## Introduction

Faults slip in a variety of modes, from dynamic earthquakes to transient slow-slip events and aseismic slip1,2. The classical picture of faults being either locked and prone to dynamic and damaging earthquakes or unlocked and quietly slipping to accommodate tectonic stress is evolving. Growing evidence indicates complex fault behaviors and interactions among and between modes of slip3. Evidence includes fault segments hosting both slow and dynamic earthquakes, as well as slow earthquakes preceding and possibly triggering the nucleation phase of dynamic earthquakes4,5,6. Answering a number of fundamental questions, such as what controls the slip mode on a fault, whether there exists a continuous spectrum of slip modes on faults, and what determines the possible evolution of a slow earthquake into a dynamic seismic rupture, requires exhaustive characterization of all slip phenomena. Interferometric Synthetic Aperture Radar (InSAR) holds the promise of continuous geodetic monitoring of fault systems at a global scale, which may well hold the key to address these questions. However, although the data exists, current algorithms are not suited for global monitoring because they require time-consuming manual intervention, and the final product requires exhaustive expert interpretation.

InSAR is routinely used to measure ground deformation due to hydrologic, volcanic, and tectonic processes7,8,9. The apparent range change in the satellite Line Of Sight (LOS) between two SAR acquisitions is, after corrections from orbital configurations and topography, the combination of atmospheric propagation delay, changes in soil moisture and vegetation, and actual ground deformation. Rapid, large-amplitude deformation signals such as coseismic displacement fields often exceed the amplitude of sources of noise10. Similarly, slow but steady accumulation of deformation over long periods of time may be quantified using InSAR either through stacking11 or time-series analysis12,13. However, detecting low-amplitude deformation related to transient sources such as slow-slip events, episodes of volcanic activity, or hydrologic-related motion remains challenging and requires significant human intervention and interpretation8,14,15. Measuring Earth surface deformation is fundamental to characterizing diverse tectonic processes, as well as surface and underground changes induced by human activities.

The most pressing issue in InSAR processing for small, millimeter-scale, deformation monitoring remains the separation between atmospheric propagation delays and ground deformation. Spatial and temporal variations in atmospheric pressure, temperature, and relative humidity modify the refraction index of the air, resulting in spatial and temporal delay variations in the two-way travel time of the radar carrier between a SAR imaging satellite and the ground16,17. Such delays directly affect the phase of an interferogram, which combines two SAR acquisitions. Atmospheric propagation delays in a single interferogram can be equivalent to tens of centimeters in apparent range change16. Current correction methods based either on empirical estimations18,19 or on independent data20,21,22,23 reduce the contribution of the stratified atmosphere—the long-wavelength atmospheric perturbation that, to a first order, correlates with topography. Nonetheless, remaining delays, corresponding to the turbulent portion of the troposphere may represent centimeters of apparent range change. Propagation delays in the atmosphere decorrelate after periods of 6–24 h, as shown by the temporal structure–function of Global Navigation Satellite System (GNSS) zenith delays24. Therefore, remaining tropospheric delays, which are coherent in space, can be considered random in the time given the time span between SAR acquisitions (e.g., 6 days for Sentinel 1, 46 days for ALOS-2). Moreover, it can be shown that, because of potential temporal aliasing17 and loss of spatial coherence of the radar phase echo, spatio-temporal filtering can lead to biased results. Therefore, deciphering a consistent, days- to month-long transient signal in the time series of InSAR data remains a critical challenge, especially when automation is envisioned.

Convolutional neural networks are central to the most recent dramatic advances in computer vision and natural language processing. Autoencoders have been developed to create sparse representations of data—the model copies its input to its output through a bottleneck that forces a reduction of dimension equivalent to a compressed knowledge representation of the original input, enabling noise removal. Of note are recent developments applied to classify InSAR data in order to detect ground uplift and subsidence, and specifically to identify volcanic unrest25,26,27,28. Although promising, these developments do not make use of the different temporal signatures of signals of interest to reconstruct denoised deformation patterns.

