How every number is made

AlterEcho answers one narrow question: how alike is the ground in two 10 km windows, one on Earth and one on Mars or the Moon, using measurements that mean the same thing on both? This page shows each step with the formula that runs, the symbols and units, and a worked example taken from the live calculation.

Measure the groundSlope and relative elevation on every 60 m cell of both windows, from real elevation models.
Compare whole distributionsNot just averages: the Wasserstein distance measures how far apart the two sets of values are.
Turn distance into similarityA tolerance taken from ordinary Earth terrain says how big a difference is.
Check it against ordinary landThe same score for 88 ordinary Earth windows shows whether a high number is special.

1 · Two windows and one question

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E · Earth window: A registered 10 km window with an integrity-checked NASA SRTMGL1.003 elevation model (about 30 m posting).
P · Reference window: One of five Mars windows (Jezero, Gale, InSight, Gusev, Eberswalde; USGS CTX stereo elevation, 20 m) or the lunar south-pole window (LRO LOLA, 20 m).
T · Comparison: Terrain shape. Temperature and field-test records are shown beside it but are not scored.
H · Epoch: The present-day surface. Jezero’s ancient lake is a past process; today’s elevation model records its result, not the lake.
V · Method: …

2 · Measuring the ground

Both windows are resampled onto the same 60 m grid, with a 1.2 km margin so edge cells have full neighbourhoods. Two quantities are scored:

Here $z$ is elevation in metres and $x,\,y$ are ground distances in metres. Relative elevation removes the vertical datum, which differs between Earth, Mars and the Moon. Local relief and roughness are measured too, but they rise and fall with slope (Spearman $\rho \ge 0.95$ ), so scoring them would count one property three times; they are shown as diagnostics only.

3 · Comparing whole distributions

$F$ is the share of cells at or below a value, $Q$ its inverse (the quantile function), and the distance is in the feature’s own units: degrees for slope, metres for relief. Two windows with the same shape of distribution shifted by one degree are one degree apart. A distribution does not record where ridges, channels or craters are, so two windows with very different landforms can still look alike here.

4 · From distance to similarity

The tolerance $\tau$ is the median distance between pairs of 88 ordinary Earth land windows, so a site that differs from the reference as much as two typical landscapes differ from each other gets $s = 0.5$ . It is a project definition, not fitted to expert labels, and it is accepted only when it is at least three times the resolution floor $\delta$ .

Feature	Role	τ at 60 / 120 / 240 m	τ, one tile left out	δ placement	δ elevation model	τ ≥ 3δ

5 · Combining the features

Slope and relative elevation each carry weight $a = \tfrac12$ . For terrain the index is simply 100 × the terrain score, and coverage $C = 1$ means all of the planned terrain evidence was measured; it says nothing about geology, temperature or safety. Missing evidence is left out of both sums and never counted as zero.

6 · Checking the score against ordinary land

$\mathcal{B}$ is the set of 88 ordinary Earth windows, $A_b$ their index against the same reference and $A_c$ the candidate’s. If many ordinary windows score highly, a high index says little; the table shows how often that happens. This is why the atlas shows “beats N of 88” beside every index, and the verdict reads the count in quarters: three quarters or more is “closer than most ordinary land”.

Reference	Median ordinary window	Ordinary windows at 90 or more	at 99 or more	Slope alone, 90+	Relief alone, 90+

Synthetic checks

Worked example from stored data

Every number below is read from the live calculation for the chosen site against Jezero.

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Reference windows

Each reference is a 10 km window placed by a written rule before its terrain was measured. Its resolution floor comes from moving it 2 km in four directions; a reference whose terrain changes by more than a third of the tolerance under that move is shown with distances only.

Reference	Elevation product	Median slope (60 m)	δ slope · relative elevation	Scored

What the atlas compares, and what it does not

Offered: Terrain shape (scored), Surface temperature (measured, not scored) and Field-test record (published research, not scored).

Planned, not offered yet. These task profiles exist in the API but are not shown in the atlas, because their task-critical evidence has not been measured comparably:

Temperature comparability audit

MODIS land-surface temperature (a 1 km area mean from orbit, clear-sky 8-day composites) and Mars 2020 MEDA ground temperature (a few square metres beside the rover, all conditions) are both skin temperatures, but they fail spatial-support and sampling equivalence. Temperature is therefore shown as context and never enters an index.

Why earlier scores clustered near 99.9

Index after each change, Jezero reference
Site

Each column changes one thing. The first Atacama window is shown for history only: it contains an industrial complex and is not scored.

Data exclusions

Holuhraun (Iceland) and the McMurdo Dry Valleys (Antarctica) are registered but not measured: both lie outside SRTM elevation coverage (60° N to 56° S).

Paper and historical methods

The technical paper docs/terraluna-technical-paper.md describes this build from the same verified outputs. Earlier method versions are still served for reproducibility: