relatively localized (median width of 31 m). The width of the inelastic failure zone is not correlated to off-fault deformation, surface geology or displacement magnitude. Instead, the extent and kinematics of inelastic failure reflect active, mylonitic deformation of the fault damage zone that is influenced by rupture velocity and fault maturity. These results highlight how a single earthquake contributes to the long-term, permanent geologic record of faulting.

Deformation kinematics near earthquake surface ruptures are of critical importance to understanding the mechanics of shallow faulting<fnref n="1" />,², fault-zone hydrology and recovery³⁻⁵, the evolution of geomorphic features used to infer earthquake recurrence intervals⁶⁻⁸ and hazard of ground rupture zones⁹,¹⁰. Although earthquakes are understood to be a primarily elastic phenomenon¹¹⁻¹⁴, some permanent deformation must occur as evidenced by the presence of topography, fault damage zones¹⁵⁻¹⁷, low-velocity seismic zones⁵,¹⁸,¹⁹ and distributed surface cracking. The juxtaposition of elastic and inelastic deformation in earthquakes opens several important questions: how much inelastic deformation occurs; what structures (that is, surface cracks) measured in the field can be tied to permanent deformation; and what factors control the magnitude and extent of inelastic failure?

Strain kinematics provide a direct indicator of surface-deformation magnitudes<fnref n="1" />,¹³,¹⁴,²⁰,²¹ that can address these questions without invoking deformation proxies, such as tying off-fault deformation (OFD) to permanent deformation²²⁻²⁴. Moreover, co-seismic strain tensors allow us to directly measure phenomena such as fault-zone dilatancy that previously were only inferred through post-seismic analysis²⁵⁻⁷. Recent advances in high-resolution lidar²¹,²³ and optical²⁰,²⁵ geodesy make it tractable to retrieve high-resolution strain tensors within and around fault zones such that we can resolve the transition between elastic and inelastic surface deformation. In this study, we quantify surface-strain tensors and inelastic deformation of the moment magnitude ($M_w$) 6.4 and $M_w$ 7.1 Ridgecrest, California, earthquakes through analysis of the full three-dimensional (3D) surface-displacement field²⁰,²¹,²⁶.

## High-resolution imaging of the Ridgecrest earthquakes

The 2019 Ridgecrest earthquake sequence produced the two largest surface-rupturing earthquakes in southern California since the 1999 Hector Mine earthquake²⁷⁻²⁹ (Fig. 1). The $M_w$ 7.1 mainshock, a right-lateral strike-slip rupture, was preceded 34 hours by a $M_w$ 6.4 foreshock on a left-lateral strike-slip fault²⁷,²⁸. We resolved the

3D displacement field of the Ridgecrest earthquakes by independently assessing the 2D horizontal and 1D vertical displacements (Fig. 1 and Extended Data Fig. 1). We derived 2-m-resolution horizontal-displacement fields from 0.5 m orthorectified optical imagery acquired by the WorldView-1, 2 and 3 satellites³⁰,³¹ (Fig. 1 and Supplementary Table 1). With the exception of one image, all of the image pairs span both the $M_w$ 7.1 and $M_w$ 6.4 earthquakes. An image acquired on 4 July 2019 before the $M_w$ 7.1 mainshock captures the two earthquakes independently (Fig. 1). We additionally used the pre- and post-event stereo imagery to generate digital surface models (DSMs) and retrieve the vertical displacement field at 2 m spatial resolution³⁰ (Fig. 1b). Given the general lack of topography around these two earthquakes, we needed only difference the heights of the DSMs to obtain a vertical displacement field (Fig. 1c and Extended Data Fig. 2).

We inverted the displacement fields for both 2D and 3D surface-strain and rotation tensors²⁰,²¹,²⁶ (Fig. 2a and Extended Data Figs. 3 and 4). The vertical displacements and surface elevations are considerably noisier than the horizontal displacements fields, so the 3D tensors are typically too noisy to resolve the threshold between elastic and inelastic strain (~0.5% strain<fnref n="1" />,²¹). Thus, our discussions later focus on the 2D surface-strain tensors. Nevertheless, we are able to discern general strain patterns, such as dilatation and shear sense, from the 3D tensors. We also extracted conventional surface-slip distribution measurements from across-fault displacement profiles: surface offsets and OFD zone widths (Fig. 3a,b). We explicitly define OFD as the distance over which the peak across-fault offset is accommodated in geodetic surface-displacement profiles (Fig. 3a, b)²².