Here, we describe a deep learning-based method to automatically detect and extract transient ground deformation signals from noisy InSAR time series. Our approach, based on a purely convolutional autoencoder, is specifically designed for removing noise in InSAR time series. In the following, we consider the evolution of the interferometric phase with time with respect to a reference both in space and time. We consider classical Small Baseline (SBAS)-like approaches for the reconstruction of the time series29,30. The time series we analyze stem from the inversion of a sequence of SAR interferograms previously corrected from orbital and topographic contributions31, with a first-order atmospheric correction derived from global atmospheric reanalysis products21,32. Our autoencoder takes as input a noisy InSAR time series reconstructed from successive SAR acquisitions, and outputs accumulated ground deformation taking place during the time-series interval, with the atmospheric noise removed.

In this work, we first introduce the notion of autoencoders before describing the architecture of our neural network. We then describe our training set and perform preliminary tests on synthetic data. We finally highlight the efficiency of our algorithm on two reconstructed time series of ground deformation, the first one derived from COSMO-SkyMed acquisitions and the second one derived from Sentinel 1A–B SAR acquisitions.

## Results

### Description and validation of the deep autoencoder

#### Autoencoder architecture

Our goal is to extract ground deformation from noisy InSAR time series. For the purpose of training our deep learning model, we assume that input time series are the combination of three physical contributions: ground deformation, the stratified component of the atmosphere, and the turbulent component of the atmosphere. In most cases, the stratified component can be corrected for using Global Atmospheric Models (hereafter referred to as GAMs, often corresponding to reanalysis products), e.g.32,33, or GNSS data, e.g.34, for instance. However, such a correction is often incomplete and part of the remaining, often turbulent, atmospheric delays may still correlate with topography. Attempts have been made to estimate tropospheric delays using multispectral radiometric data20; however, the acquisition of such independent data must be coincident with the SAR acquisition and over a terrain with minimal cloud cover for optimal performance, conditions rarely met. In addition, it can be shown that GAM-derived correction sometimes worsens the situation as the local estimate of the state of atmospheric variables may be incorrect32.

Our deep learning model must recognize transient deformation in InSAR time series in the presence of remaining atmospheric noise. To this end, it must distinguish the spatial and temporal statistical differences between deformation signals and atmospheric patterns. As mentioned above, the structure of atmospheric delays decorrelates for periods longer than 6 h24. Ground deformation related to transient tectonic events takes place over seconds to minutes for dynamic rupture and to weeks or months or even years for slow-slip events14,15,35,36, and remains until further deformation occurs. Therefore, the temporal signature of deformation signals is very different from that of atmospheric delays. We make use of this different temporal signature to learn appropriate filters to remove atmospheric perturbations and extract ground deformation in the InSAR time series.

Here, we build and train an autoencoding architecture to directly output the deformation signal, formulating the problem as a regression task. We rely on the following assumptions: (1) atmospheric delays are random in time, considering two successive SAR acquisitions, (2) ground deformation has a temporal coherence considering the rate at which SAR images are acquired, and (3) part of the atmospheric delay correlates with topography. We, therefore, use as inputs a time series of interferometric phase change and a map of ground elevation to produce a time series of cumulative surface displacements.

In order to separate deformation from atmospheric delays, we developed the deep learning architecture shown in Fig. 1. This architecture consists of 11 purely convolutional layers. The first six layers of the model are tasked with encoding signals that are persistent in time, by progressively removing the time dimension of the input. At the seventh layer, ground elevation (topography) is added as a secondary input. The remaining layers decode the ground deformation map. In short, we build a model tasked with reconstructing ground deformation given input InSAR time series and ground elevation from noisy input.

Initially developed for feature extraction by projecting high-dimensional data sets onto a lower-dimension manifold by forcing the reconstruction of the data through a bottleneck in deep learning architectures37, autoencoders have also evolved into powerful denoising38,39 and image enhancing techniques40,41. In this work, we exploit this aspect of deep learning autoencoding and tailor it to the problem of cleaning InSAR time series, building a deep learning autoencoder to effectively automate the design of filters in time and space to recover ground deformation.

