Water runs downhill on the ground, not on the page

A slope is a direction and a magnitude at one point. A drainage network is what happens when that direction is followed: start anywhere, step downhill, and keep going until the ground stops falling. Every catchment boundary on every hydrological map in the world is the set of points where the answer to where does this end up changes.

Which makes the network a purely directional object. Its shape does not depend on how steep the ground is, only on which way it faces. And the previous rung established exactly what a map does to that: a conformal projection carries the direction of steepest descent unchanged and every other projection turns it.

So the network is an invariant of a conformal map, and it is not an invariant of anything else.

What the routes do #

The field is seven Gaussian caps at stated centres and widths on a gentle basin — a construction rather than a landscape, for the same reason the generalisation work builds its curves: a real elevation model would bring a sampling interval and a vendor’s smoothing into a measurement that is about the projection.

On the equal-area page the two routes part almost immediately and keep parting. Over four starting points on a harmonic field the worst separation between the true track and the computed one is 149.6, 580.5, 664.7 and 891.8 kilometres. On the Mollweide page it is 241.9, 377.4, 548.4 and 656.6. On the conformal page it is zero — not small, but zero to the precision the coordinates are held in, at every step of every route, because the two integrations are following identical bearings and therefore taking identical steps.

On the field the figures draw, which has a divide running through it, the conformal case is not quite zero: the routes agree to a few kilometres over thousands. That is not a bearing error — the bearings still agree to 10⁻¹⁰ degrees at every point sampled — it is that difference amplified by passing close to a separatrix, where the flow is exponentially sensitive to its own direction. It is a hundredfold smaller than the equal-area case and it is a different kind of quantity, which is why the assertion behind these figures tests the bearing and reports the track.

Why the agreement is exact rather than close #

The conformal result deserves more than the observation that the numbers are small, because it is a statement about curves rather than about arithmetic.

A trajectory of steepest descent is an integral curve of a direction field: at every point there is a line, and the curve is the one that is everywhere tangent to it. Two direction fields that agree pointwise have the same integral curves as sets, whatever speed anything is parameterised at. On a conformal page the page gradient is (1/k) times a rotation of the ground gradient, and a rotation by the same angle the projection turns everything else by — so measured against the projected meridian, the direction is the same number. The two integrations are therefore following one direction field, and any difference between their outputs is the integrator’s own.

What does differ is the speed. The page gradient’s magnitude is the true one over k, so a route integrated at a fixed rate on the page covers ground faster where the map is stretched. Both routes drawn above take a fixed ground step precisely so that this difference is excluded, and the first version of the measurement did not: comparing point by point at a fixed page step put two identical tracks 4,084 kilometres apart at the same index, which looked exactly like a failure of the claim and was a failure of the comparison. The fix is to compare curves rather than parameterisations, and stating it is the honest version of a result that would otherwise read as suspiciously clean.

What the false routes actually are #

The dashed routes are not noise and they are not errors in any local sense. They are the integral curves of the direction field A⁻ᵀg, which is the gradient of the same field with respect to a different metric — the flat metric of the page, pulled back onto the sphere.

So a drainage analysis run on a projected grid is not a corrupted analysis of the Earth. It is a correct analysis of a different planet: one whose surface distances are the page’s distances, carrying the same elevations at the same places. Every route it produces is the route water would take on that planet, and every catchment it delineates is a real catchment of it.

That reframing is worth having because it says which projections will be safe without measuring them. The false planet is the true one exactly when the pullback metric is proportional to the sphere’s, which is the definition of conformality — and it says the error has nothing to do with how large the distortion is. A conformal map that inflates the poles by a factor of five gets every route exactly right; an equal-area map that never departs from true area by anything at all gets them wrong, because area is not what a trajectory is made of.

The separation grows, and it does not stop growing #

A route that is wrong by a few degrees of bearing at each step accumulates, and there is no restoring force: nothing about the false route brings it back towards the true one, because the false route is following a genuine direction field of its own — just not the field the ground has.

Two features of the curves are worth reading. They are not straight, so the error is not a constant rate: it grows fastest where the routes cross the parts of the map whose axis ratio is largest. And they have flat stretches, where the two routes happen to run in a direction near one of the indicatrix’s principal axes, along which there is no turning at all.

The divide, which is where the answer changes #

A separation of 300 kilometres does not necessarily mean anything went wrong. A route that wanders and still arrives in the same basin has told the same story about the ground. What matters is whether it arrives somewhere else — and that can only happen near a divide, which is where the true answer itself changes.

Over a grid of 196 points spanning 36 degrees of longitude and 27 of latitude, the number that change basin is 0 on the conformal page, 2 on the Lambert cylindrical and 10 on Mollweide — one per cent and five per cent of the area.

Five per cent sounds modest until it is read as what it is: five per cent of a region’s ground assigned to the wrong river. A catchment boundary is a legal and administrative object — abstraction licences, flood liability, nutrient budgets and inter-basin transfer are all defined by it — and this is a five per cent error that no amount of care in the elevation data can remove, because it was introduced after the data by the sheet it was drawn on.