## Surface-displacement-field characteristics

Our horizontal-displacement fields show that the Ridgecrest earthquakes produced a spatially complex network of anastomosing and crosscutting surface ruptures akin to those previously mapped²⁷,³². Our horizontal displacements are likewise comparable to optical correlation results derived from other sensors³³. The $M_w$ 6.4

<footnote n="1">¹Department of Earth and Environmental Sciences, University of Iowa, Iowa City, IA, USA. ²Geologic Hazards Center, US Geological Survey, Golden, CO, USA. ³Université Grenoble Alpes, ISTerre, Grenoble, France. ✉e-mail: william-barnhart-1@uiowa.edu</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 1** | High-resolution (2 m) surface-displacement fields of the 2019 Ridgecrest earthquakes. **a**, East-west (dX) horizontal displacements. **b**, Vertical (dZ) displacements. **c**, Co-seismic rupture traces derived from the surface-displacement fields, with annotations for the $M_w$ 6.4 and $M_w$ 7.1 ruptures. QFaults, Quaternary Fault Database. **d**, dX displacements of the $M_w$ 6.4 foreshock alone. **e**, North-south (dY) displacements of the $M_w$ 7.1 mainshock alone. Grey fault traces in **c** are previously mapped faults⁵⁰. The black circles in **d** and **e** indicate the intersection of the two ruptures. Black arrows highlight fault strands that slipped within each respective earthquake, and white arrows highlight fault strands that did not slip.</figcaption></figure>

foreshock produced a single, ~14-km-long left-lateral surface rupture that crossed the later surface rupture of the $M_w$ 7.1 mainshock (Fig. 1d,e).

The $M_w$ 7.1 mainshock surface rupture is predominantly defined by southeast-northwest trending right-lateral slip on a series of anastomosing faults extending ~49 km from southeast to northwest (Fig. 1). There are additionally southwest-northeast trending left-lateral surface ruptures in the northernmost and southernmost extents of the $M_w$ 7.1 rupture and 1.7 km northwest of the junction of the $M_w$ 6.4 and $M_w$ 7.1 earthquakes. The left-lateral surface rupture north of the junction was active during the $M_w$ 7.1 rupture but not during the $M_w$ 6.4 foreshock (Fig. 1d,e). At the detection threshold of our observations, we find no evidence that the $M_w$ 7.1 reactivated surface slip on any of the strands that slipped in the $M_w$ 6.4 foreshock, and vice versa.

Along the $M_w$ 7.1 rupture, we find highly localized instances of subsidence exceeding 2 m (Fig. 1b). We additionally find an ~2-km-wide region of subsidence located between two subparallel fault strands where vertical displacements exceed 2 m (Fig. 1b and Extended Data Fig. 2). This region probably reflects subsidence above a negative flower structure where the right-lateral surface rupture steps towards the right. North of this feature, a poorly defined but spatially broad (1- to 3-km-wide) region of moderate (0.5–1 m) subsidence is well resolved up to the northern terminus of the $M_w$ 7.1 surface rupture (Fig. 1b).

## Strain-field characteristics

The surface rupture of both earthquakes is characterized by both positive and negative dilatation, indicating that the surface rupture zone dilated (expanded) and contracted during co-seismic rupture
<figure><img src="image_4.png" /><figcaption>**Fig. 2** | Surface-strain overview of the Ridgecrest earthquakes. **a**, 2D dilatation strains. **b**, Distribution of individual measurements of inelastic fault-zone width (FZW) determined from dilatation (black dots) and shear (blue dots) strains exceeding 0.5% and projected onto the profile X-X'. Green dots, width of the OFD zone; solid black, blue and green profiles, the 500 m moving median and 18th/64th percentiles of the individual measurements; dark-grey shaded region, zone where the $M_w$ 6.4 rupture crosses the profile; light-grey regions, zones where the $M_w$ 7.1 rupture traversed exposed bedrock.</figcaption></figure>

(Fig. 2a). The fault zone is in positive dilatation in both 2D and 3D strain results, indicating that horizontal dilation is not balanced by vertical contraction. This observation provides a direct measurement of fault-zone dilatancy that was previously only indirectly inferred from seismic-wave travel-time changes and post-seismic deformation$^{3–5,18}$. We find dilatations as high as 25% over 100 m$^2$ areas (0.25 strain), suggesting that co-seismic dilatations exceed the presumed failure strain of surface materials (~0.5%) by nearly two orders of magnitude. Interestingly, the width of the inelastic failure zone is relatively narrow and consistent throughout the rupture: 32 m + 50 m / -21 m (Figs. 2b and 3a,b and Extended Data Fig. 5).

