# Distributions of throws, widths and scarp slopes on normal faults and their relations to fault growth: Insights from Auto_Throw code

Tiziano Giampietro, Isabelle Manighetti, Frédérique Leclerc, Yves Gaudemer

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Tiziano Giampietro, Isabelle Manighetti, Frédérique Leclerc, Yves Gaudemer. Distributions of throws, widths and scarp slopes on normal faults and their relations to fault growth: Insights from Auto_Throw code. Journal of Structural Geology, 2025, 196, pp.105393. 10.1016/j.jsg.2025.105393. hal-04993436

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# Distributions of throws, widths and scarp slopes on normal faults and their relations to fault growth: Insights from Auto_Throw code

Giampietro Tª, Manighetti Iª,*, Leclerc Fª, Gaudemer Yᵇ

## ARTICLE INFO

**Keywords:**

- Normal faults
- Scarps
- Topography
- Code
- Throw
- Scarp slope
- Fault width
- Fault growth

## ABSTRACT

Cumulative displacements on faults can tell us how they have accumulated over time, and thus how faults have grown. We approach this question from two angles, focusing on normal faults with topographic escarpment. First, we develop a code, Auto_Throw, to automate the measurements of throws on normal fault scarps. Using a piecewise linear fitting approach and basic slope and length constraints, Auto_Throw mathematically explores all possible configurations of scarps and offset surfaces on a topographic profile, so that a throw and its uncertainty are defined from the statistical analysis of tens to hundreds of measurements. Auto_Throw also measures the scarp slopes and the across-fault widths. We test Auto_Throw performance on 35 normal faults of different lengths and contexts, performing a million measurements. Second, we address the tectonic implications of the measurements. We conduct a global analysis of all fault data in order to smooth out site-dependent complexities and identify common features that may result from the fault growth process. We find that throws, scarp slopes and fault widths significantly vary along faults, yet in a similar manner for each fault quantity. Throw distributions show a common envelope shape, asymmetric with a long slip taper. Steepest scarp slopes are sub-constant along faults but systematically drop down towards fault tips. Fault width scales with throw, but a rough zone of constant width is identified off the faults. Tapering slip on a fault, shallowing of its scarp slope, and distributed deformation off its trace, could all be indicative of the fault lateral propagation.

## 1. Introduction

Faults grow as they accumulate displacements over repeated seismic cycles. On seismogenic faults that reach the Earth's surface, cumulative displacements generally offset geomorphic features. This provides an opportunity to observe and measure fault displacements that, otherwise, would be inaccessible at depth. Distributions of cumulative displacements at the surface hold invaluable information to decipher the fault history and growth over the long-term (10⁻²-10 Myrs). Where they can be dated, fault offsets also provide critical information on the fault slip rates (e.g., McPhillips and Scharer, 2023). These slip rates are in turn used to derive the seismogenic potential of the fault, i.e., to estimate how frequent and large are the earthquakes the fault produces (e.g., Tapponnier et al., 2001; Anderson et al., 1996, 2021).

Normal faults offer the best opportunity to measure cumulative displacements at the surface. As their hanging wall moves down, it is generally preserved from erosion, while the footwall upward motion

generally exposes the fault plane in the form of a topographic scarp (e.g., Wallace, 1977; McCalpin, 1987; Stewart and Hancock, 1990). Although the morphology of the scarp at its formation varies with the lithology and the strength of the faulted material (e.g., Holland et al., 2006; Ferrill et al., 2011, 2017), the ground surface prior to normal faulting is a marker that records the total fault displacements. Provided that this surface is not significantly modified after faulting (by erosion, sedimentation, adjacent faulting, anthropogenic changes, etc., e.g., Arrow-smith et al., 1996), its offset, mainly vertical, provides a fair measurement of the cumulative displacement on the fault. Furthermore, displacements can generally be measured on scarps along the entire fault length, in contrast with strike-slip and reverse faults where offset measurements are commonly sparse. Although a number of factors contribute to control or modify the morphology of the scarp as it is growing (lithology, pre-existing weaknesses, distributed deformation, erosion, sedimentation, e.g., Wallace, 1978; Stewart and Hancock, 1988; Ferrill et al., 2017), the actual fault dip is one of them, making it possible

This article is part of a special issue entitled: Paul Tapponnier published in Journal of Structural Geology.

* Corresponding author.

E-mail address: manighetti@geoazur.unice.fr (M. I).

<footnote>ª Université Côte D'Azur, Observatoire de La Côte D'Azur, IRD, CNRS, Géoazur, Sophia Antipolis, France</footnote>

<footnote>ᵇ Université Paris Cité, Institut de Physique Du Globe de Paris, CNRS, F-75005 Paris, France</footnote>
that scarps retain some information on the original fault attitude (e.g., Holtmann et al., 2023). As normal fault scarps expose the fault at the surface, they offer an opportunity to examine the fault thickness (or fault width), i.e., whether the fault is a narrow plane or a wider zone of deformation (e.g., Evans, 1990; Sibson, 2003; Berg and Skar, 2005; Berio et al., 2023; Mayolle et al., 2023). Normal faults are thus key to measure cumulative slips and other fault quantities and to examine how those relate to fault growth (e.g., Dawers et al., 1993).

Normal faults have been extensively studied since they were first described more than a century ago (e.g., Watson and Cline, 1913; White, 1950; Wallace, 1977; see below for more recent references). Their cumulative throws have been measured in different ways, and these measurements have fueled significant advances in our understanding of normal fault geometry and behavior, and fault growth in general (e.g., Watterson, 1986; Peacock and Sanderson, 1991, 1994; Dawers et al., 1993; Nicol et al., 1996, 2020b; Cowie, 1998; Manighetti et al., 2001a; Walsh et al., 2002; Schlagenhauf et al., 2008; Childs et al., 2017; Rotevatn et al., 2019). However, important questions remain. For instance, while it is generally accepted that cumulative slip varies along faults (e.g., Manighetti et al., 2001a; McPhillips and Scharer, 2018, 2023), it is unclear whether this variability results from fault behavior or geometry, context (tectonic setting, lithology, erosion), or epistemic uncertainties on offset measurements (e.g., McPhillips and Scharer, 2018, 2023). Questions also remain on how such slip variability relates to fault growth. Since fault dip and width measurements are sparse along long-term faults, it is unknown whether these latter two quantities vary along the faults, and if so, what could be learnt from these variations. Addressing these questions would require to analyze jointly dense data on cumulative displacements, fault plane dips and fault widths along the length of a large number of normal faults of different sizes and contexts. Yet, at present, no method exists to perform such measurements.

In this work, we have two objectives. Our first objective is to provide a novel approach, implemented in a Matlab code which we call Auto-Throw, to automate dense measurements of normal fault throws, fault widths, and scarp slopes in topographic data. Auto_Throw mathematically explores the multiple ways to fit a topographic profile in terms of offset surfaces and fault scarps. Doing so, it identifies the fault scarps along the profile, and measures their throw for each plausible configuration of offset surfaces. This provides tens to hundreds of throw

measurements per scarp and profile. From the statistical analysis of these measurements, Auto_Throw recovers the best vertical offset and its uncertainty. As it allows analyzing numerous topographic profiles along a fault, Auto_Throw eventually delivers the distribution of (resolvable) throws along and across the fault. In its detailed exploration of the topography, Auto-Throw also measures the steepest and mean scarp slopes and the fault width. We describe Auto_Throw in section 2. We then test it on 35 normal faults worldwide (Fig. 1, Table S1). We mainly focus on long-term faults having a long earthquake history, but we also analyze two earthquake ruptures. We target both single and composite faults (e.g., Manighetti et al., 2001a; Hodge et al., 2020). The faults span a broad range of lengths (0.1–30 km) and contexts, and are analyzed in high-resolution topographic data (≤0.5 m, 2–10 m, and 30 m, for 55, 29, and 16 % of the faults, respectively). We perform more than a million measurements of throws, scarp slopes, and fault widths, from which we derive more than 12 000 best throws, steepest and mean scarp slopes, and fault widths. Altogether these data describe in detail the distributions of throws, scarp slopes and fault widths along the 35 faults and systems. We compare our results to prior measurements where available, which demonstrates the robustness of Auto_Throw (section 3). We then discuss the performance of Auto_Throw with respect to prior work (section 4).

Our second objective (section 5) is to examine how the dense fault dataset obtained in section 3 may contribute to address the tectonic questions posed above. Our approach is to carry out a global analysis of all fault data in order to smooth out site-dependent complexities and identify common features that may result from the fault growth process. We find that throws, steepest scarp slopes, and fault widths are each distributed in a systematic manner along the faults: throws are systematically asymmetric with long slip tapers; steepest scarp slopes show a systematic decrease towards the fault tips; fault width scales with fault throw; a rough zone of constant width is systematically identified off the faults. We discuss these findings in the context of fault growth, and suggest that tapering slip on a fault, shallowing of its scarp slope, and distributed deformation off its trace, could all be indicative of the fault long-term lateral propagation.

In the following, as most offset surfaces are gently sloping, we use similarly the terms throw, vertical displacement, vertical offset, and vertical slip (see discussion in Caskey, 1995; MacKenzie and Elliott,

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Analyzed fault sites. (a) Worldwide location. (b) Faults analyzed in Bishop Tablelands. Background is 0.5 m resolution Pléiades DEM.</figcaption></figure>
2017). The code (along with the Matlab functions and toolboxes it needs), the DEM data, and the measurements (i.e., raw and analyzed profiles) for all faults are available in Giampietro et al. (2025) [dataset] (https://zenodo.org/records/14676843). We also provide for each fault a document Fault Fi_Synthesis, including a table listing the fault, DEM, and run parameters; maps of measured far-field slopes in the fault zone; and a synthesis of the results.

