Inversion for stress orientations
We employed the package developed by ref. 2, which uses the stress inversion of ref. 1. This method minimizes the difference between the direction of shear stress imposed by a normalized stress tensor on a nodal plane and the actual slip on that plane. The deviatoric part of the average background stress orientation can be retrieved given a sufficient number of focal mechanisms that rupture under a consistent stress condition. Under the further assumption that the amplitude of shear stress at failure is the same for all events, the inversion is linear. The shape ratio is defined as
$$R=\frac{{\sigma }_{1}-{\sigma }_{2}}{{\sigma }_{1}-{\sigma }_{3}}$$
1
,
where \({\sigma }_{1}\) and \({\sigma }_{3}\) represent the most compressional and tensional axes, respectively. Ref. 2 introduced the notion of fault instability in order to assess which of the two nodal planes is more likely to fail in a given stress state (comprising the stress orientations and shape ratio). This leads to higher accuracy of the shape ratio.
Focal mechanisms were taken from a refined catalog using the HASH method (refs. 22–23; extended), including earthquakes from 1981 to 2022 (Fig. 1). Only high-quality events (quality A or B) were retained. Events within 2 km of the finite-fault model (ref. 19) after the M7.1 mainshock were excluded (Fig. 1b) because near the fault the finite-fault models generate highly heterogeneous coseismic stress changes where neighboring sub-faults adjoin. Thus, most of the very-close-in aftershocks of the doublet were excluded (Fig. 8 in ref. 12). The uncertainty of each focal mechanism was utilized to generate noisy solutions for assessing the results (Supplementary Text S1).
According to synthetic tests (ref. 2), ~ 50 focal mechaisms provide adequate accuracies in both stress orientation and shape ratio which do not improve much upon including more focal mechanisms. A uniform number of focal mechanisms in each block leads to more comparable uncertainties in inverted stress orientations.
In addition to the aforementioned assumptions required to establish the inverse problem, it is desirable that the focal mechanisms be adequately diverse but also consistent with rupture in one stress field (refs. 1, 8, & 13). The diversity of focal mechanisms can be quantified by the RMS angular difference between the focal mechanisms and a “mean focal mechanism” (refs. 8 & 13). We adopt the inverted stress orientation as the “mean focal mechanism”, and measure the angular difference by the Kagan angle (ref. 18). The calculated RMS angular differences are labeled with each stress orientation (upper right corners in Fig. 2), mostly satisfying the diversity requirement (greater than ~ 35–40º if the focal mechanism errors are ~ 10–20º, according to ref. 8).
The assumption of a consistent stress field can be assessed by the misfit angle, which is the median difference between the observed slip vectors and the ones predicted by the inverted stress orientation. Ideally, diverse and noise-free focal mechanisms would look like two “butterfly wings” of P- or T-axes around \({\sigma }_{1}\) or \({\sigma }_{3}\) orientations, respectively (ref. 2). The deviation from that pattern reflects the misfit angle (Fig. 2). Our 12 blocks all satisfy the consistent-stress-field assumption (less than ~ 35–40º misfit angle if the focal mechanism errors are ~ 10–20º, according to ref. 30).
Computing representative coseismic stress change for each block
We assumed 30GPa for both Lamé parameters to compute the coseismic stress changes, based on the analytical static stress solution in a homogeneous elastic half space derived by ref. 21. The trace of the calculated coseismic stress change tensor was then removed.
The following inversion requires one representative coseismic stress change tensor for each block. We employed three different modes to capture it: 1) Following ref. 10, we averaged coseismic stress changes at all post-event hypocenters within a block, referred to as the “mean” coseismic stress mode; 2) We adopted the coseismic stress change at the centroid of post-event hypocenters as the “centroid” coseismic stress mode; 3) In the “pixel inversion”, we adopted the in-situ coseismic stress change at each pixel.
The “mean” mode results in slightly smaller coseismic stress changes than the “centroid” mode (Supplementary Fig. S5) when averaging tensors. The “centroid” and the in-situ modes would sample the near-fault heterogeneous coseismic stresses, whereas the “mean” mode would not, due to the removal of the near-fault seismicity. Thus, we opted to use only the “mean” mode in our systematic pixel inversion.