We also find that the shear strains exceed the expected failure strain of rocks (Extended Data Figs. 3 and 4). However, the width of the inelastic shear strain zone is systematically narrower (23 m + 29 m / -12 m) than the zone defined by dilatation (Fig. 2b and Extended Data Fig. 5), as was observed following the 2016 Kumamoto, Japan, strike-slip earthquake$^{21}$. This relationship indicates that co-seismic shear is more localized within the fault zone, while volumetric dilatation extends away from the primary co-seismic shear zones.

The strain fields highlight other important characteristics of near-fault surface deformation. The inelastic strain zone is systematically narrower than the zone of OFD (63 m + 66 m / -26 m) (Figs. 2b and 3a,b). The width of the OFD zone is likewise comparatively heterogeneous along the length of the earthquakes (Fig. 2b and Extended Data Fig. 5). We interpret this relationship to indicate that OFD is not related to the width of the inelastic deformation zone$^{20,21}$. The inelastic deformation zone also shows no well-defined correlation with the magnitude of co-seismic offset at the fault, meaning that the near-fault surface displacement gradients are similar for both small and large horizontal offsets (Fig. 3). Last, the width of the inelastic zone within bedrock is indistinguishable from the

rest of the rupture that is predominantly through playa sediments (Fig. 2b), indicating no clear correlation between inelastic zone width and surface geology.

The positive dilatation of the surface rupture is caused by the greatest principal strains exhibiting extension (Fig. 4). We find that greatest principal strains are only contractional (negative) in short lengths of the $M_w$ 7.1 rupture where the rupture trace steps to the right but extensional everywhere else. The extensional strains are oriented at an acute angle (<90°) to the *P* axes of the moment tensors for the two earthquakes (Fig. 4). Strains rotate across the fault zone; however, the magnitude and pattern of rotation are variable and the principal strain orientations vary in part as a function of local fault strike (Fig. 4b,c).

## Strain localization and mylonitization

The surface-strain field of the Ridgecrest earthquakes provides new constraints on the nature of fault-zone deformation kinematics within a single earthquake rupture. The rupture traces of both earthquakes dilate co-seismically, and our measurements provide direct quantification of combined shear and dilation at shallow faulting depths$^{24}$. Fault-zone dilation was previously documented for a transpressional strike-slip rupture$^{20}$ and inferred from post-seismic contractional dilation recovery of a purely strike-slip rupture$^3$. These occurrences suggest that co-seismic volumetric dilation is a ubiquitous characteristic of earthquakes and that the volume increase is not purely a function of the transtensional nature of the Ridgecrest earthquakes. Because dilation in the Ridgecrest earthquakes exceeds the failure strain of rocks, only a fraction of this dilation is likely to be recovered via fault-zone relaxation$^3$.

One striking characteristic of the Ridgecrest co-seismic strain field is that the zone of inelastic failure (>0.5%) is relatively narrow (<50 m) and exhibits no correlations to OFD, peak fault
<figure><img src="image_5.png" /></figure>

**Fig. 3** | Illustrations of the relationships among displacement, strain-zone width and OFD. **a,b,** Across-fault profiles (see locations in Fig. 2a) and interpretations of 2D dilatational strain (red dots) and fault-parallel offset (black dots). **a,** Example of profiles where OFD and inelastic strain zone are similar widths. **b,** Example where OFD and inelastic strain zone have different widths. The dashed horizontal lines indicate the ±0.5% dilatation strain threshold. The red spline shows the fit to the strains from which the permanent deformation zone is inferred (dark-grey region). The black lines show the across-fault displacements, and the light-grey region shows the OFD zone width. **c,** Comparison of inelastic-deformation-zone width with across-fault offset magnitude.

<figure><img src="image_6.png" /></figure>

**Fig. 4** | Surface-strain orientations and mylonite-like surface deformation patterns. **a,** 2D dilatational strains overlain by principal strain orientations and magnitudes (white, extension; black, shortening; red, surface rupture). **b,c,** Trends of the greatest principal strains (white lines and numbers, in degrees) and the earthquake surface rupture (red lines and numbers, in degrees) within the profiles of **a.** **b,** Strain trends in the western profile. **c,** Strain trends in the eastern profile. The black arrows indicate the trend of the *P* axis of the *M*<sub>w</sub> 7.1 mainshock. **d,** Interpreted photograph of a dextral mylonitic shear zone (Naxos, Greece) exhibiting an S-C fabric (white and red lines) with similar orientations to surface strains shown in **b** and **c**.

