Distributions & summary statistics

How to cite

The questions: what is the spread of each measured output across all the simulations — and which input parameter shifts it? And what are the typical values, class by class, at a glance?

Where Model vs reality scatters every simulated point and Per-event boxplots summarises the field data, this page looks at the shape of the distinct element method (DEM) results themselves. The top panel is a histogram of one measured quantity over every model stage, split into translucent overlapping distributions — one per value of a chosen model parameter, each rising from zero so their shapes can be compared directly. Beneath it sit two summaries of the same data, both as a mean and a one-standard-deviation band for each scarp class — the six shapes shown in Figure 2: one pooling every model stage into a single value per class, one tracking how those values move as slip accumulates — the paper's Figure 8. What the three measures are, and where on a scarp each is taken, is Figure 5.

Unfamiliar terms?

Mean, standard deviation and histogram are covered under statistics; the six scarp classes — Monoclinal, Pressure Ridge, Simple and their … Collapse variants — and the field datasets are in the glossary too.

The vertical black lines

The thin black lines standing in the histogram are field measurements — one line per individual measurement, not one per earthquake, so an event's internal spread is visible: the 1952 Kern County compilation contributes eleven deformation-zone widths and sixteen vertical displacements, and every FDHI-flatfile and SURE event in the export — Kaikoura, Chi-Chi, Wenchuan, Kashmir and the rest — contributes each of its own measurements.² Where the model's histogram and the field's needles overlap, the simulations bracket reality — the same conversation as the paper's Figure 15, which overlays Kern County lines on exactly this kind of distribution.¹

Three controls

• Measure switches the histogram between scarp height, deformation zone width and scarp dip, re-binning as it goes (¼ m, 1 m and 5° bins respectively). Field needles follow where the field measured the same quantity; on scarp dip they disappear — the field datasets carry no comparable value, so nothing is drawn.
• Hue By re-splits the distributions by any of the model's inputs — scarp class, sediment density, depth, fault dip, sediment strength and more — which is how the paper's Figures 9–12 walk through the parameter study one hue at a time.¹
• Population chooses between every model stage (the default — distributions pooled over the whole run of every experiment, the convention of the paper's histogram figures) and final state per trial (one row per experiment, its end state: 3,434 rows). Typical values run higher on final states, because scarps grow as a run progresses — both views are legitimate answers to slightly different questions.³

Both summary panels obey the Measure control, so they always describe the quantity on display. Population applies to the histogram and the pooled panel; the Figure-8 panel is computed across every model stage by definition, so it does not move with that control.

On the Figure-8 panel, hovering a class brightens its three lines and clicking one isolates it — the y-axis rescales, which is how you can tell the other classes are filtered out rather than dimmed. Click the same line a second time to bring all six back.

Two ways of summarising, and why both are here

The lower two panels answer different questions, and the paper's Figure 8 is the second of them.

Typical values per class pools every model stage into one mean and one standard deviation per scarp class. It answers what does a monoclinal scarp typically look like? — useful, and what this page showed on its own until September 2026.

Mean ± σ as slip accumulates is Figure 8. For each 0.05 m increment of slip it takes the mean and sample standard deviation across the model stages that fall in that increment, per class — so each class becomes a curve rather than a point. That is what lets the paper report a near-linear relationship of mean scarp height and the amount of slip at depth, deformation zone width growing as slip accumulates, and scarp dip showing only a limited relationship with the slip at depth — the paper's own phrase, and a weaker claim than the other two: for the collapse variants dip does move appreciably. None of those are readable from a single pooled number: a monoclinal scarp averages 1.53 m across its whole life, but grows from close to zero to about 3.5 m as slip runs from 0 to 5 m.

The recipe is the authors' own. Their Figure-8 analysis code was recovered in September 2026, and the pipeline reproduces its bins, its means and its sample standard deviations exactly.³

Deliberate departures from the typeset figures

The histograms show counts, not the probability scale of the paper's Figure 15, so tall and short classes keep their true proportions.

On the Figure-8 panel, three smaller differences. The paper begins each class at a hand-chosen amount of slip; this panel shows every increment for which a standard deviation can be computed, so some curves start earlier. Where the paper draws a fitted polynomial through the means, this draws the binned means themselves — dense enough, at up to a hundred points per class, to carry the same shape. And for scarp dip the authors' code reads a Convert_Scarp_Dip column that the published dataset does not carry, so that one series applies the same method to the dataset's Scarp_Dip instead — the other three measures come from the same columns the authors used.

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What the printed figures cannot do

Figures 9–12 fix one hue per panel and one binning per figure; Figure 8 shows one aggregation of one population. Here the same underlying data answers all of those at once: flip the hue to ask which parameter shifts this distribution, flip the measure to ask it of a different quantity, flip the population to see whether the answer depends on pooling model stages or taking end states — and the field needles stay overlaid throughout, keeping the model-vs-reality comparison in view.

Where this comes from

This page covers chart families 3 and 4 in the project's inventory — the faceted histograms of the paper's Figures 9–12 and 15, and the per-class mean ± standard-deviation summary of Figure 8.¹ The field needles come from a dedicated per-measurement export described on the Data page, with its populations pinned by the project's tests.³

Where to go next

• Per-event boxplots — the field data's own spread, event by event, with the model alongside.
• Model vs reality — every simulated point and every field point on one canvas.
• Slip regression — the one dashboard that does arithmetic on the simulations rather than displaying them.

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. notes/chart-families.md in the source repository maps family 3 to Figures 9–12 (histograms of one output, hue = one model parameter) and Figure 15 (the same with historic-event reference lines), and family 4 to Figure 8. Figure 8's analysis code sits outside the two legacy notebooks; the authors supplied it separately in September 2026, and the pipeline reproduces it. ↩↩↩

2. The reference-line export unions the FDHI flatfile, the SURE database and the Kern County compilation, one row per field measurement, keeping whichever of the two measured quantities each row carries: 2,392 + 203 + 21 = 2,616 rows, pinned by subprojects/python/tests/test_historic_events.py in the source repository. ↩

3. Means and standard deviations for both populations — all model stages and final state per trial — are tabulated in notes/dashboard-5-build-spec.md in the source repository, computed from the shipped DEM data (346,834 stage rows; 3,434 trials). ↩↩↩