#### Training on synthetic data

Because deep learning models require large amounts of data and there exists no ground truth for InSAR time series, we rely on synthetic data to train the deep autoencoder. The synthetic data are randomly generated cumulative surface deformation time series mimicking nine successive maps of range change. These cumulative deformation maps include surface displacements in the LOS due either to a slipping fault (either strike-slip or dip-slip) with random latitude and longitude (position in a virtual box), depth, strike angle, dip angle, and width (based on Okada’s model42) or to an inflating or deflating point source (Mogi’s model43). Deformation onset occurs at a random time as a pulse with a random duration within the time series, excluding the first and last time steps, which are taken as nondeforming references by the model (see Supplementary Fig. S1). The model is therefore tasked with finding cumulative deformation in the seven middle time steps of the time series arising from a wide variety of transient processes. We then corrupt each map of these ground deformation time series with different noise signals. At each time step, we create both turbulent and stratified synthetic atmospheric delays. Spatially correlated Gaussian noise mimics delays from atmospheric turbulence of various length scales44,45 (Fig. 1, top row) and a quadratic function of pixels’ elevation mimics the atmospheric delays that correlate with topography16,46 (also randomly generated47). Lastly, we add random pixels, in patches and isolated, to mimic incoherent pixels and unwrapping errors commonly encountered in real data. Each of the steps of the time-series results from a random realization of noise built following these assumptions.

We train two independent models with the synthetic time series of deformation, one tasked with recovering point source deformation and the other with recovering deformation on faults. All other phase delays are identified as noise. Both models are trained to map synthetic noisy time series to the synthetic cumulative displacements. We trained our deep autoencoder with 2.5 × 107 randomly generated time series for which we provide as input the apparent LOS deformation time series, corrupted by the sum of synthetic noise described above. The training includes a LOS with random orientation (30–45° incidence and any azimuth), so that the model is directly trained for various SAR satellite configurations and for any fault azimuth. The output is the target ground deformation accumulated during the time series. All 482,185 trainable parameters are adjusted during that training phase with the Adam variation of stochastic gradient descent48 (see Supplementary Fig. S2 for the training curves).

We note that our deep autoencoder only considers time series of nine time steps, as a good compromise on the input duration, such that the input time series are long enough for the model to learn the temporal differences between signal and noise. When applying our models to longer time series of n time steps, we apply the algorithm using a sliding window with a width of nine time steps and obtain n − 8 images of cumulative deformation. In this way, our model acts as a moving integral of actual deformation.

#### Performance on a synthetic data set

Once trained, we test the deep autoencoder on synthetic realizations of time series that have not been used to train the model. We randomly generate 105 time series of nine time frames using the same procedure as that described for the training phase. For each of the 105 time series, we evaluate the signal-to-noise ratio (hereafter referred to as SNR) as the ratio of signal power to noise power. We then apply the deep autoencoder to these time series in order to evaluate the performance of the model. We evaluate the resulting, cleaned time series using the structural similarity index49 (SSIM, see “Methods”), a standard denoising evaluation metric, which makes a nonlocal comparison between two images, and is bound between −1 and 1.

We find that the deep autoencoder applied to synthetic data accurately reconstructs deformation signals on faults, even in circumstances very challenging to expert interpretation (SNRs well below 1; Fig. 2). For SNRs above 20%, our algorithm provides a very accurate reconstruction, as shown by the SSIM between model output and deformation ground truth (0.7 < SSIM < 1.0). For low SNRs (10% and below), no signal can be visually observed, while the structural similarity is still correct and the overall deformation signal is recovered down to SNRs of ~0.5%, below which our model starts to fail. Supplementary Fig. S1 shows the pairwise distributions of different properties of the synthetic data as well as the SNR and performance of our model. For comparison, the performance on synthetic data of the same architecture as in Fig. 1 but trained on single time steps is shown in Supplementary Fig. S3, and the performance of a simple temporal filter is shown for comparison in the Supplementary Fig. S4. The model is trained on single patches, but interestingly performs almost as well on synthetic time series with more complex fault geometry (see Supplementary Fig. S5). We note that for point sources of deformation, the limit of our model is ~20% SNR (Supplementary Figs. S9 and S10), but that such signals are also much harder to distinguish from the noise for the eye. Therefore, our architecture allows us to exceed the ability of the expert eye to detect signals in noisy time series of deformation, provided their noise structure resembles the training set.