## 2. Auto_Throw code: overview and use

### 2.1. Prior work

For a long time, measurements on normal faults have been made on the field, using different instruments which, over time, have gained in practicality and resolution: telescoping fiberglass stadia rod (e.g., Wal-lace, 1977; Bucknam and Anderson, 1979), theodolite (e.g., Peltzer et al., 1988; Tapponnier et al., 1990a; Manighetti et al., 1997, 1998; Piccardi et al., 1999), kinematic GPS (e.g., Rood et al., 2011). In the last few decades, digital topography has become available with increasing coverage and resolution, allowing the remote analysis of normal fault scarps. This has fueled the development of automated approaches to remotely measure normal fault offsets in topographic data (Hooper et al., 2003; Hilley et al., 2010; Wilkinson et al., 2015; Billant et al., 2016; Haddon et al., 2016; Johnson et al., 2018; Stewart et al., 2018; DuRoss et al., 2019; Sare et al., 2019; Hodge et al., 2019; Wolfe et al., 2020; Bello et al., 2021; Scott et al., 2022; Riedl et al., 2022). As in prior field measurements, most of these approaches extract topographic pro- files across the fault, calculate linear regressions through footwall and hanging wall lines on either side of the scarp, and measure the vertical separation between the intersection of these lines and a vertical plane at the assumed fault location (Amos et al., 2010; Wilkinson et al., 2015; Haddon et al., 2016; McPhillips and Scharer, 2018; Johnson et al., 2018; Stewart et al., 2018; DuRoss et al., 2019; Bello et al., 2021; Scott et al., 2022; Riedl et al., 2022). A few other approaches identify specific points along the topographic profile, commonly the crest and toe of the scarp, and measure the scarp height between those points (Morewood and Roberts, 2001; Manighetti et al., 2001a; Bello et al., 2021; Brigham and Crider, 2022). Other studies have measured offsets through the recon- struction of displaced non-planar surfaces (Billant et al., 2016; MacK- enzie and Elliott, 2017). A few other codes use scarp-like numerical templates to fit the topography and derive scarp features including their height (Hilley et al., 2010; Sare et al., 2019; McPhillips and Scharer, 2023).

These recent approaches offer an invaluable toolbox to identify fault scarps in topographic data and measure their throws. However, they share a number of issues. In most cases, the regressions taken to repre- sent the offset surfaces are defined manually (Amos et al., 2010; Rood et al., 2011; Wilkinson et al., 2015; Haddon et al., 2016; Middleton et al., 2016; McPhillips and Scharer, 2018; Johnson et al., 2018; Wolfe et al., 2020; Nicol et al., 2020a; Bello et al., 2021; Hampel et al., 2021; Riedl et al., 2022; Salomon, 2023). Generally, only one measurement is made on a profile, which implicitly assumes that the chosen regressions are a perfect representation of the original offset surfaces. The position of the fault and thus of the offset measurement is defined by the user, generally at a single site (e.g., Amos et al., 2010; Johnson et al., 2018). Un- certainties on offsets are derived from analytical errors (e.g., Mattson and Bruhn, 2001; Middleton et al., 2016; Riedl et al., 2022), repeated measurements (McPhillips and Scharer, 2018, 2023; Bello et al., 2021), or comparison of measurements in different topographic datasets (Thackray and Staley, 2017). In some approaches, prior manual fault trace mapping is needed as a reference or calibration (Billant et al., 2016; Stewart et al., 2018; Hodge et al., 2019; Bello et al., 2021; Scott et al., 2022; Riedl et al., 2022), the topographic data need to be pre-processed to smooth out non-tectonic features (Wilkinson et al., 2015; Billant et al., 2016; Hodge et al., 2019; Scott et al., 2022; Riedl et al., 2022), the user is asked to input a significant amount of

information (e.g., Johnson et al., 2018), and the results need some post-processing to discriminate fault- and non-tectonic-scarps (Hilley et al., 2010; Sare et al., 2019; Scott et al., 2022). We developed Auto_Throw to cope with some of these issues. More specifically, Auto_Throw aims to limit user intervention and the quantity of input information. It mathematically describes the morphology, derives fault metrics from this description, and provides robust uncertainties on the measure- ments. We discuss its performance compared to prior work in section 4.

### 2.2. Auto_Throw overview

We describe Auto_Throw using a normal fault example, F1, in vol- canic Bishop Tablelands, USA (Fig. 1), which we analyze in 0.5 m res- olution and 1 m vertical accuracy Pléiades optical and topographic data (Fig. 2a).

Auto_Throw is written in Matlab (Matlab, 2019). It can be run without any prior mapping of the fault. In the visualized topographic data, which can be zoomed in, the user traces the overall fault trend by picking two (single line) or more (polyline) points (Fig. 2a). The code then traces and extracts evenly-spaced topographic profiles along the fault, centered on and perpendicular to the mean fault strike at the profile location. The profile length ($L_p$) and spacing ($Sp$) are defined by the user.

#### 2.2.1. Identification of fault scarps and offset surfaces along a profile

Auto-Throw treats each profile separately. In each profile, it iden- tifies and discriminates fault scarps and offset surfaces (referred to as flats, i.e., the sloping surface above or below the scarp) based on basic slope and length thresholds input by the user (Fig. 2b).

1) A plausible minimum fault scarp slope, Min-Dip$_{Scarp}$, in degrees;

2) The minimum across-fault, horizontal length of a scarp, Min-$L_{Scarp}$, in meters. It is set as the integer larger than the DEM resolution;

3) A plausible maximum slope of the offset surfaces, Max-Dip$_{Flat}$, in degrees. In common situations, it is a few degrees and can be set up by default (e.g., Hanks and Andrews, 1989);

4) The minimum length to define an offset surface, in meters (Min-$L_{Flat}$). By default, the code sets Min-$L_{Flat}$ as 10 % of the profile length $L_p$.

Using these parameters, Auto_Throw applies a piecewise linear fitting approach to explore the range of scenarios that succeed fitting a topographic profile with a variable number of lines of different lengths, called “segments” (Fig. 2b and 3). Each segment is defined as the best- fitting regression among its elevation data points. If a segment fulfills the conditions slope $\geq$ Min-Dip$_{Scarp}$ and length $\geq$ Min-$L_{Scarp}$, it is considered as a scarp. Conversely, if a segment fulfills the conditions slope $\leq$ Max-Dip$_{Flat}$ and length $\geq$ Min-$L_{Flat}$, it is considered as a flat (Fig. 2b).

As the location and number of segments (i.e., flats and scarps) along a profile are unknown, Auto_Throw explores the full range of possibil- ities. Using a signal processing method (findchangepts Matlab function, equations in https://fr.mathworks.com/help/signal/ref/findchangepts. html; details in caption of Fig. 3), the code identifies the change points that may separate each pair of segments (Fig. 3). A change point is where both the slope and the mean elevation of the two segments on either side change abruptly. As the number of change points is unknown, and introducing too many change points may produce overfitting, the code penalizes the introduction of too many change points by mini- mizing the following quantity (Lavielle, 2005; Killick et al., 2012):

$$J(K) = \sum_{\text{segments}} \left\{ \sum_{\text{points}} [z_i - (ax_i + b)]^2 \right\} + KT_h \quad (1)$$

where.
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

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

(caption on next page)
<figure><img src="image_7.png" /><figcaption>**Fig. 2.** Basics of Auto_Throw and scarp measurements. (a) Automatic extraction of topographic profiles of length Lp and spacing Sp across a normal fault scarp (F1 fault, Pléiades topographic data, location in Fig. 1). (b) Slope and length thresholds used in Auto_Throw, and definition of scarp and flat segments. (c) Definition of the widths measured in the study. The example profile is populated with change points identified at Th 2. Fault width is the length between the internal tips of the two flats on either side of the scarp, measured at lowest Th (here, Th 2). Scarp width is derived from the final best throw and mean scarp slope both calculated over the population of modal offsets (thus, scarp width is not related to change points shown in figure). The width of the residual non-flat zone is the fault width minus the scarp width. See text for discussion.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 3.** Piecewise linear fit of a topographic profile across a fault scarp. The piecewise function is the Matlab Findchangepts. m function, which finds the points that separate sections of a signal with sharp differences in a given statistical property (mean, std, etc.) and minimize the error sum of squares. We use here the « linear » option that looks for a linear fit of the signal between change points. In the figure, only 8 out of 370 calculations are shown, at increasing Th (all measurements in Giampietro et al., 2025 [dataset]). Segments identified as a flat or a scarp are in red or blue, respectively. Throw is measured at each scarp segment base (within uncertainties discussed in text), for each calculation at a given Th value. As Th increases, the number of change points, K, decreases. At lowest Th, the large number of change points provides a high-resolution description of the morphology. Several sections with different slopes are generally identified within a scarp, and the steepest scarp slope is measured (black in a). The fault width is the non-flat zone between the internal tips of the two flats on either side of the scarp (light blue in a-b). It is averaged over the two measurements at the two lowest Th values. As Th increases and K decreases, the description of the morphology simplifies to eventually recover only its largest-scale features, i.e., average far-field slopes and overall scarp attitude. The scarp width is eventually calculated using the final throw and mean scarp slope, both calculated among the dense population of modal offsets. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>
* zᵢ is the actual elevation at point xᵢ, axᵢ + b is the modelled elevation at point xi, a and b being specific to each segment;

* Th is a threshold value, expressed as a positive integer, in square meters;

* K is the number of change points along the profile.

Th balances the number of change points (Fig. 3): the lower the Th value, the higher the number of change points needed to make the sum of the squared residuals as low as possible (Eq. (1)). When Th = 0, a K value considering every single elevation point as a change point is needed to minimize the residual (to 0). At each Th, the calculation starts with K = 0 and finds the best straight line through all data points. It then moves to K = 1 and finds the location of the single change point that minimizes J (Th, 1). It then goes to K = 2 and finds the pair of change points that minimizes J (Th, 2). If J (Th, 2) is smaller than J (Th, 1) then the fit with two change points is significantly better than with one change point. If not, it means that introducing a second change point does not improve the fit. Therefore, sticking with one change point is the best way to fit the data. In the former case, it moves to K = 3, and if J (Th, 3) is smaller than J (Th, 2), a third change point significantly improves the fit. And so on.

To capture all possible scenarios of flats and scarps along a profile, the code explores a large range of Th values (and thus of change points), from a minimum $Th_{min}$ fixed to 2 (as 0 and 1 would lead to over-fitting), to a maximum value $Th_{max}$ set by default to the sum of squared residuals (with respect to the mean) of the original signal. $Th_{max}$ may also be adjusted by the user. The code then performs the whole range of calculations for all Th values between $Th_{min}$ and $Th_{max}$ (examples in Fig. 3; all measurements in Giampietro et al., 2025 [dataset]).

At each Th value, the code thus determines the number and location of change points and segments that best fit the profile. The location, length and slope of the segments are compared to the length and slope threshold conditions described above, which leads to identify and locate the flats and scarps (Fig. 3).