displacement or surface materials (Figs. 2b and 3c). The narrow inelastic strain zone of the Ridgecrest earthquakes, instead, is consistent with active deformation within a fault damage zone. The width of the inelastic zone is likely influenced by two factors: the rupture velocity of the earthquakes and the maturity of the active faults. Previous work showed that stress concentrations at a propagating rupture tip narrow as the rupture velocity increases towards the limiting Rayleigh wave velocity, meaning that the inelastic strain zone should decrease with increasing rupture velocity³⁵⁻³⁸. At the same time, inelastic failure zones, as represented by damage zones, systematically widen with accumulated slip on a fault¹⁶,³⁹. Within a single rupture, the relative width of the inelastic failure zone is

thus influenced by the earthquake rupture velocity and the distance from the hypocentre³⁶, and these factors are superimposed on the absolute width of the inelastic failure zone that is governed by the long-term slip history of the fault itself³⁶. In the Ridgecrest sequence, seismic back-projection models indicate that rupture velocities varied from 0.6 to 1.5 km s⁻¹ (ref. ⁴⁰). The inelastic-strain-zone width, in turn, decreased with increasing rupture velocity from 32 m + 14/-6 m at 0.6 km s⁻¹ to 31 m + 13/-7 m at 1.4 km s⁻¹ and finally to 29 + 20/-11 km s⁻¹ at 1.5 km s⁻¹, as predicted from previous studies³⁵⁻³⁸ (Extended Data Fig. 6). The inelastic-strain-zone widths of the 2013 Baluchistan, Pakistan and 2016 Kumamoto, Japan, earthquakes, however, were nearly an order of magnitude
wider (>175–250 m) (refs. 20,21) despite comparatively faster rupture velocities of ~3 km s⁻¹ (refs. 41–43). The faults that ruptured in those events, however, were more mature. Offset geomorphic markers along the Hoshab fault that hosted the Baluchistan earthquake indicate cumulative slip magnitudes of 100–200 m (ref. 31), leading to an expected damage-zone width >100 m (ref. 39). The Ridgecrest faults are understood to be immature structures²⁷,⁴⁴,⁴⁵, so the absolute width of the inelastic failure zone should be narrow despite the low rupture velocities. Similarly, the Baluchistan earthquake encompassed a longer rupture (~200 km versus ~49 km) wherein fault damage zones are expected to increase in width³⁶.

The orientations of principal strains within the high-strain, inelastic deformation zone are another important characteristic of the strain field (Fig. 4). The largest principal strains are extensional and oriented at acute angles to the seismological P axes and local fault strike. These strain orientations form an S–C foliation that is commonly described in mylonites (Fig. 4b,c). Here, the C foliation, or the principal-slip-plane foliation, is demarked by the fault strike and is subparallel to the shear strains (oriented 45° to the principal strains, Fig. 4). The extensional axes, in turn, demark an S foliation that is oblique (~30°) to the main shear fabric. Given these characteristics, we can relate the shallow fault-zone kinematics to the development of a mylonite zone in a mixed brittle–ductile regime⁴⁶,⁴⁷, providing further justification for the imposition of elasto-plastic rheologies in dynamic rupture simulations. These results for single earthquake ruptures mirror field observations from deeper fault zones that suggest ductile shear localization occurs in concert with brittle dilatant fracturing⁴⁷ (Fig. 4d). The notion of brittle–ductile mylonitic deformation near the Earth’s surface further supports the hypothesis that the characteristics of shallow slip deficits reflect reduced co-seismic slip magnitudes due to frictional changes, rather than the initiation of broadly distributed deformation²,⁴⁸,⁴⁹. Future detailed surface-strain studies are needed to rigorously compare strain characteristics with earthquake slip distributions, local geology and co-seismic rupture characteristics (that is, rupture velocity) and systematically test these hypotheses.

## Online content

Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41561-020-0628-8.

Received: 31 January 2020; Accepted: 29 July 2020;
Published online: 7 September 2020

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## Methods

**Horizontal displacements: sub-pixel cross correlation.** We derived co-seismic horizontal displacements of the Ridgecrest earthquakes through sub-pixel cross correlation of satellite optical imagery from the DigitalGlobe satellites Worldview-1 (WV01), Worldview-2 (WV02) and Worldview-3 (WV03) (Supplementary Table 1, Fig. 1, S1). We first acquired pre- and post-event in-track stereo image pairs. Pre-event stereo images were acquired between September and December 2014, and post-event images were acquired between July and August 2019. We then constructed 0.5 m spatial resolution DSMs from each in-track stereo pair using the open source Surface Extraction with TIN-based Search-space Minimization (SETSM) algorithm$^{20,51}$. In addition to generating a DSM, SETSM orthorectifies the non-orthorectified base images using the constructed DSM at the 0.5 m resolution of the DSM. We use the SETSM-generated orthorectified images because they entail smaller orthorectification errors in the panchromatic images that introduce topographically correlated offsets in the sub-pixel offset results. For WV01 and WV02 images, we applied corrections to mitigate CCD (charge-coupled device) boundary artifacts using the NASA Ames StereoPipeline software. We applied no other a priori image corrections to the orthorectified images.