Inversion for full deviatoric stress tensors before and after the doublet
The stress orientations before and after an earthquake, coupled with the coseismic stress change, can be used to estimate the absolute deviatoric stresses, enabling the reconstruction of the full pre-event and post-event deviatoric stress tensors at seismogenic depths (refs. 10–11). Analytic methods, such as those by refs. 7–8, and inversion approaches (refs 10 & 32) have been introduced in the literature, as reviewed by ref. 11. The analytic methods simplify the problem into a 2-D model, relating the rotation of the stress orientation to the ratio of stress drop to the maximum shear stress (ref. 8) or the shear stress on the fault (ref. 7). Their assumptions about the coseismic stress change have been substantiated for simple strike-slip faults (ref. 32); however, they are not valid in off-fault areas. The inversion approach used here (following refs. 10 & 17) accounts for the full 3D coseismic stress tensors and is capable of addressing complexities in off-fault areas. While the initial application (ref. 10) still divided the fault into several cross-fault segments, our blocks avoid extending over both sides of the fault to eliminate the singularities and uncertainties of coseismic stress changes derived from unsmoothed slip models.
Since we treated the 12 blocks as an ensemble prior to the Ridgecrest doublet to enhance stability, the inversion approach from ref. 10 was adapted as follows:
$$\left[\begin{array}{c}\varDelta {\varvec{S}}_{1}\\ \begin{array}{c}\varDelta {\varvec{S}}_{2}\\ \dots \end{array}\\ \varDelta {\varvec{S}}_{12}\end{array}\right]=\left[\begin{array}{ccc}{\varvec{S}}_{1}^{after}& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & {\varvec{S}}_{12}^{after}\end{array}\begin{array}{c}{-\varvec{S}}^{before}\\ ⋮\\ {-\varvec{S}}^{before}\end{array}\right]\left[\begin{array}{c}{C}_{1}^{after}\\ \begin{array}{c}{C}_{2}^{after}\\ \dots \end{array}\\ \begin{array}{c}{C}_{12}^{after}\\ {C}^{before}\end{array}\end{array}\right]$$
2
,
where \(\varDelta {\varvec{S}}_{i}\) and \({\varvec{S}}_{1}^{after}\) are the representative deviatoric coseismic stress change and the normalized post-event stress tensor in block \(i\), respectively. \({\varvec{S}}^{before}\) is the normalized pre-event uniform stress tensor for all 12 blocks. Each of these stress tensors comprises six components arranged vertically. Although only five components of a deviatoric stress tensor are independent, we have constructed the matrix with six components to equally weight the misfits among them as in ref. 17. After re-normalizing the stress orientation tensors by their eigenvalue ranges, \({C}_{i}^{after}\) and \({C}^{before}\) represent the differential stresses (twice the maximum shear stresses) in post-event block \(i\) and for the pre-event unified 12 blocks, respectively. To solve Eq. 2, we utilized the non-negative least-square function LSQNONNEG, implemented in MATLAB.
Pixel inversion
We applied a systematic inversion by partitioning the study area into small “pixels” at 0.01º intervals in latitude and longitude. For each pixel before and after the Ridgecrest doublet, we utilized the 45 focal mechanisms before and after the doublet (picked from the quality A or B mechanisms in Fig. 1) closest to the pixel to constrain its stress orientations. Near the fault, the coseismic stress changes appear highly heterogeneous, primarily due to the unsmoothed slip as well as the bends between fault segments. As such, we opted to use only the “mean” mode in our systematic inversion. Given the substantial size of the combined matrix (Eq. 2) of 91\(\times\)101 pixels, amounting to 55146\(\times\)9192 elements, it is practically challenging to perform the non-negative inversion of Eq. (2) on such a scale. Therefore, we inverted for each pixel individually. Thus, here the pre-event shape ratio (Fig. 5a) and stress (Fig. 5d) can vary across the study region.
Quantifying the variability of stress orientations
To quantify the variability of post-event stress orientations, we first determined their centroid tensor using the MATLAB function KMEANS. We represented each stress orientation as a vector comprising 9 components and employed the ‘cosine’ distance metric within KMEANS. We computed the mean Kagan angle between the centroid and each post-event stress orientation to quantify the variability of stress orientations. This procedure reveals increased stress heterogeneity following the doublet for both regionalization approaches.