### 2.2.2. Measuring vertical offsets along a profile

At each Th value, flat and scarp segments are identified along the profile, and throw is measured across each of the scarps. If the scarp is a single segment (Fig. 2b), the two flats on either side are projected onto a vertical line at the base of the scarp segment (see below for uncertainties on this position), and the vertical offset is measured between the two flat intercepts. Measuring throw in this way is relevant where flats have a fairly similar and shallow slope on either side of the scarp, which is the common case (e.g., Caskey, 1995). If flat slopes differ significantly and/or are steep, the code may be adapted to include a more complex throw calculation (Caskey, 1995; Hampel et al., 2021). If the scarp divides into two or more segments, different situations may occur depending on whether adjacent segments are separated by a flat (Fig. S1). Whatever the situation, throw is measured at the base of each identified segment (uncertainty below).

Three sources of uncertainties affect each throw measurement (Fig. S2).

* the uncertainty on each elevation point, i.e., the DEM resolution, which affects the flat and scarp regressions;

* the uncertainty on the on-fault position where throw is measured. By default, the code considers the fault position between the base and the mid of the scarp (e.g., McCalpin, 1987; Rood et al., 2011; McPhillips and Scharer, 2018; Johnson et al., 2018), but this range can be modified;

* the uncertainty on the actual fault dip at depth, Dip<sub>fault</sub>. By default, the code considers a dip range between 60° and 90°, but this range can be modified.

To calculate the total uncertainty from these combined sources, we use a Monte Carlo-like approach (e.g., Thompson et al., 2002; Stewart

et al., 2018): the code runs a series of trials by varying the position of each point that contributes to the calculation, randomly within the ranges above (Gaussian range). The final error is the standard deviation of the entire series of measurements. We have tested the code with 100, 500, 1000, 2000, 3000, 4000, 5000, 10 000, 15 000, 20 000, 50 000, and 100 000 trials and observed that the errors stabilize after 10 000 runs. We also made a sensitivity analysis to explore which source of uncertainty contributes most to the offset variability (Fig. S3). The on-fault position where throw is measured has the greatest impact in case of flats with different slopes.

A fourth source of uncertainty exists however, that cannot be quantified: the lack of knowledge on whether the measured throw possibly under- or over-estimates the actual offset, for instance due to erosion or sedimentation having modified the original scarp or flat morphology. While Auto_Throw cannot address this epistemic uncertainty, its capacity to explore multiple scenarios and conduct statistical analyses of dense sets of measurements provides insights on this issue (see next section).

### 2.2.3. Statistical analysis of vertical offsets along a profile

At each Th value, flats and scarps are identified, and throw is measured across each of the scarps. As the value of Th is varied, the number of change points also varies, so that the flat and scarp configuration may change (Fig. 3). Since the code generally explores large Th ranges (Fig. 3), it delivers many throw measurements per profile, commonly several tens to hundreds, that can vary in both location and amplitude. To analyze them, Auto_Throw builds a bivariate histogram showing the location and frequency of throws with a given amplitude (Fig. 4). The most robust offsets are those found repeatedly at about same location and amplitude. To isolate them, the code applies a frequency cut-off and retains only the throws with similar amplitude at a same location that represent more than the fraction of the total offset population indicated by the cut-off. We refer to the retained offsets as the modal offsets (or throws).

In the general case, the modal offsets are tightly grouped at about the same location on the profile (Fig. 4a). They thus characterize a single fault scarp. The code averages their amplitude and position to calculate the best throw and its best position, with their uncertainty the standard deviation of these calculations (unless measurements are few, see Fault F11&F12_Synthesis). In some cases, however, two or more groups of modal offsets are found at different locations along the profile (Fig. 4b). If the distance between two groups is lower than Min-L<sub>Flat</sub> (or another threshold set by the user), they are considered to represent the same fault scarp. The code thus proceeds as above. Conversely, if the distance between two groups is greater than Min-L<sub>Flat</sub>, the two groups are considered as representing two distinct scarps (Fig. 4b). The code then treats them separately and derives the two best throws and positions as before.

Since flats are identified and measured at each Th value, we end up with a dense collection of flat slope measurements. Auto_Throw uses these measurements to derive the average flat slope along the profile, and the flat slope difference on either side of the scarp (Fig. 5).

### 2.2.4. Measuring fault scarp slope and fault width along a profile

Auto_Throw commonly identifies different segments within a scarp (Fig. 3), and the slope of each of these scarp segments is measured at each calculation step (each Th value). For each scarp, Auto_Throw eventually retains two slopes: i) the steepest scarp slope, a single slope value among the total population of measurements, generally revealed at the lowest Th value(s) (Fig. 3a); ii) the mean slope, taken as the average of all segment slopes obtained in the calculations of the modal offsets (average of, generally, hundreds of slope measurements).

As Auto_Throw excels in identifying flat surfaces, it also excels in isolating non-flat, i.e., rough, surfaces. It is actually common that the code identifies a rough zone on one or both sides of the scarp, between the internal tips of the two flats enclosing the scarp (Fig. 2c and 3). As
<figure><img src="image_9.png" /><figcaption>Fig. 4. Statistical analysis of throw measurements. (a) and (b) show cases with a single and two scarps, respectively. Topographic profiles are shown at the top, and bivariate histograms at the bottom. These histograms show the frequency of throws with a given amplitude and location, among the hundreds of measurements per profile (color scale is number of measurements). To avoid outliers, each axis only presents measurements within 2 σ from overall mean. The frequency cut-off isolates the most robust offsets, found repeatedly at about same location and amplitude (modal offsets). In case of a single scarp (a), the modal offsets (here 157 measurements) are tightly grouped and the final best throw and its best position are derived from averaging the 157 offsets and locations, with uncertainty the standard deviation of these calculations (median or modal values can also be calculated). If several scarps are crossed by a profile (b), two or more groups of modal offsets are found at different locations (here 2 groups, each made of more than 150 measurements). Depending on whether the distance between the two groups is lower or larger than Min-L<sub>Flat</sub> (default threshold distance), they are considered to represent the same or two different scarps, and are thus treated together or separately, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>

this rough zone is immediately adjacent to the fault and surrounds it, it is
likely genetically related to the fault and its interplay with geomorphic
processes (discussion in section 5.4). We thus feel interesting to measure
it. We provide three types of measurements: i) at each Th, Auto_Throw
measures the across-fault length of the zone between the internal tips of
the two flats enclosing the scarp, which we refer to as the “fault width”
(about as in Mayer, 1984; Mayolle et al., 2021; Gaudreau et al., 2023,
Fig. 2c). Hundreds of fault width measurements are so obtained per
scarp and profile, but we consider only the two fault width measure-
ments done at the two lowest Th values, and retain their average. This is
because calculations at lowest Th values populate the profile with many
change points that allow small topographic variations (i.e., roughness)
to be represented, unlike calculations at larger Th which smooth out the
topography (Fig. 3); ii) once Auto-Throw has run its 10–100s calcula-
tions per scarp and profile, it derives the “scarp width” from the final
throw and mean slope, both are derived from 10 to 100s measurements
(Fig. 2c); iii) once the fault and scarp widths are determined, the user
can decide to derive the width of the “residual non-flat zone” on either
side of the scarp by subtracting the scarp width from the fault width
(Fig. 2c).

2.2.5. Deriving along-fault distributions of throw, scarp slope, and fault/ scarp width

From the calculations above, the user knows, for each profile, the
number, location, throw, slope, and width (along with uncertainties) of
the fault scarps crossed by the profile. Interestingly, plotting the scarp
positions provides an automatic mapping of the fault trace(s) (Fig. S4b).

To assess the robustness of the measurements, we define quality
factors. Best throws/mean slopes that result from one measurement are

ignored (unless indicated); those derived from less than 10 modal offsets
are considered weakly constrained and assigned a quality factor Q3;
those derived from 10 to 30 modal offsets are considered fairly well
constrained and assigned a quality factor of Q2; those calculated with
more than 30 modal offsets are well constrained and assigned a quality
factor of Q1.

Using these information Auto-Throw delivers the along-fault distri-
bution of: i) throws; ii) steepest and mean scarp slopes; and iii) fault and
scarp widths (Fig. 6). Under assumptions on the fault dip, Auto_Throw
can also derive the fault heave (horizontal, extensional across-fault
component; e.g., Caskey, 1995; Riedl et al., 2022) and dip-slip (i.e.,
downdip) components along the fault.

If several scarps are identified along some of the profiles, Auto-
Throw can calculate the summed slip across those scarps.

2.2.6. Expert assessment of the measurements

Even though Auto-Throw provides robust throw, slope and width
measurements on fault scarps (see section 3), we recommend that they
are eventually examined by an expert to detect possible inconsistencies
including measurements biased by local morphological complexities.
We have done this exercise for every measurement we present below,
using basic criteria such as those shown as examples in Fig. 7. We have
been deliberately conservative. In the following, we analyze the results
with and without expert evaluation, and show they are consistent.

2.3. Sensitivity of Auto_Throw to input parameters

On fault F1, we performed 23 runs to examine the influence of each
of the parameters used in Auto_Throw. A first run, TestRef, is taken as a
a)

<figure><img src="image_10.png" /><figcaption>**Fig. 5.** Far-field slopes measured by Auto_Throw. Example is F1 (Test<sub>Ref</sub>). At each Th value along a profile, the code measures the slope of each offset surface on either side of the scarp. (a) Average far-field slope along each profile, derived from averaging all slope measurements in dense modal offset population along the profile. (b) Far-field slope difference on either of the scarp derived, along each profile, from averaging and subtracting the two series of slopes (modal offsets) either side of the scarp. Overall, the slopes of the offset surfaces are shallow, and fairly similar across the scarp. See all measurements in Giampietro et al. (2025) [dataset].</figcaption></figure>

b)

<figure><img src="image_11.png" /></figure>

reference, while in each of the other 22 runs, one input parameter is varied while the other parameters are unchanged. The runs are described in supplement document S5, and their results shown individually in Fault F1_Synthesis and collectively in Fig. 6. Below, we synthesize the major outcomes.