In the following, we show the application of our autoencoder to two case studies that have been independently analyzed by InSAR experts.

### Application to real data

#### Extracting deformation from a slow earthquake along the North Anatolian Fault, Turkey

Our deep autoencoder is trained to isolate and reconstruct cumulative ground deformation signals in 48 × 48 pixels series of nine time steps. However, a fundamental property of purely convolutional deep learning models is that the filters they learn do not depend on input size. As a result, we can create an autoencoder with exactly the same architecture as the model described in Fig. 1, but with an input size matching the number of pixels in the InSAR time series of interest. Because the parameters of the model do not depend on the input size, we can copy every parameter (i.e., weights and biases of the filters) of the model trained on synthetics to the new model, which can then be applied to InSAR data of any size.

Here, we apply the model to a time series built from images acquired by the COSMO-SkyMed constellation over the central section of the North Anatolian fault in Turkey (Fig. 3). This major plate boundary fault accommodates the motion of rotation of the Anatolia plate with respect to Eurasia and has ruptured in large, moment magnitude (Mw) 7 earthquakes multiple times over the past century50. An 80-km-long section of the fault has been slipping aseismically, at least since 1944, Mw 7.3, earthquake located near the small town of Ismetpasa51. In situ measurements based on creepmeters indicate that this fault experiences transient aseismic slip episodes52,53,54.

Rousset et al. produced an ~1-year-long time series from COSMO-SkyMed SAR acquisitions and detected a significant slow-slip episode that lasted 1 month during 2013 with a maximum of 2 cm of fault-parallel slip14. Average long-term velocity maps covering the whole region derived from InSAR data show aseismic slip over an 80-km-long section of the fault. This average relative displacement was found to result from successive transient events14,53, which were not apparent in data from older constellations of SAR satellites due to the long time span between acquisitions. In the InSAR time series processed by Rousset et al., large atmospheric delays are apparent, despite careful correction of atmospheric delays using ECMWF reanalysis products14,21. Therefore, knowledge of the fault location was key in the interpretation of the surface displacement field. We revisit the same time series in order to assess if our model is able to recover the known surface slip in real-time series of data. We stress that we do not provide the location of the fault to the model. With no human intervention and no a priori knowledge of the local tectonics and fault location, the model automatically isolates and recovers clean deformation signals where expert analysis previously found signals attributed to tectonic activity (Fig. 3). Importantly, the recovered deformation is obtained after training only on synthetic data and with no further fine-tuning on real data. Our model finds up to 1.5 cm LOS relative displacement across the fault, which we interpret as the signature of surface slip, as previously found14.

Fault-perpendicular cross-sections illustrate that even in regions where a slip would not have been convincingly identified by an expert (Fig. 4), our model recovers 2 mm of slip, extending the previous estimate of the along-strike length of this slip event. Rousset et al. identified a 5-km-long slow-slip event while the deep learning model determines that the portion that slipped was 8.5 km in length. Interestingly, the new 2 mm slow slip we find is on a segment adjacent to the previously identified 1 cm slow slip, and the two segments are separated by a kink on the fault, suggesting a potential interplay between fault geometry and slip55,56. What we presume to be the remaining noise can be seen to the north-west and to the south-east of the slow-slip event in the output of the deep denoiser (see Supplementary Fig. S8 for cross-sections). We suspect these errors may arise from errors in the elevation model that propagated in the time series.

We finally note that our current model interprets wavelengths longer than a kilometer as noise, although experts might interpret those as the signature of slip at depth. This limitation however is related to the size of pixels with respect to the size of the training samples. The same network architecture trained on larger synthetics would circumvent this limitation (at the cost of increased computation and training time). An alternative approach consists of rescaling input data (see Supplementary Fig. S7) to ensure consistency of the model output in deformation wavelength (which is the case here for the North Anatolian Fault event).