We additionally orthorectified an image acquired 4 July 2019 in between the $M_w$ 6.4 foreshock and $M_w$ 7.1 mainshock. We orthorectified this image using the Ames StereoPipeline and a pre-event DSM$^{52}$. Last, we included three panchromatic images orthorectified by DigitalGlobe acquired in July 2018 in our analysis with no topographic corrections applied to these images. These images introduce substantial orthorectification errors in regions of steep topography west of the $M_w$ 7.1 surface rupture, but they allow for co-seismic offset retrieval along the central portion of the $M_w$ 7.1 surface rupture where earlier pre-event scenes decorrelate.

We cross-correlated overlapping pre- and post-event orthorectified images generated with SETSM using AMPCOR$^{31,53}$, a correlation window of 16 pixels, and step sizes of 4 pixels, resulting in 2-m-output spatial resolution of the horizontal-displacement fields (Fig. 1, S1). The horizontal displacements derived from each satellite swath (Supplementary Table 1) were then referenced to the scene pair spanning 28 November 2014 to 13 July 2019 because this pair spatially covers both the earthquake and a substantial region with 'zero' displacement.

To gain perspective of the consistency of our 2D displacement field with other results, we compared our displacement field derived from WorldView imagery with one derived from Planet Labs imagery (Milliner and Donnellan$^{30}$) (Extended Data Fig. 7). We combined independently constrained displacement fields of the $M_w$ 6.4 and $M_w$ 7.1 by summing the two displacement fields of ref. $^{33}$, and then we resampled our 2-m-resolution displacement field to the 87 m spatial resolution of the Planet Labs results. Differencing these two displacement fields shows that there are no substantial or systematic residuals, indicating that our displacement field results are consistent with independently derived results (Extended Data Fig. 7).

**Vertical displacements: DSM differencing.** We derived co-seismic vertical displacements of the Ridgecrest earthquakes using the 0.5 m DSMs generated by SETSM during our image orthorectification step (Fig. 1c and Extended Data Fig. 2). In strike-slip earthquakes, horizontal surface offsets can cause misregistration of pre- and post-event topography, which in turn leads to vertical displacement biases when pre- and post-event datasets are simply differenced$^{30,54}$. However, when the ground surface is relatively flat, as it is for much of the valley where the Ridgecrest sequence occurred, horizontal translation of topography does not introduce biases$^{30}$. This allows us to directly difference the DSMs and retain the spatial resolution of the input topography without using an approach that requires decimation of the observations, such as with the iterative closest point approach (for example, Nissen et al.$^{21}$; Scott et al.$^{21}$) (Extended Data Fig. 2). To minimize potential effects of both DSM and topographic misregistration, we first extracted a subregion of topography from each DSM that was located as far as possible from the surface rupture. We then co-registered this subregion to a single 2 m optical DSM that itself was registered to ICESat (Ice, Clouds and Land Elevation Satellite) altimetry measurements$^{52}$ using the iterative closest point algorithm in the NASA Ames StereoPipeline$^{55}$. Finally, we applied the resulting transform to the entire pre- or post-event DSM. In this methodology, each of our pre- and post-event DSMs is registered to a consistent elevation reference frame. We then differenced the pre- and post-event pairs to retrieve the vertical displacement field (Fig. 1b).

**2D surface strain.** After deriving co-seismic surface displacement fields, we inverted the surface displacements for 2D surface-strain tensors (Fig. 2a and Extended Data Fig. 3). The general approach to estimating the 2D strain tensor is to invert surface displacements for the displacement gradient tensor, $L$, and offset of the reference frame (effectively, the geometric centre of the region of interest) and then to decompose $L$ into the symmetric infinitesimal strain ($\epsilon$) and antisymmetric rotation ($\omega$) tensors$^{20,56}$:

$$L = \epsilon + \omega \quad (1)$$

$$\epsilon_{ij} = 1/2(L_{ij} + L_{ji}) \quad (2)$$

$$\omega_{ij} = 1/2(L_{ij} - L_{ji}) \quad (3)$$

We retrieve the shear strain and the rotation magnitude directly from $\epsilon$ and $\omega$, and we retrieve the principal strains and strain orientations through a singular value decomposition of $\epsilon$.

