Slip regression

How to cite

The question: what law links slip on the fault to the vertical displacement it produces at the surface — and, run backwards, what slip would explain a displacement measured in the field?

This is the one dashboard that does arithmetic on the simulations rather than just displaying them. The cloud is every model stage from the distinct element method (DEM) experiments, plotting slip against the vertical displacement of the hanging wall — how far the upthrown side actually rose. Colour is fault dip.

Unfamiliar terms?

Slip, fault dip, hanging wall, r² and back-projection are all defined in the glossary, which also sets out why each slope lands on sin(fault dip).

The fits

Seven black lines cross the cloud — one ordinary least squares fit per fault dip, each computed over that dip's own experiments. Their slopes run from 0.3436 at 20° to 0.9445 at 70°, with r² between 0.998 and 0.999.¹

Those slopes are not arbitrary. Each one lands on sin(fault dip): a fault tilted at 20° converts about a third of its slip into uplift (sin 20° = 0.342), one at 70° converts almost all of it (sin 70° = 0.940). That is the physical content of the paper's Equation 2, and the paper uses the relationship precisely because it lets model results and field measurements be compared directly.² The near-perfect fit is a consistency check rather than a discovery — the simulations drive the hanging wall along a plane at that dip, so the geometry is built in. Panel (d) of Figure 3 shows that geometry: the dip angle, the seeded fault, and the wall whose motion is the slip.

The Kern inference

The useful part is running that law backwards. Given a vertical displacement someone measured in the field, the fit says what slip must have produced it — back-projection, slip = (vertical − intercept) / slope.

The stars are sixteen vertical-displacement measurements from the 1952 Kern County earthquake compilation, placed on the chosen dip's fit line. Kern is the worked example rather than a special case: a California event with well-documented surface-rupture measurements and a known fault geometry — its 30° dip is a direct field measurement, reported in the classic Buwalda & St. Amand (1955) survey.⁵ On the fit for that measured dip, the sixteen displacements imply slips spanning 0.16 to 2.74 m.¹ The paper's own figure reaches the same conclusion — up to about 3 m of near-surface slip, consistent with independent published estimates.³

One assumption rides along

What field geologists measured at Kern is scarp height, and the inversion treats that as equal to vertical displacement.³ For simple and monoclinal scarps that holds closely; for pressure ridges the scarp stands higher than the fault alone lifted it, which is exactly what Us − Ud measures.

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What the printed figure cannot do

The paper's Figure 14 shows this analysis once, at one fault geometry. Here the geometry is yours to move.

• The fault-dip checkboxes filter the cloud, the fit line and the stars together, so you can isolate one dip and see its band cleanly, or compare two.
• The Kern Dip (measured: 30°) parameter slides the stars from one fit line to another — re-reading the same sixteen field measurements under a different fault geometry. For Kern itself the measured 30° dip is the right setting; the other positions show how the same measurements would read at a site whose fault dips differently.
• Hovering picks out a single dip's band and its line.

That second control is worth playing with, because it makes the fits' generality visible. The linear slip–displacement relationships are not specific to Kern County — they are properties of the simulations, and they extrapolate to any rupture site with those fault dips. That is exactly why a well-measured event makes the right demonstration.⁵ Move the parameter from the measured 30° to 45° and the implied slips shrink from 0.16–2.74 m to 0.12–1.95 m.¹ A steeper fault converts more of each metre of slip into uplift, so less slip is needed to explain the same step at the surface. The static figure has to show one geometry; this one lets you read any site's geometry off the same fits.

Where this comes from

This is chart family 6 in the project's inventory, which maps it to the paper's Figure 14 and Equation 2.⁴ It is the only family that needed an analytical pre-compute step: the per-dip fits and the back-projected slips are calculated in the data pipeline and exported as their own small tables, rather than being recomputed in the browser, so the numbers on this page are the numbers the project's tests pin.¹ Those exports are listed on the Data page.

Where to go next

• Response curves — the same simulations, asking how every other measured quantity grows with slip.
• Model vs reality — how the modelled range compares with field measurements overall.
• Glossary — the slip-to-uplift relationship in plain language.

How to cite

Please cite as:

Chiama, K., Bednarz, W., Moss, R., Plesch, A., and Shaw, J. H. (2025). "Quantifying relationships between fault parameters and rupture characteristics associated with thrust and reverse fault earthquakes." Earthquake Spectra, 41(5), 3977–4014. DOI: 10.1177/87552930251346434

If you use the underlying data itself, cite the archives it comes from as well — the DEM experiments are deposited open-access on DesignSafe-CI, and the field compilations carry their own citations. The full list, with DOIs, is under How to cite this data.

Any DOI issued for this site or its source code identifies the software and the website, and does not replace the citations above.

Related work, for the wider project's 3D models — not a source for anything shown here:

Chiama, K., Plesch, A., and Shaw, J. H. (2025). "Along-Strike Variability of Surface Deformation on Thrust and Reverse Fault Ruptures: Insights from 3D Distinct Element Method Models." Seismological Research Letters 96(6), 3473–3489. DOI: 10.1785/0220250173

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1. Coefficients, counts and ranges computed from the shipped DEM and Kern data by the project's own dem_regression, dem_regression_lines and kern_inferred_slip views, and pinned by subprojects/python/tests/test_regression_views.py in the source repository. See also notes/dashboard-4-build-spec.md. ↩↩↩↩

2. Chiama et al. (2025): Equation 2 "effectively describes the relationship between vertical displacement, slip, and fault dip across all of our DEM models", which "allows us to directly compare the Kern County data and DEM model results". ↩

3. Chiama et al. (2025): Kern County displacements "yield a near-surface slip of up to 3 m", consistent with independent estimates of 1–3 m and 1–4 m from earlier studies; the relationship uses scarp height "which we assume to equal vertical displacement". ↩↩

4. notes/chart-families.md in the source repository. ↩

5. Buwalda & St. Amand (1955), the classic field survey of the 1952 rupture, reports the 30° fault dip directly. The per-dip fits themselves are properties of the simulations, not of Kern — they apply to any rupture site with the modelled dips. ↩↩