#### Extracting ground deformation signal at the Coso geothermal system, California

In a second example, we use our deep learning architecture to detect surface deformation caused by underground pressure changes. As above, our model is trained on several million examples of synthetic noisy InSAR time series. In this case, surface deformation is modeled by a point pressure source using Mogi’s equation of elastic deformation57, corrupted as before by synthetic atmospheric delays. Mogi pressure sources are used extensively for the modeling of volcanic inflation and deflation signals, e.g. refs. 58,59. Further, the combination of multiple sources allows one to model complex subsidence/uplift patterns.

After training exclusively on synthetic data, we apply our model to real data from the Coso geothermal field (California, USA), again without further training (details on the InSAR processing are in the “Methods” section). Because InSAR time series may be very noisy, even after correcting predicted atmospheric effects32, analysis of inflation or subsidence of less than a few centimeters per year in InSAR have relied to date on deriving long-term cumulative deformation60, such that random atmospheric delays cancel out. Detecting transient subsidence and uplift signals in SBAS time series below a few centimeters remains challenging, just as it does for faulting.

As with identifying deformation on faults, our model is able to disentangle actual ground deformation from atmospheric noise at short time scales, with a resolution of a few millimeters. In Fig. 5 we show the application of our deep denoising model to a time series over Coso in 2016. Contrary to what could be inferred from long-term cumulative deformation, we find that ground subsidence at Coso is primarily due to transient episodes of deformation. The cumulative deformation from these transients we detect accounts for most of the cumulative deformation observed in the data (see Supplementary Figs. S11S14 for details and for other examples of transient deformation). Interestingly, we find a number of transient events that are constituted of well-separated deflation sources, in agreement with geochemical observations showing that the geothermal field is constituted of isolated reservoirs61.

## Discussion

As the properties of the atmosphere cannot be measured at the same spatial and temporal resolution as SAR acquisitions, InSAR time series still contain large-amplitude atmospheric delays, on the order of centimeters, in spite of recent marked improvements in atmospheric correction and processing strategies23,32. For this reason, expert processing and analysis is required to interpret InSAR data. Furthermore, since the onset of the Sentinel 1 mission, the amount of available InSAR data has grown at a pace that is already challenging the ability of the community to process and analyze it, and the upcoming NISAR mission will increase the amount of available InSAR data several fold. Therefore, significant effort has been put into developing strategies to build time series with such vast data sets, e.g., refs. 13,30,62. Nonetheless automatic, autonomous InSAR interpretation methods are poised to become essential, if just to leverage the increasing spatial and temporal resolution of the data.

We note that several avenues of improvement should enhance the ability of our neural network to detect finer and finer deformation signals in the future. First, we did not include sources of noise representative of ionospheric perturbations. The total electronic content of the ionosphere introduces a differential delay in interferograms that can bias analysis further16. Although this effect is more pronounced for L-band SAR satellites16,63, long-wavelength ionosphere delays can be problematic for large images acquired with C-band SAR systems such as Sentinel 164. Although these delays can be corrected by using techniques such as the range split-spectrum method64,65, the structure of the remaining noise associated with imperfect corrections must still be evaluated and could then be used in the training of our model. Second, we considered atmospheric turbulence to be isotropic and equivalent everywhere in the image (i.e., noise is second-order stationary) while some anisotropy can be observed in the phase delay of some interferograms. However, such anisotropy depends on the scale of the image observed, which would involve complex considerations in the construction of an adequate tropospheric noise model to train our model. In general, any improvement in the forward modeling of the nature of noise in InSAR should lead to a significant improvement in the detection capability of the models. Finally, the receptive field of the autoencoder and the pixel size of the input InSAR data restrict the size of the deformation signal that can be deciphered. For instance, interseismic deformation related to loading of a fault by plate motion extends over 10 s of kilometers, e.g., refs. 11,12,66. Additional developments may be necessary for the detection and cleaning of long-wavelength deformation patterns.