The inversion for $L$ and reference frame offset in 2D has six unknowns (two components of offset and four components of $L$), so the inversion is perfectly determined for three displacement observations (two components of displacement for each single observation). We solve to $L$ using a regularization approach wherein we calculate the mean displacement gradient tensor for groups of more than three observations$^{20,21,56}$. In this approach, we extract displacements from cells within a grid of prescribed dimensions (10×10 m cells, or 25 observations points per cell), and then invert for $L$ through a regularized inversion where we impose minimum moment regularization so that there is no expectation of spatial smoothing within or between grid cells$^{57}$. We chose a uniform weighting coefficient of 0.25 from an $L$-curve trade-off curve tested on 10 different strain boxes drawn from low- and high-strain regions. The resulting strains and rotations describe the mean strain and rotation within each cell (that is, strain across a 100 m$^2$ area in the case of 10×10 m cells). This inversion is overdetermined, which provides some redundancy and less sensitivity to individual noisy displacements. In general, the regularized inversion approach yields a strain field where we can identify strain thresholds that suggest transition between elastic and inelastic deformation.

The choice of both strain cell dimensions and regularization coefficient have the potential to bias our inversion for $L$ and interpretation of the inelastic-strain-zone width. For example, choosing a larger strain cell box may lead to a more robust inversion, but it also affects the spatial resolution of the strain results and potentially smooths the permanent strain zone width. Accordingly, we tested a range of both strain cell sizes and regularization weights to determine what effect our choices of 10×10 m and 0.25 might have on our derived strain results (Extended Data Figs. 8 and 9). In each case, we highlight strain profiles across the rupture. In general, increasing the spatial dimensions of the strain cell (Extended Data Fig. 8) leads to a smoother and less noisy strain field with fewer independent strain measurements within a given area. However, at the strain cell resolutions tested (triangulation (highest possible resolution) to 30×30 m boxes), the choice of strain cell size does not notably impact the width of the inferred permanent strain zone. Given that we cannot discriminate inelastic deformation of the wall rock from the across-fault offset in strain cells that cross the surface rupture, the dimensions of the strain cell itself reflect the minimum inelastic-strain-zone width we can resolve.

Similarly, changes in the weighting coefficient (Extended Data Fig. 9) have no detectable effect on the inferred inelastic-strain-zone width or the spatial distribution of surface strains. Variations in weighting coefficient appear to affect the inferred peak strain magnitudes; therefore, we do not apply meaningful interpretations to the peak strain magnitudes we find in this study.

**3D surface strain.** We additionally characterize the 3D surface-strain field using vertical displacements derived from our differed DSMs (Extended Data Fig. 4). The inversion approach is the same as that described for 2D surface strains, except that we augment the Green's function in the inversion for $L$ with the elevation of each data point, and we include vertical displacements in the data vector$^{20}$, such that $L$, $\epsilon$ and $\omega$ are 3×3 tensors. This inversion has 12 unknowns, and each observation point contributes three independent observations (dX, dY and dZ), so four or more observations points are necessary for the perfectly and overdetermined inversions, respectively. In general, the noise of the vertical displacements leads to noisy strain results where there is a z component that we cannot interpret at the ±0.5% strain threshold for inelastic deformation.

**Surface-slip distribution and OFD zone.** We extract two descriptors of the Ridgecrest surface ruptures from the horizontal-displacement fields: the surface-slip distribution and the OFD zone (Figs. 2b and 3a,b and Extended Data Fig. 5). To retrieve these values, we first mapped a surface fault trace of ground ruptures from the horizontal surface-displacement field (Fig. 1d). We then discretized the rupture traces into 10 m intervals and extracted east-west and north-south displacements from fault-normal profiles that extend 500 m on either side of the fault (Fig. 3a,b). For each profile, we first fit reprojected each profile into fault-normal and fault-parallel components. We then fit a line to the displacements on either side of the fault and projected the line into the fault trace. The distance between these two lines is taken as the across-fault offset (Fig. 3a,b). We excluded displacements within 100 m of the fault when doing this to reduce the impacts of OFD on the inferred across-fault displacement. To measure the zone of OFD, we measured the horizontal distance in each profile between where the linear fits diverge from the displacement profile near the fault in the fault-parallel component (Figs. 2b and 3a,b and Extended Data Fig. 5). We measured this quantity manually.