Fig. 6 demonstrates the robustness of the measurements, which remain similar overall in the 23 tests. The parameters most impacting the results are.

i) the profile length ($L_p$). Using a long profile is needed to fully capture far field slopes and measure the total throw across a fault or fault zone, and the fault width;

ii) the minimum scarp slope (Min-Dip$_{Scarp}$). We recommend using low Min-Dip$_{Scarp}$ (8–10°) to ensure capturing the scarp along most of its length. Min-Dip$_{Scarp}$ should be larger, however, than the steepest flat slope in the site;

iii) the minimum flat length (Min-$L_{Flat}$). Varying this parameter allows changing the resolution at which the user examines the morphology and recovers the scarps and their throws. A long Min-$L_{Flat}$ allows recovering the far field slopes and hence measuring the total throws across the fault or fault zone, and the fault widths. It also smoothes out the small-scale features of the morphology. By contrast, a short Min-$L_{Flat}$ allows describing the morphology in detail. As such, it helps identifying nested and closely-spaced scarplets or fault strands and measuring their individual throw. It also helps recovering the top-to-base slope variability of the scarp, the roughness of some zones, and small-scale morphological complexities that may bias the measurements;

iv) the Th range. A large Th range, at least up to 100–200 in cases similar to F1, puts more weight on large-scale descriptions of the morphology and smoothes out measurement variability, making it appropriate to measure total throws and overall scarp attitudes. By contrast, to recover details of the fault zone morphology and measurement variability, low Th ranges (<~100) are more appropriate, especially if combined with a low frequency cut-off.

The other parameters may be chosen in a more standard manner. In particular, if the flat slopes (Max-Dip$_{Flat}$) do not vary much in the site, a single average value may be appropriate and set up as default in the code. The fault dip range at depth hardly impacts the measurements and uncertainties. A frequency cut-off of 25 % is a good compromise to recover total throws but lower cut-offs may be needed if morphological details are sought.

### 3. Testing Auto_Throw on a large population of faults and data

We apply Auto_Throw to 35 normal faults and systems worldwide of various lengths (~0.1–30 km), tectonic setting, rock lithology, and erosion/sedimentation contexts, using topographic data of different types and resolutions (Fig. 1, Table S1). We aim to test Auto_Throw on faults with various complexities, including earthquake ruptures, and to compare our results with prior data wherever possible.
<figure><img src="image_12.png" /><figcaption>**Fig. 6.** Throw, scarp slope and fault/scarp width data obtained in the 23 runs on F1 fault (Fig. 1b). Results from all runs shown to the left, and results from most relevant runs shown to the right (Tests Ref, 1–3, 7, 9, 12, 14–18, 21–22, see Document S5). Measurements are projected on mean fault trend, and discriminated based on our expert evaluation. Fault tips are in blue (arrows or dots). (a) Fault scarp in topographic, 0.5 m resolution, Pléiades data. (b) Manual fault map (not used in calculations) in Pléiades DEM and optical imagery. (c–d) Along-fault profile of throw. (e–f) Along-fault profiles of steepest (red) and mean (black) scarp slopes. (g–h) Along-fault profiles of fault (black) and scarp (green) widths. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 7.** Examples illustrating basic criteria used in our expert assessment of the Auto_Throw measurements. For each fault metrics, left and right panels show examples of measurements estimated as satisfying and as less satisfying, respectively. Wherever possible, we compare measurements on nearby profiles. Measurements considered as not fully satisfying are represented in transparency in the result figures, with, generally, additional information on whether the measurement under-, over-, or poorly represents the actual quantity.</figcaption></figure>

<figure><img src="image_14.png" /></figure>

## 3.1. Normal faults in Bishop Tablelands, California, USA (0.5 m DEM)

A dense network of active normal faults dissects the young volcanic tuffs (~760 ka, e.g., van den Bogaard and Schirnick, 1995) of the Bishop Tablelands (e.g., Dawers et al., 1993, Fig. 1b). The area is desertic and submitted to low erosion (1.2–2.5 m/Ma, Goethals et al., 2009). In addition to F1 described before, we analyze 3 faults having a fairly simple geometry and scarp morphology (F2, F3, F4, 1.5–5 km-long; Fig. 1b and 8b-d). We measure their total throws in a single run. Overall, notwithstanding the quick calculations and the various local complexities of the faults and offset surfaces (such as fumarole mounds in F4), the total throws, steepest (and mean) scarp slopes, and fault widths are successfully measured in the 3 fault cases (Fig. 8b-d).

We then apply Auto_Throw to a more complex fault zone, F5, ~5 km-long (e.g., Dawers and Anders, 1995; Goethals et al., 2009, Fig. 1b and 8e), that includes two principal faults, M1 and M2, and 8 most prominent secondary faults splaying off M1 or M2 (Fig. 9a). We first run 3 calculations across the whole fault zone (runs 1–3) with variable Min-DipScarp and Max-DipFlat. The results bracket the total throws and fault widths, and depict the variability of the scarp slopes (Fig. 8e). We then run calculations aiming to recover throws, scarp slopes and fault

widths on each of the 10 faults (run 4). Vertical offsets and scarp slopes are best recovered on longest faults (length >1 km: M1, M2, f7, f8, f10; Fig. 9), whereas fault widths can only be measured on M1 and M2. We eventually compare the two approaches. Summing individual fault throws provides a total offset that is generally lower than the total throw measured across the whole fault zone (Fig. 8e). The two throws are similar where F5 has a simple, single trace, and differ where the fault zone includes several branches. The largest difference is in the M1-M2 relay zone, where the summed offset is 30–35 % lower than the total throw measured across the fault zone.

Finally, we measure vertical offsets on the Fish Slough faults where throws were measured in earlier work (Scott et al., 2022) (Fig. 1b and 8f). The zone includes sections of 3 main W-dipping normal faults, FSI 1, FSI2 and FSI 3, cutting sloping surfaces. Locally, the morphology of the fault scarps and offset surfaces is complex (faults: nested scarplets, multiple branches, lateral segmentation; flats: folding at fault tops, steep slopes, local erosion and sedimentation; Fig. 8f and 8a). In a first approach (runs 1–4), we measure the total throws across each of the 3 main faults and the overall system, as such following the “coarse-scale mapping” analysis of Scott et al. (2022) (Fig. S6b, Fault FSI_Synthesis). Where fault traces, fault organization, and offset surface morphology are
<figure><img src="image_15.png" /><figcaption>**Fig. 8.** Distributions of throws (black), steepest scarp slopes (red), and fault widths (blue) along faults analyzed in Bishop Tablelands. Top figures show the faults and profiles in the Pléiades DEM. In the plots, most robust measurements are in solid symbols. When throw profiles are asymmetric, they are presented with the largest slip to the right (to compare to later Fig. 16a). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>

simple, our results are similar to those of Scott et al. (2022) (details in Fig. S6 caption). In a second approach (run 5), using shorter profiles and adapted parameters, we measure fault throws at a finer scale, as such following the “fine-scale mapping” approach of Scott et al. (2022; Fig. S6c). Most measurements are consistent with those of Scott et al. (2022) (details in Fig. S6 caption). However, as in Scott et al. (2022),

offsets summed across individual faults (Fig. S6c, pink) are lower than total throws across the whole fault zone (Fig. S6c, black).

We have completed the FSI2 measurements north of the area examined above so as to get data along the entire fault length. In the following, we designate FSI2 as F6 (Fig. 8f and S6).
<figure><img src="image_16.png" /><figcaption>**Fig. 9.** Distributions of throws (black), steepest scarp slopes (red), and fault widths (blue) along the F5 fault system in Bishop Tablelands. (a) Manual mapping of the fault system (not used in calculations), with the different faults discriminated and the profiles shown. (b–f) Top figures show the faults in the Pléiades DEM. Other caption elements as in Fig. 8. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>

## 3.2. Normal faults in Asal rift, Republic of Djibouti, East Africa (10 m DEM)

We target the ~10 km-long, en-echelon, normal fault system that bounds the Asal rift to the north (Tapponnier et al., 1990b; Manighetti et al., 1998; Doubre et al., 2007a, 2007b; Fig. 10a). The Asal rift is a desertic region submitted to negligible erosion and sedimentation (Cattin et al., 2005; Varet, 2022). The fault system includes three SW-dipping faults, A1, A2, A4 (here renamed F7-A1, F7-A2 and F7-A4), cutting young basaltic flows (<~200 ka, Manighetti et al., 1998), and whose throws were measured in prior work (Manighetti et al., 2001a). In the same DEM as that used in Manighetti et al. (2001a; 10 m-resolution, 1–2 m vertical accuracy), we performed a single calculation using a set of ~200 profiles crossing the 3 faults and average parameters made to recover the 3 scarps as best as possible over the entire length of the system (Fig. 10a and Fault F7_Synthesis).

The throw and scarp slope measurements are fairly straightforward on the 3 faults, with only a few local complexities (Fig. 10). Measuring the fault width is more challenging, with only 20–25 % of the measurements estimated as relevant (Fig. 10).

As the prior offset measurements were not tabulated nor georeferenced, we extract them from the paper. We then project our throw measurements onto the mean fault system trend defined in Manighetti et al. (2001a), and adjust the two slip profiles laterally based on the positions of the fault tips. The two sets of throw measurements are

similar, both along each fault (Fig. 11b) and summed over the whole system (Fig. 11c).

## 3.3. Normal faults in northern East African Rift, Ethiopia (30 m DEM)

We now test Auto_Throw in lower resolution topographic data, namely an Alos Global Digital Surface Model (AW3D30) of 30 m resolution (vertical accuracy ~ 5 m, Nikolakopoulos et al., 2020). The DSM covers the northern part of the East African Rift, a desertic region submitted to negligible erosion and sedimentation (Varet, 2022). At this site, Manighetti et al. (2001a) measured cumulative throws on a number of normal faults, using SRTM, ~30 m resolution and ~5 m accuracy topographic data. We apply Auto_Throw to 5 of these faults (R50-54 in Manighetti et al. (2001a), here renamed F8_R50-R54; Fig. 12). The faults are ~4–15 km long, west-dipping, cut < 1–2 Ma thick basaltic piles (Manighetti et al., 2001b) and belong to a right-stepping, en echelon, NNE-trending, ~60 km long system (Plate 3 in Manighetti et al., 2001a).