The initial application of our method on InSAR time series enables the direct observation of a slow earthquake, refining previous estimates, autonomously and without prior knowledge. In particular, we expect that the ability to systematically observe fault and pressure source deformation at a global scale will further the understanding of hydrologic, volcanic, and tectonic processes, and may bring us closer to bridging the observational gap that exists for transient surface deformation.

## Methods

### Autoencoder architecture

Here, we provide additional details regarding the autoencoder architecture developed for ground deformation extraction from InSAR time series (Fig. 1). This architecture consists of 11 purely convolutional layers. The first six layers of the model are tasked with encoding signals that are persistent in time, by progressively removing the time dimension of the input. At the seventh layer, topography (a digital elevation model) is added as a secondary input, before the remaining layers decode the ground deformation map. Because our model is comprised of purely convolutional layers, it can be applied to arbitrarily sized inputs (in terms of the spatial dimension, not the time dimension because of the pooling in time).

At each layer, the input is passed through 64 different filters to form as many channels, which are simplified representations of the data. During the encoding, 3D filters (two dimensions of space and one in time) of size 3 × 3 × 2 are applied to the data, until time has been completely eliminated by max-pooling operations. During decoding, 2D filters of size 3 × 3 are applied to the data (two dimensions of space) and summed for the output layer to reconstruct the cumulative ground deformation.

For each of the 64 filters within the encoder layers, each of the input filtered channels are summed and passed through a biased leaky reLU activation function. Each layer has a number of trainable parameters given by nkernel × ninput × noutput + noutput, with nkernel the convolutional kernel size (product of its shape in all dimensions), ninput the number of input channels to the layer, and noutput the number of output channels of the layer. This procedure gives 3 × 3 × 2 × 64 × 64 + 64 trainable parameters for each encoding layer, except for the first one, which has 3 × 3 × 2 × 1 × 64 + 64 parameters, and 3 × 3 × 64 × 64 + 64 trainable parameters for each decoding layer, except for the layer where ground elevation is added as an additional channel, that has 3 × 3 × 65 × 64 + 64 trainable parameters, and except for the last decoding layer that has 3 × 3 × 64 × 1 + 1 trainable weights. This gives our deep autoencoder a total of 482,185 trainable parameters, a modest amount when compared to natural image classification networks such as AlexNet67, which has 62,378,344 trainable parameters. The last layer of our model has a linear activation instead of a leaky reLU, such that positive and negative deformations can be equally output for the final reconstruction. Final reconstruction is a single image of the cumulative deformation that occurred during the nine time steps used as input. Our model was implemented on GPUs using the keras and tensorflow python libraries.

### Evaluation metrics

To assess the performance of our model on synthetic test sets, we use the SSIM, a common denoising performance metric in image processing. This measure of resemblance between two images is nonlocal and compares intensity, luminance, and contrast of the two images in moving windows, resulting in a metric closer to perceived similarity. We use the formulation and parameters from the original paper49:

$${{{{{{{\rm{MSSIM}}}}}}}}({{{{{{{\bf{X}}}}}}}},{{{{{{{\bf{Y}}}}}}}})=\frac{1}{M}\mathop{\sum }\limits_{j=1}^{M}{{{{{{{\rm{SSIM}}}}}}}}({{{{{{{{\bf{x}}}}}}}}}_{j},{{{{{{{{\bf{y}}}}}}}}}_{j})$$
(1)
$${{{{{{{\rm{SSIM}}}}}}}}({{{{{{{\bf{x}}}}}}}},{{{{{{{\bf{y}}}}}}}})=\frac{(2{\mu }_{x}{\mu }_{y}+{C}_{1})(2{\sigma }_{xy}+{C}_{2})}{({\mu }_{x}^{2}+{\mu }_{y}^{2}+C1)({\sigma }_{x}^{2}+{\sigma }_{y}^{2}+{C}_{2})}$$
(2)
$${\mu }_{x}=\mathop{\sum }\limits_{i=1}^{N}{w}_{i}{x}_{i}$$
(3)
$${\sigma }_{x}={\left(\mathop{\sum }\limits_{i = 1}^{N}{w}_{i}{\left({x}_{i}-{\mu }_{x}\right)}^{2}\right)}^{1/2}$$
(4)
$${\sigma }_{xy}=\mathop{\sum }\limits_{i=1}^{N}{w}_{i}\left({x}_{i}-{\mu }_{x}\right)({y}_{i}-{\mu }_{y}),$$
(5)