**Inelastic strain zone.** We measured the inelastic-strain-zone width (Fig. 2b and Extended Data Fig. 5) by first fitting a smoothing spline to the across-fault profiles of strain. We use strains of ±0.5% to define the threshold of inelastic deformation on the basis of reporting in previous studies$^{1,21}$. This value is derived from laboratory experiments and from comparing the ratio of yield stress ($\sigma_{yield}$) to Young's Modulus ($E$) (for example, Scott et al.$^{21}$):

$$\epsilon_{yield} = \sigma_{yield}/E \quad (4)$$
After fitting smoothing splines to the strain profiles, we found where the smoothing spline intersected the ±0.5% strain threshold on either side of the fault and measured the distance between these intersections (Fig. 3a,b, red profiles). Given that the zone of inelastic strain is systematically wider for dilational strains than for shear strains, our discussion and conclusions are based on dilational strains.

Where we report average values and uncertainties of OFD and inelastic-strain-zone widths (from both dilatation and shear, Fig. 2b and Extended Data Fig. 6), these values reflect the median value and 16th and 84th percentiles of all measurements. We use these statistical descriptions because we do not expect these widths to vary as a Gaussian distribution.

## Data availability

Derived 2 m DSMs used for image orthorectification and vertical differencing are available at Open Topography (https://doi.org/10.5069/G9TM789K and https://doi.org/10.5069/G9ZC8119). Horizontal and vertical displacement fields and complete 2D and 3D strain maps are available through the US Geological Survey Science Base-Catalog (https://doi.org/10.5066/P9QRZ6NR). No raw imagery is shown in this manuscript. Raw imagery details are given in the Supplementary Information and can be obtained independently through the appropriate provisions of the NextView license or commercial access.

## Code availability

MATLAB scripts used to generate co-seismic strain fields can be accessed through https://github.com/williamBarnhart/strainCodes.

## References

51. Noh, M.-J. & Howat, I. M. Automated stereo-photogrammetric DEM generation at high latitudes: Surface Extraction with TIN-based Search-space Minimization (SETSM) validation and demonstration over glaciated regions. *GISci. Remote Sens.* **52**, 198–217 (2015).

52. Willis, M. J. et al. CaliDEM: Ridgecrest, CA Region 2m Digital Surface Elevation Model (Digitl Globe and Open Topography, 2019); https://doi.org/10.5069/G998854C

53. Rosen, P. A., Hensley, S., Peltzer, G. & Simons, M. Updated repeat orbit interferometry package released. *Eos* **85**, 47 (2004).

54. Oskin, M. E. et al. Near-field deformation from the El Mayor–Cucapah earthquake revealed by differential LIDAR. *Science* **335**, 702–705 (2012).

55. Beyer, R. A., Alexandrov, O. & McMichael, S. The Ames Stereo Pipeline: NASA’s open source software for deriving and processing terrain data. *Earth Space Sci.* **5**, 537–548 (2018).

56. Allmendinger, R. W., Reilinger, R. & Loveless, J. Strain and rotation rate from GPS in Tibet, Anatolia, and the Altiplano. *Tectonics* **26**, TC3013 (2007).

57. Menke, W. *Geophysical Data Analysis: Discrete Inverse Theory* (Academic Press, 2012).

## Acknowledgements

W.D.B. was supported in part by NSF/USGS SCEC Awards 17086 and 16147. We acknowledge the DigitalGlobe/NextView licensing agreement through which we accessed pre-earthquake and post-earthquake WorldView imagery. Any use of trade, firm or product names is for descriptive purposes only and does not imply endorsement by the US Government. This research was supported in part through computational resources provided by The University of Iowa.

## Author contributions

W.D.B. and R.D.G. designed the study. W.D.B. processed horizontal and vertical displacements fields, derived strain fields and generated interpretable materials. R.D.G. facilitated data access. J.H. processed imagery spanning the $M_w$ 6.4 foreshock. All authors contributed to the writing of the manuscript.

## Competing interests

The authors declare no competing interests.