Measuring throws on these faults is challenging as most of them show an antithetic secondary fault in their footwall, very close to the scarp top, that isolates a sharp horst (Fig. 12). The latter hinders the identification of the footwall flats. Additional difficulties arise from the faults being closely spaced or adjacent to other faults (Fig. 12). The offset surfaces have variable slopes, generally differing by 3–4° on either side of a scarp, but showing larger differences at local sites. To cope with these complexities, we run up to 2 calculations per fault, using variable
<figure><img src="image_17.png" /><figcaption>**Fig. 10.** Distributions of throws (black), steepest scarp slopes (red), and fault widths (blue) along the F7 fault system in Asal rift, Republic of Djibouti. (a) DEM showing faults and profiles. (b–d) Measurements on individual faults. (e) Slip summed on whole fault system. Other caption elements as in Fig. 8. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>

Min-L<sub>Flat</sub> and Th ranges (Fault F8_Synthesis).

Overall, throws are well measured on all faults except R52 (Fig. 12). The steepest scarp slopes are also well measured (Fig. 12). The fault width is more poorly measured, with about a third of relevant measurements per fault (Fig. 12). To compare our throws with prior ones, we proceed as in prior section. Overall, the two sets of measurements are similar, for both individual faults (Fig. S7b) and whole system (Fig. S7c). The few differences likely result from uncertainties on measurement collocation.

### 3.4. Offshore normal faults in Lesser Antilles (10 m bathymetric data)

We apply Auto_Throw to some Lesser Antilles submarine faults in bathymetric data. We use bathymetric data of 10 m resolution and 2 m vertical accuracy (Leclerc et al., 2016). The data include a few artefacts that produce 50 m wide and 5 m high artificial ridges. We target the SW-dipping Rodrigues and Redonde faults (here named F9 and F10; Fig. 13) which bound the NW-trending Les Saintes graben (Feuillet et al., 2002; Leclerc et al., 2016; Hughes et al., 2021; Henri et al., 2022). The faults offset 2–3 Ma-old volcanic rocks (basalts and pyroclastic units; Henri et al., 2022) capped with limestones and sediments (Hughes et al., 2021). Erosion and sedimentation are active in this offshore area, with erosion rates of 14–46 m/Ma (Hughes et al., 2021) and local mass-wasting processes (Leclerc et al., 2016).

F9 is a ~11 km long normal fault, cutting a fairly flat morphological surface. It includes two major segments (F9a and F9b, Fig. 13a–c) separated by a relay zone. We measured the total throws across the fault (Fig. 13a). Those are robustly constrained except locally in the north where a few throws are under-estimated due to antithetic faults in the hangingwall hampering the identification of the flats. The artefacts in

the bathymetric data are smoothed out in the statistical analysis and do not affect the offset measurements. The steepest scarp slopes are well-constrained along the fault (Fig. 13). Fault width is fairly well measured except where antithetic faults dissect the hanging-wall (Fig. 13).

F10 is ~5 km-long, cuts the same morphological surfaces and shows the same complex geometry as F9. A greater complexity arises from the existence of many synthetic and antithetic secondary faults in the F10 hangingwall close to its scarp (Fig. 13d). This led us to adapt the code parameters in 3 subsets of profiles, and to run two calculations, one to recover total throws, and one to discriminate throws on adjacent faults (Fault F10_Synthesis). From the combined measurements, the throws are well determined except where secondary faults densely dissect the F10 hangingwall (Fig. 13d). The steepest scarp slopes are well measured (Fig. 13d). The fault width is only sparsely measured, mainly due to secondary faults in the hanging-wall (Fig. 13d).

### 3.5. Edgecumbe surface earthquake rupture, New Zealand (2 m DEM)

We now use Auto_Throw to measure vertical offsets on a historical normal earthquake rupture, the 1987 Mw 6.5 Edgecumbe earthquake, New Zealand (Fig. 14a and b). The earthquake broke a normal fault system in the Taupo rift and produced or reactivated a number of scarps at the surface, cutting through unconsolidated sediments. Measuring coseismic throws on these scarps is challenging because they were reported to be lower than 2.5 m (Beanland et al., 1989; Berrill et al., 1987) and cutting through farmland with fields, fences, and trees imprinting the topography (Fig. 14a and b). Delano et al. (2022) calculated 2 m resolution Sfm topographic data on the scarp zone, and used it to measure the coseismic and cumulative vertical slips on the most
<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 11.** Auto_Throw measurements on F7, compared to prior data. (a) View of F7 fault zone in 10 m topographic data. Fault zone tips in orange. (b) Throws measured on each fault with Auto_Throw (color) and by Manighetti et al. (2001a, grey). (c) Summed throws on whole fault zone determined with Auto_Throw (blue) and by Manighetti et al. (2001a, grey). Most robust Auto_Throw measurements in solid symbols. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>**Fig. 12.** Distributions of throws (black), steepest scarp slopes (red), and fault widths (blue) along faults analyzed in northern East African Rift. Other caption elements as in Fig. 8. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Fig. 13.** Distributions of throws (black), steepest scarp slopes (red), and fault widths (blue) along faults analyzed in Les Saintes site. Other caption elements as in Fig. 8. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>

prominent scarps (here cumulative refers to a few seismic events only). Measurements were made manually. Cumulative throws were measured in a post-earthquake DSM (0.5 m vertical accuracy), while coseismic throws were measured by differencing two pre- and post-earthquake DMs (Diff DSM, 0.75 m vertical accuracy).

We focus on the surface rupture of the main Edgecumbe fault (F11, Fig. 14a and b). Its trace is clear in the post-earthquake DSM except at its westernmost tip where we could not identify the scarp mapped by Delano et al. (2022) (Fig. 14a). The rupture trace is more subtle in the Diff DSM, yet still visible except in its westernmost part (Fig. 14b). To cope with the subtle topographic expression of the rupture and the non-tectonic features in the topography, we have adapted the code parameters, using a very shallow Min-Dip<sub>scarp</sub> and a very low Th range (Fault F11_Synthesis).

Overall, offsets measured in the post-EQ DSM compare well with those of Delano et al. (2022) (Fig. 14a). However, they have systemat-ically larger uncertainties (~0.3–0.8 m versus ~0.1–0.3 m in Delano et al., 2022), whereas, in the zone of largest slip, they are lower by 0.5–1 m (Fig. 14a). For a few profiles, this difference may arise from the small number of change points identified by Auto_Throw in such a smooth topography, making it unable to discriminate eroded scarp tops or bases from far field slopes. However, in most profiles, the far field slopes are well defined and so are the throws. Furthermore, while the vegetation hampers a relevant offset measurement in a few cases, those have been discarded in our expert evaluation. Therefore, it is not clear why the Delano et al. offsets are significantly larger than ours in the zone of largest slips. As Delano et al. (2022), the code does not measure any relevant throw west of profile 15, suggesting that the fault ends at short distance away from profile 15 (Fig. 14a). To the throw data of Delano et al. (2022), we add some scarp slope measurements which confirm a very shallow-dipping scarp (Fig. 14a). The fault width cannot be measured, due to a too small number of change points in the smooth topography (Fig. 14a).

Throws measured in the Diff DSM also compare well with those of Delano et al. (2022) (Fig. 14b). We measure a largest coseismic throw of ~3 m, versus ~2.5 m in Delano et al. (2022). In the westernmost and

eastermost parts of the rupture, the code is unable to identify a fault scarp and/or to recover a relevant throw measurement, even at places where Delano et al. (2022) provide an offset. Due to the low Th, the fairly low resolution of the DSM and the complex morphology, uncertainties on our throw measurements are fairly large, 0.5–1.4 m, more than those estimated by Delano et al. (Fig. 14b). The few scarp slope measurements are consistent with those in the post-EQ DSM (Fig. 14b). Only a few fault width measurements are relevant (Fig. 14b).

### 3.6. Borah Peak surface earthquake rupture, Idaho, USA (0.1 m DEM)

We apply Auto_Throw to another earthquake rupture, the 1983 Mw 6.3 Borah Peak normal fault earthquake, Idaho, USA. We focus on the ~3 km-long Arentson Gulch splay fault of the Lost River fault (F12, Fig. 14c), where very high-resolution (0.1 m) topographic data have been acquired (Bunds et al., 2019), and throws measured semi-manually (Bello et al., 2021). The splay fault has a segmented trace, and we analyze its principal, ~2 km-long, continuous scarp (Fig. 14c). Besides metric offsets, another challenge is that the earthquake ruptured a pre-existing scarp, altogether cutting a steep hillslope in unconsolidated sediments (Crone et al., 1987; DuRoss et al., 2019, Fig. 14c). Discriminating nested flats and scarps is thus challenging. This led us to run three calculations with sub-sets of profiles with variable Min-Dip<sub>scarp</sub> and Max-Dip<sub>Flat</sub>, all at a low Th range (Fault F12_Synthesis). The runs well define offset surfaces and throws in general (Fig. 14c). Because of the low Th ranges, uncertainties on throws are fairly large, from 0.1 to 1.5 m. The runs also robustly constrain the steepest and the mean scarp slopes, and the fault width (Fig. 14c). Our throw measurements are consistent with those of Bello et al. (2021), although generally larger by a few tens of cm (Fig. 14c). In the zone of largest slip and farther east, our measurements are consistent with the few “Quaternary” offsets measured by Bello et al. (2021), suggesting they represent cumulative slips including several earthquake events (Fault F12_Synthesis). When we exclude them to retain only the 1983 coseismic offsets, our measurements combine with those of Bello et al. (2021) to produce a dense coseismic slip profile (Fig. 14c).
<figure><img src="image_23.png" /><figcaption>**Fig. 14.** Distributions of throws (black), steepest scarp slopes (red), and fault widths (blue) along earthquake ruptures in Edgecumbe (a–b) and Borah Peak (c) sites, and comparison with prior data. Other caption elements as in Fig. 8. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>

## 4. Analysis of fault measurements in terms of Auto_Throw performance

Applying Auto_Throw to various topographic data (Pléiades, Sfm, ALOS, bathymetric data; 0.1–30 m horizontal resolution and ~0.06–5 m vertical accuracy; Table S1), we made more than 1.4 millions of measurements on 35 faults and systems spanning a wide range of lengths and contexts, and derived more than 12 000 final throws, scarp slopes and fault widths. Most of these final measurements are relevant (expert evaluation), and consistent with prior measurements. This demonstrates the robustness and efficiency of Auto_Throw.

The greatest strength of Auto_Throw is its ability to mathematically explore the range of flat and scarp configurations that successfully fit a topographic profile. The exploration is based on two basic geomorphic constraints - the minimum slope of a scarp and the maximum slope of an offset surface, that are generally standard in natural cases and hence can be assigned default values in the code. While a few other studies have similarly used slope constraints (Hodge et al., 2019; Scott et al., 2022), the novelty of Auto_Throw is to use them along with length thresholds to explore the full range of scenarios in which segments of various lengths and slopes combine to fit the topographic profile while they discriminate scarps and offset surfaces (flats). With hundreds of scenarios per profile, different scarp and flat locations and attitudes are considered, and their performance recorded at each step. Eventually, the performance of all configurations is compared and the most robust configurations are identified, retained, and statistically analyzed to derive the most likely

throw(s) and its uncertainties. These uncertainties are more global than in prior studies (see discussion in Bello et al., 2021) as they include, not only the aleatory errors on every parameter and output (such as flat regression), but also the actual variability of scarp and flat configurations along the profile, with its impact on the throw variability. Also, as no manual or visual interpretation or manipulation is made, there is no subjectivity in the uncertainty calculation (see discussion in Gold et al., 2012; Scharer et al., 2014; Salisbury et al., 2015). Furthermore, several runs can be made, using different profile orientations and lengths, slope constraints, and calculation parameters, each providing a realistic, mathematical description of the morphology across a scarp at a given place. This is especially needed when offset surfaces have a significant slope aspect (horizontal angle between topographic gradient and normal to fault trace). At each site, the combination of such different runs constrains the offset variability and thus, the epistemic uncertainty on the throw (see discussion in MacKenzie and Elliott, 2017). Finally, as Auto_Throw allows analyzing serial profiles along a fault, it also quantifies the true variability of the throw along the fault.

Another strength of Auto_Throw is its capacity to work with a minimum of user actions. Provided that non-tectonic features do not dominate the zone of interest, no pre-processing of the topographic data is needed, as artefact measurements due to vegetation, roads, fences, etc. Are eventually smoothed out in the global statistical analysis of the several tens to hundreds of measurements per profile. Prior manual mapping of the fault trace is not needed either. Moreover, the positions of the final best throws actually map the trace of the fault as an
additional output (Fig. S4b). The profile location and trend are auto-
matically set, based on a rough drawing of the fault strike or the user
decision to measure throws in a specific direction (for instance in case of
oblique slip; see MacKenzie and Elliott, 2017). The user can decide to
not post-process the results (in our study, results are similar overall
whether they are based on raw or post-processed data, see below).
However, we recommend that the expert examines the final measure-
ments to verify their relevance.

While most available approaches measure throws only, Auto_Throw
measures three fault parameters: i) throw, ii) scarp slope (steepest and
mean), and iii) fault width (along with scarp width). These three mea-
surements are independent (so long as there are more than three
breakpoints in the profile). All measurements are made along the fault so
that dense along-fault profiles of throws, scarp slopes and fault widths
can be obtained. Under some assumptions on the fault dip at depth, the
heave and dip-slip components of the fault motion can also be
calculated.

Auto_Throw allows measuring throws at different resolutions: using
standard slope parameters, long profiles and large Th ranges is most
appropriate to capture large-scale far field slopes and measure total
throws across a fault zone. One or two rapid calculations may be enough
to provide a fair measurement of the total throw distribution and its
uncertainties along the fault. In contrast, if more details are needed, such
as vertical offsets on individual nested scarplets or fault strands, or
throws across steep flats, very shallow scarps, or complex fault
morphology, more specific slope parameters and shorter Th ranges are
generally necessary. In complex fault zones, we recommend using both
approaches to ensure that both total and individual throws are
recovered.

In the fault cases we analyzed, ~95 % of the far-field slopes (flats) are ≤5° and within less than 5° of one another, and their slope aspect is sub-perpendicular to the fault strike. This ensures that the throw measurements are relevant (MacKenzie and Elliott, 2017; Johnson et al., 2018). If flats have a steeper slope or slope aspects are larger, the equation for the throw calculation can be modified as proposed in previous work (Hanks and Andrew, 1989; Caskey, 1995; Thompson et al., 2002; MacKenzie and Elliott, 2017; Hampel et al., 2021). To help the user know where the throw calculation may need to be adjusted, Auto_Throw provides maps of both the average flat slope and the flat slope difference on either side of a scarp along each profile (example in Fig. 5; all measurements in Fault_Fi-Synthesis). Interestingly, the measurements of the flat slopes and the mapping of their spatial variability provide insights to analyze and possibly quantify the erosion and/or sedimentation which may have affected the far-field surfaces since or while they were faulted (e.g., Wallace, 1977; Kokkalas and Koukouvelas, 2005; Tucker et al., 2011). When specific slope parameters and low Th ranges are used, degraded fault tops and debris slopes at scarp base may be mistakenly interpreted as offset flats; while this hampers the fair measurement of the throw, it provides valuable insights on the fault scarp degradation.

Auto_Throw allows measuring the scarp slope variability in great
detail both across the scarp (i.e., slope of the several sections that form
the scarp from top to bottom, see Fig. 3) and along the fault. Scarp slopes
have rarely been measured, only sparsely (McCalpin, 1987; Wilkinson
et al., 2015; Pucci et al., 2021), or along a few faults (Hodge et al., 2019).
Yet they are important as they may retain information on the fault ge-
ometry, rock lithology, scarp degradation, strain distribution, and even
scarp age in poorly indurated rocks (e.g., Wallace, 1977; Bucknam and
Anderson, 1979; Avouac and Peltzer, 1993). Also, a steepest scarp sec-
tion or a slope break across a fault scarp may be indicative of one or
several past earthquakes or creep on the fault (Wallace, 1980, 1984;
Mayer, 1984; Avouac and Peltzer, 1993; Arrowsmith et al., 1996; Kok-
kalas and Koukouvelas, 2005; Holtmann et al., 2023). The scarp slopes
measurements provided by Auto_Throw may thus be helpful to progress
in the understanding of the interplay between tectonic and geomorphic
processes.

Finally, Auto_Throw provides a novel approach to measure the width
of a fault. We discuss it below.

5. Analysis of fault measurements in terms of tectonic implications

We have analyzed a fault population spanning a wide range of
lengths (0.1–30 km) and contexts (Table S1). Faults have different ages,
yet are all young, less than 1–2 Ma. Most are long-term features but two
of them are coseismic ruptures. Most faults cut indurated bedrock of
various types (rhyolitic tuffs, basaltic piles, pyroclastic units, lime-
stones), but two of them cut unconsolidated sediments. Most are located
in sites where erosion and sedimentation are low to insignificant (Bishop
Tablelands, Asal, EAR).

On this fault population, Auto_Throw delivers 3 independent data:
throws, scarp slopes, and fault widths. We discuss these metrics below.

5.1. Fault displacement-length scaling

Along with the metrics delivered by Auto_Throw, we have measured
the fault lengths in the image and topographic data (Table S1). They are
generally slightly greater than the lengths over which measurements
were possible. In Fig. 15, we have compiled the published data we could
find on largest cumulative displacements (Dmax) and lengths (L) for
faults worldwide, of any slip mode (references in figure; note that
Lathrop et al., 2022 provide a similar type compilation for normal
faults). We have then reported the maximum throw and length data we
measured on long-term faults. The Auto_Throw data appear consistent
with the overall slip-length scaling relation long established for faults
worldwide (references in figure). Not only this further supports the
relevance of the Auto_Throw measurements, but this confirms the strong
empirical observation that, overall, net slip scales with fault length on
long-term faults. This overall scaling holds information on the fault
growth process, as we discuss it in section 5.5. In greater detail, the data
show some variability, due to various factors such as errors in the
measurements, fault growth stage, lithology, and tectonic history. We
report to Lathrop et al. (2022) for a comprehensive analysis of these
factors and their impact on displacement-length scaling.

5.2. Common pattern of throw distribution along faults

The throws we measured markedly vary along the faults (Figs. 8–14,
black). Variations occur at small length and slip amplitude scales, and at
the scale of the entire fault length. We focus on the latter. We calculate
the median position of the vertical offsets greater than 90 % of the
largest throw on the fault. This position is not at mid-fault for 90 % of the
profiles, revealing their systematic asymmetry, with largest slip on one
end of the fault. In Fig. 16a, the zone of largest throw has been placed to
the right for all asymmetric profiles, while profiles have been normal-
ized to both the fault length and the average throw. The slip function
averaged over the profiles demonstrates the systematic asymmetry of
the throw distributions (Fig. 16a; calculation of the median in Fig. S8):
the zone of largest slip is skewed toward one fault end while throw
progressively tapers down to ~0 at the other fault end, drawing a
triangular asymmetric envelope slip distribution. This asymmetry is
found on both individual and composite faults, as on both long-term and
coseismic ruptures. Results are unchanged when raw data, i.e., with no
expert evaluation, are analyzed (Fig. S9). The other ~10 % of the faults
have a more symmetric throw distribution (Fig. 16b; raw data in
Fig. S9), yet still triangular overall. Throws thus seem to vary along
faults in a systematic, common manner. We come back to this finding in
section 5.5.

5.3. Common patterns of steepest scarp slope distribution along faults

The scarp mean slope is a complex and convolved product of various
<figure><img src="image_24.png" /><figcaption>**Fig. 15.** Maximum displacement - fault length scaling for long-term faults worldwide. Maximum displacement refers to the largest cumulative offset measured along a fault. We have compiled all the data we could find in the literature, for normal (N), reverse (R), and strike-slip (SS) faults worldwide, and added our own measurements (throws for faults longer than 1 km). The scaling is ~linear over a broad range of fault lengths from $10^{-2}$ to $10^6$ m. References in reference list, data in Giampietro et al. (2025) [dataset] (Alghuraybi et al., 2021; Balsamo et al., 2016; Bazin et al., 2010; Bramham et al., 2021; Cartwright et al., 1995; Corti et al., 2019; Delogkos et al., 2017; Densmore et al., 2004; Elliott, 1976; Ellis and Barnes, 2015; Finch and Gawthorpe, 2017; Ghalayini et al., 2017; Gillespie et al., 1992; Gross et al., 1997; Hollinsworth et al., 2019; Jenkins et al., 2021; Johri et al., 2014; Karp et al., 2012; Khalil and McClay, 2017; Kicono, 2005; Kim et al., 2000; Krantz, 1988; Lamarche et al., 2005; Lathrop et al., 2021; Mansfield and Cartwright, 2001; McClymont et al., 2009; McGlue et al., 2006; McGrath, 1992; McLeod et al., 2000; McMillan, 1975; Morley, 2017; Muraoka and Kamata, 1983; Opheim and Gudmundsson, 1989; Pan et al., 2021; Peacock, 1991; Reilly et al., 2017; Rippon, 1984; Roberts and Michetti, 2004; Roche et al., 2017; Schlische et al., 1996; Scholz and Cowie, 1990; Shunshan et al., 2011; Siegburg et al., 2020; Soliva et al., 2005, 2006; Soliva and Schultz, 2008; Torabi et al., 2019; Vétel et al., 2005; Villemin et al., 1995; Walsh and Watterson, 1987; Wedmore et al., 2020; Whipp et al., 2014; Wibberley et al., 1999; Wilkins and Gross, 2002; Willemse, 1997; Williams et al., 2022; Worthington and Walsh, 2017; Yielding et al., 1996; Zygouri et al., 2008).</figcaption></figure>

factors including fault dip, the geometry of the pre-faulted landscape, the rock type and strength, possible deformation within the scarp, and the erosion and sedimentation the scarp has experienced. Rather, the steepest scarp slope is expected to be the most preserved part of the scarp, generally called the free face. While it can also be partly affected by non-tectonic factors, it is the most reliable part of the scarp. We thus focus on steepest scarp slopes.