with C1 = (0.01L)2, C2 = (0.03L)2, L the range of the pixel values, xi the pixel values of patch x of image X, and wi weights given by the unit Gaussian function with a standard deviation of 1.5 pixels. The SSIM values we report here are the average SSIM of aligned patches x and y of size 8 × 8 from the two compared images X and Y.

We use a standard definition of SNR, as the ratio of signal power to noise power:

$${{{{{{{\rm{SNR}}}}}}}}({{{{{{{\bf{X}}}}}}}})=\frac{{{{{{{{\rm{P}}}}}}}}({{{{{{{{\bf{X}}}}}}}}}_{{{{{{{{\rm{signal}}}}}}}}})}{{{{{{{{\rm{P}}}}}}}}({{{{{{{{\bf{X}}}}}}}}}_{{{{{{{{\rm{noise}}}}}}}}})}=\frac{{{{{{{{{\rm{RMS}}}}}}}}}^{2}({{{{{{{{\bf{X}}}}}}}}}_{{{{{{{{\rm{signal}}}}}}}}})}{{{{{{{{{\rm{RMS}}}}}}}}}^{2}({{{{{{{{\bf{X}}}}}}}}}_{{{{{{{{\rm{noise}}}}}}}}})}$$
(6)
$${{{{{{{\rm{RMS}}}}}}}}({{{{{{{\bf{X}}}}}}}})=\sqrt{\frac{1}{9}\frac{1}{{N}^{2}}\mathop{\sum }\limits_{t=1}^{9}\mathop{\sum }\limits_{i,j=1}^{N}{x}_{ijt}^{2}}$$
(7)

where Xsignal is a nine frames long time series of synthetic deformation (e.g., deformation on a fault patch), and Xnoise is a nine frames long time series of synthetic noise, as described in the main text.

### COSMO-SkyMed data processing over Turkey

We use the ISCE framework to combine COSMO-SkyMed acquisitions into coregistred interferograms and then filter and unwrap these interferograms14,68. After interferogram generation, we use the ERA-Interim atmospheric reanalysis to perform a first-order correction of the atmospheric phase delay21,32. Finally, we use the New Small Baseline Subset (NSBAS) method implemented in the GIAnT toolbox69 to construct a time series of phase change, e.g.12 (extended details about the processing and the data set can be found in ref. 14).

### Sentinel 1 InSAR time series over Coso

We process the Synthetic Aperture Radar (SAR) images collected along ascending track 64 of Sentinel 1A–B from October 2015 to July 2019. We build 244 unwrapped interferograms (Supplementary Fig. S15) using the ISCE package68. We coregister SAR images with a network-based enhanced spectral diversity approach70 and correct for atmospheric perturbations using ERA-5 ECMWF global reanalysis of atmospheric data21. We apply a phase-preserving filter and multilooking (i.e., averaging of adjacent pixels) so that the final pixel size is about 70 m in range and azimuth71. Potential unwrapping errors are corrected using CorPhu72. Interferograms are unwrapped using a branch-cut method73 in areas for which coherence exceeds 0.5. We subtract a best-fitting ramp (i.e., linear function in range and azimuth) to each interferogram to correct long-wavelength perturbations due to orbital errors or ionospheric content, in order to focus on local, kilometer-scale deformation. Finally, we computed the optimal time series of displacement with the NSBAS approach as it is implemented in GIAnT74. Any pixel for which one interferogram could not be unwrapped is not included in the study. This restrains the spatial coverage of our InSAR time series, but ensures maximum (and equivalent) redundancy to all pixels.