## Additional information

**Extended data** is available for this paper at https://doi.org/10.1038/s41561-020-0628-8.

**Supplementary information** is available for this paper at https://doi.org/10.1038/s41561-020-0628-8.

**Correspondence and requests for materials** should be addressed to W.D.B.

**Peer review information** Primary Handling Editor: Stefan Lachowycz.

**Reprints and permissions information** is available at www.nature.com/reprints.
<figure><img src="image_7.png" /></figure>

**Extended Data Fig. 1 | North-South Horizontal Displacements.** Mosaic of north-south horizontal displacement field spanning both Mw6.4 foreshock and Mw7.1 mainshock (east-west and vertical displacements are shown in Fig. 1a, b).
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

**Extended Data Fig. 2** | Vertical Displacement Comparison between Differencing and ICP approaches. a) Vertical displacement field generated by differencing pre- and post-seismic DSMs (negative is co-seismic subsidence). b) Vertical displacement field generated by applying a windowed ICP approach to pre- and post-seismic DSMs (negative is co-seismic subsidence). c) Overlapped comparison of vertical displacements from profile X-X' highlighted in panels a and b. n is the total number of observations within the profile.
<figure><img src="image_10.png" /><figcaption>**Extended Data Fig. 3** | 2D Horizontal Strain Fields. a) S1 (greatest principal strain) component of the 2D strain field. b) S2 (smallest principal strain) component of the 2D strain field. c) Shear strain component of the 2D strain field.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Extended Data Fig. 4** | 3D Complete Strain Fields. Each component of the full 3D strain field derived from horizontal and vertical displacements. **a)** S1 (greatest principal strain), **b)** S2 (intermediate principal strain), **c)** S3 (smallest principal strain), **d)** Shear between the north-east directions, **e)** shear between the north-vertical directions, **f)** shear between the east-vertical directions, **g)** dilatation.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Extended Data Fig. 5** | Map-view Comparison of Inelastic Strain Zone Width and Off-Fault Deformation Zone. Map view depictions of fault zone width (FZW) as defined by the inelastic zone width (as defined by dilatation and shear strains exceeding +/- 0.5%) and the off-fault deformation (OFD) zone width that is mapped from across-fault offsets (Fig. 3a, b). The along-rupture location of these measurements are shown in Fig. 2b. **a)** Dilatation zone width, **b)** shear zone width, **c)** off-fault deformation zone width.</figcaption></figure>
<figure><img src="image_13.png" /></figure>

**Extended Data Fig. 6** | Comparison of Rupture Velocity to Inelastic Strain Zone Width. Comparison of median and 16^th^/84^th^ percentile inelastic strain zone widths to earthquake rupture velocities (a) reported by Yang et al. (2020) for segments I-III shown in panel b. The strain zone widths show an inverse relationship with rupture velocity. The black star in b indicates the epicenter of the M<sub>w</sub> 7.1 mainshock.
<figure><img src="image_14.png" /><figcaption>**Extended Data Fig. 7** | Comparison of WorldView and Planet Horizontal Displacement Fields. Comparison of horizontal displacements derived from WorldView imagery in this study to horizontal displacements derived from Planet Labs imagery (Milliner and Donnellan, 2020). All displacements are resampled to the 87-meter resolution of the Planet Labs results. **a**) E-W component from this study, **b**) E-W component of Milliner and Donnellan (2020), **c**) difference between **a** and **b**, **d**) N-S component from this study, **e**) N-S component from Milliner and Donnellan (2020), **f**) difference between **d** and **e**.</figcaption></figure>
<figure><img src="image_15.png" /></figure>

**Extended Data Fig. 8 | Effects of Strain Cell Size on Surface Strain Characteristics.** Illustration of how strain inversion cell size impacts strain fields from the 2019 Ridgecrest earthquakes. The top row shows the resulting shear strain field when inverted using triangulation (**a**), and 4x4m (**b**), 10x10m (**c**), and 30x30m (**d**) averaging cells respectively. **e) Strains within the black profile box.** Regions in the anelasticity zone exceed the expected failure strain (0.5% strain), while regions in the elasticity zone do not fail. Changes in the strain cell resolution serve to create a smoother strain field with smaller peak strain values, but they also do not significantly impact the interpreted permanent strain zone width. The values in the profile are plotted as the absolute value of the shear strain, in logarithmic space.
<figure><img src="image_16.png" /></figure>

**Extended Data Fig. 9 | Effects of Inversion Regularization on Surface Strain Characteristics.** Illustration of the sensitivity of strain inversions to regularization coefficients of 0.01 (a), 0.1 (b), 0.25 (c), and 0.5 (d) for the same region shown in Extended Data Fig. 8. e) Difference between panels d and a (0.5 vs. 0.01) showing that more regularization (0.5) leads to smaller peak strain values than when little regularization (0.01) is imposed, but that the spatial characteristics of strain do not change. f) Strains within the black profile box. Regions in the anelasticity zone exceed the expected failure strain (0.5% strain), while regions in the elasticity zone do not fail. The values for each of the weighting values are similar such that the different strain values are overlapping and cannot be distinguished from each other.