First, we note that the maximum steepest scarp slopes we measured at surface are fairly similar to the fault dips revealed locally at depth by

seismological data. These fault dips are 55–60° in the Bishop Tablelands as revealed by the 1986 Chalfant earthquake (Lienkaemper et al., 1987; Smith and Priestley, 2000), 70–80° in the northern Asal rift as revealed by seismicity distribution (Doubre et al., 2007a, 2007b), 45–70° in the northern East African Rift as revealed by seismicity distribution and focal mechanisms in the zone and nearby areas (Delvaux and Barthes, 2010; Muluneh et al., 2018), and 55–75° in Les Saintes as revealed by the 2004 Les Saintes earthquake (Salichon et al., 2009; Courboulex et al., 2010).
<figure><img src="image_25.png" /><figcaption>**Fig. 16.** Distributions of throws, steepest scarp slopes, fault widths, and widths of residual non-flat zones along the faults. Profiles are normalized to fault length and mean amplitude. Left column (a, c, e, g): data on the ~90 % of faults and systems (longer than 1 km) having an asymmetric throw profile (asymmetry as defined in text; median position of largest offsets considered with a 5 % uncertainty). All throw profiles are presented so as to have their largest slip to the right, and the profiles of the other fault quantities are shown in the same orientation. Right column (b, d, f, h): data on the ~10 % of faults (longer than 1 km) having a fairly symmetric throw profile. In all plots, only best constrained data are shown (all data in Fig. S9), along with the running average curve among the profiles (median curve in Fig. S8). See text for discussion.</figcaption></figure>

On all faults, steepest scarp slopes significantly vary along the fault length (Figs. 8–14, red). Moreover, the slope variations somehow mirror the along-fault variation of the throws. To examine this issue, we normalize the slope profiles of the fault population with asymmetric throw profiles, and plot them together (Fig. 16c; median in Fig. S8). The steepest scarp slope profiles show a fairly similar, systematic asymmetry, with slopes steeper in the zone of largest slips and decreasing either side. The decrease is modest however along most of the fault, and more pronounced over the final ~20 % of the fault ends where scarp slopes drop down abruptly. The slope profiles thus resemble a gently sloping plateau of fairly constant slope over ~60 % of the central fault length, bounded by steep gradients of dropping slope (Fig. 16c). The same overall pattern is found on faults having a symmetric throw profile (Fig. 16d). Results are unchanged when raw data (no expert evaluation) are analyzed (Fig. S9). The steepest scarp slopes thus seem to vary along

faults in a systematic, common manner. Since this pattern is systematic, it is unlikely to result from fault- or site-dependent factors alone. However, we examine this issue below.

In the Bishop Tablelands, Asal and EAR sites, faults cut jointed volcanic rocks, and one may argue that scarp faces and steepest slopes reflect the geometry of the joints and of the volcanic blocks (e.g., Holland et al., 2006; Weismüller et al., 2019). We do not have detailed field observations to examine this hypothesis but scarps near those we analyzed do not show such a feature, except locally (Fig. S10). This issue likely depends on fault size. On small scarps, pre-existing blocks and joints may play a significant role in setting and changing the scarp slope. But as scarps grow, block size and attitude are swamped by scarp size (see Asal case in Fig. S10). To examine this issue, we analyze scarp slopes on largest faults only, i.e., with throws ≥80 m (Fig. 17a; see also Fig. S11a where points are colored by dataset location). The envelope
<figure><img src="image_26.png" /><figcaption>c)</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>b)</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>d)</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>**Fig. 17.** Distributions of steepest scarp slopes in subsets of data (fault length ≥1 km). (a) Along largest faults (with net throws ≥80 m; data discrimination in Fig. S11a); (b) Along faults analyzed in sites where erosion has been shown to be negligible (data discrimination in Fig. S11b); (c) Along faults analyzed in sites where sedimentation has been shown to be negligible (data discrimination in Fig. S11c); (d) Along faults analyzed in highest-resolution DEMs (≤2 m, data discrimination in Fig. S11d).</figcaption></figure>

shape is similar to that shown in Fig. 16c and d, still revealing strong scarp slope gradients at both fault ends. Interestingly, on such large faults, scarp slopes start dropping on fault sections still ≥50 m high.

Erosion and sedimentation have been shown to be low in most of the sites we analyzed (Goethals et al., 2009; Varet, 2022), suggesting that the bedrock fault scarps are well-preserved overall. We point out, however, that low erosion does not necessarily translate to an absence of local scarp-modifying processes, such as topple. Fig. 17b and c shows the steepest scarp slope distributions on the faults located in the sites where erosion and sedimentation, respectively, have been estimated to be low, overall (see Figs. S11b-c for data discrimination). The envelope shape still reveals strong scarp slope gradients at both fault ends. Thus, if scarp-modifying processes affect some of the scarps, these processes cannot be discriminated in the data.

One may argue that the DEM resolution may bias the measurements of scarp slopes towards smaller values when throws are small. However, most of the topographic data we analyzed have a resolution lower than 2 m. Fig. 17d shows the normalized steepest scarp slopes on the faults analyzed in the highest resolution data (≤2 m; see Fig. S11d for data discrimination). The envelope shape is unchanged, still revealing strong scarp slope gradients at both fault ends.

Warping, folding and flexuring of the ground surface above the upper tips of a growing normal fault are ubiquitous in extensional setting (e.g., Withjack et al., 1990; Hardy and McClay, 1999; Berg and Skar, 2005; Holland et al., 2006; Coleman et al., 2019), including in the Bishop Tablelands (e.g., Ferrill et al., 2016). In such a situation, the observed “scarp” is not the fault scarp but the fold dip or the slope of the warped zone, which are typically less steep than the fault scarp (e.g., Ferrill et al., 2016). Although we did not analyze the topography off the faults, some of the profiles we extracted indeed suggest some warping over the fault ends where shallow scarps are observed (Fig. S10, and data in Giampietro et al., 2025 [dataset]).

In Fig. S12, we provide manual maps of some of the faults we

analyzed. On these faults and others, we note that the scarp slope drop occurs about where off-fault tip splay networks have developed. These off-fault networks demonstrate that deformation is distributed over fault ends, as commonly observed on other faults worldwide (e.g., Perrin et al., 2016b; Nicol et al., 2017; see also Fig. S10). It is likely that such a strain distribution induces minor cracks and faults and dilation within the master fault scarp itself, as observed in Bishop Tablelands (Ferrill et al., 2016) and elsewhere (e.g., Withjack et al., 1990; Berg and Skar, 2005; Holland et al., 2006). This cracking and dilation of the fault scarps could contribute to their smoothing and result in a lower slope.

Finally, along-fault variations of fault dip at depth could also induce lateral changes in surface scarp slopes, with lower slopes likely indicating lower dips. Available data on fault dips are too few to examine this hypothesis for the analyzed faults. We note that significant along-fault dip changes have been reported on a number of faults worldwide (for normal faults, e.g., Giba et al., 2012; Brunsvik et al., 2021; Luo et al., 2022; Braunmiller and Wetmore, 2024; Locchi et al., 2024), with dip gradients as large as 10–15° per km, and dips shallowing towards fault ends in some cases (all slip modes: e.g., Avouac et al., 2014; Barnhart et al., 2014; Iezzi et al., 2018; Woo et al., 2020; Ishikawa and Bai, 2024).

From the above discussion, we are confident that, overall, the measured steepest scarp slopes are reliable metrics of the faults and reveal some of their intrinsic properties. We suggest that surface warping, strain distribution, and fault dip changes may occur, alone or in conjunction, to produce the observed scarp slope gradients at fault ends. It was beyond our scope to study these fault ends in detail, but Auto_Throw provides useful data to conduct such an analysis in future work, in combination with field observation of scarp structure and distributed deformation, and geophysical imaging of fault dips.

## 5.4. Common pattern of fault width distribution along faults

Fault widths vary along the faults, and the variations are significant
(Figs. 8–14, blue). Overall, these variations mirror those of the throws. We normalize the width profiles of the fault population with asymmetric throw profiles and plot them together (Fig. 16e; median in Fig. S8). This reveals that the fault width profiles exhibit a systematic asymmetry, mirroring that of the throw profiles, with the fault width being larger in the zone of largest throw and decreasing either side. Similarly, the width profiles on faults with symmetric throw profiles have a fairly symmetric envelope shape (Fig. 16f). Results are unchanged when raw data are analyzed (Fig. S9). Fault widths thus seem to vary along faults in a systematic, common manner.

As the fault width includes the scarp width, and the latter partly depends on the throw (though not only, see section 5.3), we feel more relevant to analyze the width of the “residual non-flat zones” as defined in section 2.2.4 (Fig. 2c). A visual inspection of the topographic data suggests that the roughness of these zones is a convolved product of erosion (mainly rounded scarp top), sedimentation (mainly debris slope at scarp base), and deformation (small faults and cracks, as seen in Fig. S13). Auto-Throw does not allow to discriminate these factors, and we do not have field observations to do it. However, from the remote measurements that we have, we find that the width of the residual non-flat zones is fairly constant along the faults (Fig. 16g and h). It is also fairly constant among the faults, reaching 100–300 m at most on the long-term faults (Fig. 18a and b), and 1–10 m at most on the coseismic ruptures (Fig. 18a and b). These widths and their constancy are reminiscent of observations made on damage zones worldwide. Damage zones are basically sites of intense fracturing and cracking off a master fault, and they have been reported to be no wider than 150–300 m along

faults with net slips lower than a few hundreds of meters (e.g., Savage and Brodsky, 2011; Mayolle et al., 2019, 2021; Torabi et al., 2020), as is the case of the faults analyzed here. Along earthquake ruptures, the damage zone is thinner, 1–10 m at most (e.g., Little et al., 2021; Gaudreau et al., 2023). Further work is needed to discriminate the tectonic and geomorphic factors at play in the rough zones detected by Auto-Throw, and to examine their possible link to tectonic damage zones.

