The slope of a field that was measured

The first three rungs of this ladder establish something sharp about maps and fields. A conformal projection carries the direction of steepest descent exactly and an equal-area one turns it by a median of nine degrees; every quantity a reader takes off the contours is wrong except the contour value itself; and a drainage network is a conformal invariant, so routes integrated on a conformal page are the routes on the ground and routes on an equal-area page part from them by hundreds of kilometres.

Every one of those results was obtained on an analytic field. The surfaces are sums of low-degree solid harmonics with a few Gaussian caps, their surface gradients are closed forms, and nothing anywhere is sampled or interpolated. That was deliberate and it was stated as a shortfall when the anchor was carved: a field estimated from observations has a gradient noisier than itself by a factor that depends on the sampling, and a divide computed from one moves whether or not a projection is involved.

This is the payment, and the first thing it turns up has nothing to do with maps.

Differencing divides the noise by the spacing #

The arithmetic is one line and it is the whole essay.

A central difference estimates the derivative as (f(x+h) − f(x−h)) / 2h. If each sampled value carries independent noise of standard deviation σ, the numerator carries noise σ√2 and the estimate carries σ / (h√2). The noise in the derivative is inversely proportional to the spacing.

The other error goes the other way. The same difference is exact for a quadratic and wrong at third order for anything else, so its discretisation error is h²f‴/6, which falls as the square of the spacing.

Two error terms, one rising as 1/h and one falling as h², so their sum has a minimum. Setting them equal gives

$$h_{\text{opt}} \;\propto\; \left(\frac{\sigma}{f'''}\right)^{1/3}.$$

Neither half of that is new — it is the standard analysis of numerical differentiation of noisy data, and it is why every serious terrain package smooths before it differentiates. What is new here is the number, and the fact that the number belongs beside the projection numbers rather than in a separate discipline.

The measured exponent is 0.375 rather than 0.333, and the gap is not noise in the fit. The optimum is found on a discrete list of spacings, and the third derivative of this field is not a constant across the window, so the coefficient of the h² term drifts as the optimum moves. What the measurement establishes is the sign and the order: coarser, as the cube root, over three decades, with no exponent near a half or near one.

What it looks like on the ground #

The distribution matters as much as the median. The sampling error is not a uniform blur: it is smallest where the field is steep, because the same absolute noise is a smaller fraction of a larger gradient, and it is worst where the field is nearly flat. A hillshade computed from a noisy model looks clean on the slopes and speckled on the plateau, and that is a property of the arithmetic rather than of the terrain.

This is the same shape as a coordinate with a width — an error stated as an input, propagated exactly, and reported as a distribution rather than a bound. The difference is that a coordinate’s width propagates through a linear map and shrinks; a value’s width propagates through a difference and grows.

Where the survey overtakes the sheet #

Now the two errors can be put beside each other, which is what the shortfall was really about.

The horizontal line is a property of the projection and the rising curve is a property of the survey, and neither knows the other exists. Their crossing is the only number in this ladder that tells anybody what to do:

Below the crossing, the advice of the first three rungs holds. Use a conformal sheet, and the direction of steepest descent survives to arithmetic noise. The gain is real, it is eleven degrees of median bearing on this field, and it costs nothing but the choice.

Above the crossing, the advice is irrelevant. Not wrong — the equal-area sheet still turns the gradient by eleven degrees — but swamped, because the data is already turning it by more. Reprojecting the model before differentiating it is work with no measurable benefit, and the effort belongs on the survey instead.

A recommendation about projections that does not state the noise of the field it is about has left out the larger of the two terms about half the time. That is the finding, and it is uncomfortable for a site that has spent three rungs producing exactly such recommendations.

The divide moves without a map #

The third rung’s headline result is that a drainage network is a conformal invariant: routes and divides survive a conformal page exactly and part from the ground on an equal-area one, sending 2 of 196 places into a different basin on the Lambert cylindrical and 10 on Mollweide.

That measurement had no control. It compared a page against the ground and reported the difference; it never asked how much the divide moves for reasons that have nothing to do with any page.

Twenty-one of 324 is 6.5 per cent, against the 5 per cent Mollweide managed on the same kind of measurement. At σ = 10⁻³ it is 15 of 324, and at σ = 10⁻⁴ it is one. The data can move a divide further than the map does, and the reason is the same in both cases: a divide is a curve where the gradient’s direction is balanced between two basins, so it is exactly where a small angular error has an unbounded effect on which basin a place ends up in.

The control is the part worth keeping. With σ = 0 and the same 0.2° spacing, no place at all is reassigned — so at that interval the discretisation is not the problem and the interpolation is not the problem. It is the noise, and it acts through the derivative rather than through the values.

The control does not stay clean as the grid coarsens, and that is worth recording rather than hiding. At 0.8° the noiseless sampling already moves 2 places of 324 and at 1.6° it moves 4, because a divide traced on a coarse grid is a divide traced with a truncated gradient. So the noise term and the discretisation term both move the divide, they move it in the same places, and the figure above is drawn at the interval where the second is zero so that the first can be seen on its own.

The optimum belongs to the field, not to the arithmetic #

The cube-root law fixes how the best spacing responds to the noise. What sets its level is the field, and the two fields this ladder uses are far enough apart to show it.

The reason is in the closed form: the optimum goes as (σ/f‴)^⅓, so a field with a third derivative four times smaller wants a grid about one and a half times coarser at the same noise, and a field with structure at every scale — which is what real terrain has — wants the finest grid its noise will bear and never reaches a minimum at all.

This is the Richardson result turning up in a derivative. A coastline has no length until a scale is named because its structure continues below every scale; a landscape has no slope until a scale is named for the same reason, and the “slope” quoted from a model is a slope at the model’s resolution and at no other.

Smoothing is the same knob under another name #

The standard remedy in terrain analysis is to smooth the model before differentiating it, and it is worth saying exactly what that does in the vocabulary above.

Convolving with a kernel of width w and then differencing at spacing h produces noise of order σ/(h√N), where N is the number of independent values the kernel averaged — which is (w/h)² for a two-dimensional kernel. So the noise term becomes σh/w² and the discretisation term picks up the kernel’s own bias, of order w²f‴. Setting those equal gives a best kernel width that also goes as the cube root of the noise, and delivers the same minimum error.

Smoothing and coarsening are the same operation seen from two ends, and the choice between them is about what else the model is for: a coarsened grid is smaller and loses the fine detail permanently, while a smoothed one keeps its resolution and lies about it. Neither buys accuracy that is not there.

The one thing that does buy it is more observations, and it buys it slowly. Averaging n independent measurements at each node divides σ by √n, and the optimum spacing then improves as n^{-1/6}. Sixty-four times the survey effort halves the grid a slope can usefully be taken from.

Where the model stops #

The noise is white and independent between nodes. Real elevation models are not like that. A photogrammetric surface has errors correlated over several cells, an interferometric one has long-wavelength ramps, and a contour-derived model has errors that are structured by the contour interval itself. Correlated noise differences better than white noise of the same variance — neighbouring errors cancel in the difference — so every number here is a pessimistic bound for a real survey and an optimistic one for a survey with a systematic tilt.

The field is stated and is not a landscape. It is a sum of harmonics and caps, its third derivative is bounded and mild, and real terrain has a spectrum that runs to the resolution of any instrument pointed at it. A field with power at every scale has no h² limb at all, because there is always structure below the sampling, and the whole U-curve becomes a statement about the spectrum rather than about the arithmetic.

σ is an input and is not calibrated against anything. The essay states it and propagates it exactly, in the field’s own units, and never claims that a particular σ is what a particular instrument delivers. Attaching real numbers to it would make the crossing a statement about a product rather than about a mechanism.