### 5.5. Insights on fault growth

The analyzed faults share common properties, despite all their differences (Fig. 16). We suggest that these common properties result from the fault growth process.

Asymmetric, fairly triangular slip profiles have already been described, on normal (e.g., Peacock and Sanderson, 1996; Nicol et al., 1996; Cowie and Shipton, 1998; Cartwright and Mansfield, 1998; de Joussineau et al., 2001; Manighetti et al., 2001a, 2009, 2015; Schlagenhaus et al., 2008; Weismüller et al., 2019), reverse (e.g., Ellis and Dunlap, 1988; Shaw et al., 2002; Davis et al., 2005), and strike-slip faults (e.g., Pachell and Evans, 2002; Nicol et al., 2017; Manighetti et al., 2020). A systematic asymmetry has also been demonstrated in the slip-length profiles of large earthquakes (Manighetti et al., 2005). Our observations confirm these earlier findings. The common shape of the slip profiles disproves the hypothesis that the asymmetry of the slip distributions and the long slip tapers result from local or site-dependent processes only, such as rock material changes (e.g., Burgmann et al., 1994), local fault interactions (e.g., Burgmann et al., 1994; Willemse

<figure><img src="image_30.png" /><figcaption>**Fig. 18.** Width of residual non-flat zone versus throw. (a) Most robust measurements of width and throw data on individual faults ≥1 km (data on Edgecumbe and F4a-b not included as they are poorly constrained). (b) Largest width of residual non-flat zone versus largest throw. Fault population as in (a) with additional exclusion of F5 (because largest width is at a major relay zone that do not exist in other faults). See discussion in text.</figcaption></figure>
et al., 1996; Nicol et al., 1996, 2010), or differential erosion along the fault (e.g., McPhillips and Scharer, 2023). Slip tapers have also been suggested to result from a lateral decrease in fault slip rate (Amos et al., 2010; Veliz et al., 2018; Nicol et al., 2020a,b), but other studies have shown that slip rates are almost uniform along faults (McPhillips and Scharer, 2018, 2023). Long-term slip tapers may thus rather result from the fault becoming younger, i.e., growing and propagating laterally, in the direction of progressive offset decrease (Peacock and Sanderson, 1991, 1994; Wu and Bruhn, 1994; Nicol et al., 1996; Manighetti et al., 2001a, 2004; Bennett et al., 2005, 2006; Amos et al., 2010; Ferrill et al., 2017).

Fault growth is generally thought to be bimodal, with long phases of dominant slip accumulation at fairly constant length, and shorter phases of dominant lateral fault lengthening with smaller slip accumulation (Armijo et al., 1996; Morewood and Roberts, 1999; Contreras et al., 2000; Poulimenos, 2000; Goldsworthy and Jackson, 2001; Meyer et al., 2002; Walsh et al., 2002; Gawthorpe et al., 2003; Childs et al., 2003; Taylor et al., 2004; Bennett et al., 2005; Nicol et al., 2005, 2017, 2020b; Schlagenhauf et al., 2008; Giba et al., 2012; Jackson and Rotevatn, 2013). If the fault is free to propagate laterally, it develops a fairly symmetric triangular slip profile (Nicol et al., 1996; Manighetti et al., 2001a; Schlagenhauf et al., 2008), as we observe on a few faults (Fig. 16b). But if the fault lengthening is hindered or arrested by a barrier (any material able to sustain large stresses without breaking, e.g., King, 1986; Burgmann et al., 1994; Manighetti et al., 2004), the slip profile becomes asymmetric, with larger slip near the barrier and slip gently tapering in the direction of free propagation (Nicol et al., 1996; Manighetti et al., 2001a, 2004; Schlagenhauf et al., 2008). This is the case most frequently observed in our data (Fig. 16a).

We found that steepest scarp slopes systematically decrease along the ~20 % long, distal parts of the faults, where throws also decrease. In the fault growth context described above, these distal fault sections are the youngest segments of the fault, and possibly the ones to grow the most in preparation for the subsequent lateral propagation of the fault. Prior work has shown that up-fault warping especially occurs during the early phases of fault growth, i.e., on young faults or fault sections (Ferrill et al., 2016; Coleman et al., 2019). The youthfulness of the distal fault segments could thus explain the sharp decrease in their scarp slopes, due to the warping of the ground surface above their growing plane.

Irrespective of how a fault reaches a triangular slip distribution, the latter makes the stress markedly variable along the fault (Manighetti et al., 2001a, 2004, 2015). Meanwhile, the overall displacement-length scaling (Fig. 15) requires the stress drop for faults of different lengths and contexts to be roughly constant (Manighetti et al., 2001a, 2004). To reconcile these paradoxical observations, it has been proposed that off-fault damage adjusts the stress on the fault plane: strain and stress diffusion would occur in a large damage zone (called outer damage zone, e.g., Perrin et al., 2016a) off the fault and relax stresses that would otherwise accumulate on the fault. Such stress and strain removal from the fault would explain the development of the long linear slip tapers (Manighetti et al., 2001a, 2004, 2005; Cappa et al., 2014). Meanwhile, the width of the outer damage zone is expected to increase as the fault lengthens, eventually forming a fan-shaped network of secondary splay faults and fractures at fault tips (Manighetti et al., 2004; Perrin et al., 2016a, 2016b). We do observe such networks at most fault ends (Figs. 6b, S12, and S13).

Tapering slip on a fault, shallowing of its scarp slope, and broadly distributed deformation off that fault, could thus be all indicative of the long-term fault lateral propagation.

## 6. Conclusions

We developed a Matlab code, Auto_Throw, to automate throw measurements in topographic data on normal faults showing a scarp at surface. Auto_Throw mathematically explores the variety of linear segments that can fit offset surfaces (flats) and scarps along a topographic

profile. Offset surfaces and scarps are so defined in all possible ways, making a throw or scarp slope eventually derived from the statistical analysis of several tens to hundreds of measurements. Auto_Throw also measures the fault and scarp widths.

The application of Auto_Throw to 35 normal faults and systems of different lengths and contexts reveals that throws, scarp slopes and fault widths significantly vary along faults, yet, in a systematic manner. Throw distributions show a similar envelope shape, triangular and asymmetric for 90 % of the faults. Steepest scarp slopes are sub-constant along faults except over their ~20 % long ends where slopes decrease abruptly. The fault width distribution mirrors that of the throws. When the scarp width is subtracted from the fault width, a rough zone of constant width is revealed around the fault. Taken together these results shed light on the fault growth processes, especially the fault lateral propagation.

Since throws, scarp slopes and widths vary along faults, a fault cannot be properly described nor understood using local measurements or average estimates. Dense profiles of multi-parameter measurements with uncertainties are needed along the entire fault, and Auto_Throw can efficiently assist in producing these measurements. At minimum, Auto_Throw could help estimating the along-fault variability of the fault quantities and represent it as a global “uncertainty” as defined by McPhillips and Scharer (2023). As proposed by these authors, this global uncertainty representing the lateral variability of the fault quantities could then be incorporated into the uncertainty of any local measurement. This would provide a more realistic vision of what the local measurement really represents. Applied to slip rate determination, for instance, this would make it clear that a local slip rate is unlikely to represent the actual fault slip rate unless the global uncertainty is taken into account. Similarly, as a local offset amplitude is unlikely to represent the largest fault offset, deriving an earthquake magnitude from a local slip measurement (as done in most paleoseismological studies) is unlikely to be relevant unless the global uncertainty is taken into account.

The generic, simple, long wavelength variations and co-variations of displacements, scarp slopes and fault widths along faults suggest that long time-averaged accumulation of earthquakes on faults tends to smooth out individual earthquake complexities. How this smoothing occurs over seismic cycles is an open question. Our findings suggest that fault sources in earthquake models may be more appropriately represented through their generic long-wavelength patterns than from their small-scale heterogeneities (e.g., fault plane roughness), as it has been commonly done so far (see discussion in Beeler, 2023).

## CRediT authorship contribution statement

**Giampietro T:** Writing – original draft, Software, Methodology, Formal analysis, Data curation. **Manighetti I:** Writing – review & editing, Writing – original draft, Validation, Supervision, Project administration, Methodology, Funding acquisition, Data curation, Conceptualization. **Leclerc F:** Writing – review & editing, Writing – original draft, Validation, Supervision, Methodology, Data curation, Conceptualization. **Gaudemer Y:** Writing – review & editing, Writing – original draft, Validation, Supervision, Software, Methodology, Conceptualization.

## Funding sources

This work has been done in the framework of the research project FAULTS_R_GEMS (ANR-17-CE31-0008) funded by the French Research Agency (ANR).

## Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
Isabelle Manighetti reports financial support was provided by French National Research Agency. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

We dedicate this work to our colleague, mentor, and friend, Paul Tapponnier. Paul was the advisor to two of us and a unique scientific mentor and guide to all of us. We learned so much from him. We are deeply grateful and thankful to him. Paul shared with us his incredible passion for tectonics, faults, and geomorphology. Paul had three guides that he taught us: observation, quantification, and rigor. We hope Paul would have appreciated our efforts to provide rigorous measurements and attempts to understand these faults he worked on with us a long time ago in East Africa, Antilles, and the Western USA. We will miss Paul clearly.

We thank Chelsea Scott and Ramon Arrowsmith for great discussions and moments in the field in Bishop Tablelands site. We thank Jean-Paul Ampuero for insightful discussions and his help on polishing the writing of the paper. We thank Jérémy Billant for early discussions on offset measurement methods. We thank Mike Oskin, Andy Nicol and four anonymous colleagues for their thorough comments on preliminary versions of the work.

## Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jsg.2025.105393.

## Data availability

We are sharing the data and code in a FAIR repository whose DOI and full reference are provided in the paper.

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