The crossing is for one projection, one field and one window. The Lambert cylindrical’s 11.84° over this window is the projection’s own contribution; a different sheet or a different part of the world moves the horizontal line, and a rougher field moves the curve. What does not move is that there is a crossing and that it is at a noise level a real survey can be on either side of.

Why this belongs on this ladder #

There is a reasonable objection to the whole essay: numerical differentiation of noisy data is not cartography, and none of the arithmetic above involves a projection.

The answer is the crossover figure. Everything else on this ladder is a comparison between a page and the ground, and every such comparison is a statement of the form this much error comes from the sheet. A number of that form is actionable only beside the other terms in the budget, and until this rung the ladder had no other terms — so the finding that an equal-area map turns a gradient by nine degrees was true, measured, and impossible to act on, because nothing said whether nine degrees was large.

The same gap is visible elsewhere on the site and is being closed the same way. A real edge has a width exists because the resampling measurements had been made across mathematical discontinuities no sensor produces, and the widths changed the answer. The average of noisy positions moves exists because the projection’s nonlinearity had been priced without an instrument’s scatter to price it against.

A projection error is a term in a budget, and a budget with one term in it is not a budget.

Who found it, and when #

The 1/h behaviour of numerical differentiation is as old as numerical analysis and is in every first textbook on the subject. Its appearance in terrain analysis is much more recent and much less settled: slope algorithms have been compared against each other since the 1980s, usually by fitting them to synthetic surfaces with no noise at all, which is precisely the measurement that cannot see this effect. The literature on smoothing before differentiating a digital elevation model is large, empirical, and mostly expressed as a recommended filter width rather than as a spacing.

What this rung contributes is the comparison. The projection literature and the terrain-analysis literature both compute slopes, both know their own error sources well, and neither publishes the number that says which of the two is larger for a given survey. It is one figure, and it took the field’s own noise to be stated as an input rather than assumed away.

Why the optimum belongs to nobody who could publish it #

The finding that the best spacing is a property of the field rather than of the algorithm has an awkward consequence for how such advice is given, and it explains the shape of the existing literature.

A recommended filter width is a recommendation about a field. The smoothing that minimises the error depends on the noise level and on how fast the terrain’s own curvature varies, and both are properties of the particular survey and the particular ground. So there is no width that is right in general, and every published recommendation is a width that was right for whatever the author was working on.

Which is why the literature is empirical and unsettled. Papers compare filters on datasets, report what worked, and the recommendations do not agree — not because the work is careless but because the quantity is genuinely local. A comparison run on flat farmland and one run on alpine terrain are answering different questions with the same words.

The transferable thing is the trade-off, not the answer. Noise divided by spacing falls as the spacing grows; the terrain’s own detail is lost as the spacing grows; the optimum is where the two curves cross, and that structure is the same everywhere. A practitioner given the structure can find their own crossing from two numbers they can estimate — their survey’s noise and their terrain’s roughness — and a practitioner given somebody else’s width has been given an answer to a different question.

And the estimate needs no ground truth. The noise level comes from repeat measurements or from the survey’s own specification; the roughness comes from the data at a spacing where noise is negligible. Both are available to anybody holding the dataset, and neither requires knowing the true surface.

Where the ladder goes next #

The anchor now covers a field’s first derivative on a map, on the ground, and from a survey. Two things it still treats as exact are worth naming.

The sinks are the exact field’s. Every basin above is defined by the peaks and pits of the analytic surface, and the noisy field’s own critical points were never found. A sampled field has spurious pits — thousands of them, in practice, which is why every hydrological package begins by filling them — and the count of spurious pits is a function of σ and h in the same way the bearing error is.

And nothing here is anisotropic. The noise is the same in every direction and the grid is the same spacing in both, so the sampled gradient’s error ellipse is a circle. On a real projected raster it is not: the pixel is a rectangle whose aspect ratio is h/k, the two differences are taken over different ground distances, and the noise in the gradient is then anisotropic in a way that depends on the projection — which would put the sheet back into a term this rung has just finished removing it from.