The rings sit in a thin band, which is the honest shape of the result and the reason the count is small rather than large. The width of that band is the distance a divide has moved, and it is set by how far the false trajectories run before they commit.

Why a large error moves so few points #

The two measurements in this essay look inconsistent until they are put side by side. The bearing is wrong by degrees over most of the sheet, the routes end up hundreds of kilometres from the truth — and only two points in 196 change basin on the Lambert cylindrical.

Both are true and the reconciliation is geometric. A basin is a region, and a region is robust: a route that leaves the true track by 300 kilometres and stays inside the same catchment has produced the same answer to the only question being asked. The set of points where an answer can change is the set within one wander’s distance of a divide, and a divide is a curve — a set of measure zero in a region, thickened by however far the false routes stray before they commit.

So the count of changed points is not a measure of how wrong the map is. It is the product of two things: how far the routes wander, which is the projection’s contribution, and how much divide there is per unit area, which is the field’s. A terrain of many small catchments would give a large count on a nearly-conformal map; a terrain of two enormous ones would give a small count on a badly distorted one.

Which means the honest way to quote the result is the pair — the separation and the count — and not either alone. A paper reporting only the count would understate the mechanism, and one reporting only the separation would overstate its consequences.

What was computed, and how #

Both integrators take a fixed step of ground distance, deliberately.

The natural alternative is to step a fixed distance on the page, which is literally what an algorithm walking a projected raster does. It is also a worse experiment: a page-stepped route covers ground at a rate that varies with the scale factor, so its track and the truth’s would be far apart at the same index while being the same curve, and the comparison would mostly measure the magnitude error the previous rung already priced. Matching the parameterisations puts the whole of the difference into the direction.

The read bearing is measured from grid north — the image of the meridian at that point — so no part of the difference is the convergence of meridians, which is a real effect belonging to a different ladder.

The sinks are found rather than declared. Descending from a grid of 256 starting points and clustering the endpoints gives three sinks with 107, 107 and 42 catches, and none of them sits at the centre of a cap: a hollow inside a basin settles where two gradients cancel, not where either is zero. A figure that had marked the cap centres as the sinks would have been drawing the input and calling it the output.

The check that makes the whole thing a measurement rather than a demonstration is the pair:

• on a conformal page, every one of the 196 points must reach the same basin as on the ground;
• on a page that is not conformal, some of them must not.

If the first failed, the claim about conformal invariance would be false. If the second passed, the test would be too coarse to see anything and every number in this essay would be an artefact of a grid too small to resolve a divide.

Where the model stops #

Real drainage is not steepest descent. Water has momentum, the ground has permeability, and a real network is shaped by erosion over geological time rather than by the instantaneous gradient of a surface. What is measured here is the operation that hydrological software actually performs on a raster, which is steepest descent with a tie-breaking rule.

Depressions are a separate problem, and a larger one. A real elevation model is full of pits that trap flow and have to be filled or breached before any network can be extracted, and the choices made there move divides by far more than a projection does. Nothing here bears on that; the field used has three sinks and they are all genuine.

The conformal result is exact only for the continuous field. A raster on a conformal grid still resamples, and resampling invents values whose gradients are not the field’s. The claim is that the projection contributes nothing on a conformal page, not that a real pipeline on one is exact.

One region, three projections, one field. The five per cent is this field over this window on Mollweide. What generalises is the mechanism and the ordering, not the number.

The generalisation #

The advice that falls out is one line, and like the rest of this anchor it is the reverse of the usual one.

Extract a drainage network on a conformal projection. Not because it is more accurate in general — it is worse for every area statistic anybody would compute afterwards — but because the network is made of directions, and a conformal map is the only kind that carries a direction unchanged.

And then compute the catchment areas on an equal-area projection, from the boundaries the first map produced. That is two projections for one analysis, which sounds like a counsel of perfection and is in fact the standard practice of anybody who has been caught: the boundary is a directional object and the area is an areal one, and no map is faithful to both.

The deeper statement is about what kind of thing a network is. A drainage network is a conformal invariant of the field it comes from — it survives any map that preserves angles, including ones that wildly distort area, distance and shape. That puts it in the same class as the family of curves an isothermal coordinate system draws, and it is the reason the network can be computed on a badly distorted sheet and still be right, provided the distortion is of the right kind.

Who found it, and when #

Steepest descent as the definition of a flow path is as old as contour maps; the divide as the locus where the answer changes is Playfair’s, in effect, from 1802.

The computational version arrives with O’Callaghan and Mark’s flow-accumulation algorithm of 1984, which walks a raster from each cell to its steepest downhill neighbour and is still the core of every drainage tool in use. That algorithm inherits the projection of whatever grid it is handed, and its literature discusses the eight-direction quantisation, the pit-filling and the resolution at length.

The projection dependence is largely absent from that discussion, which is what happens when an operation is defined on a raster rather than on a surface: the sheet the raster lives on stops looking like a modelling choice and starts looking like the ground.

A sentence of that account deserves emphasis, because it explains why the projection dependence is missing from a literature that is otherwise careful. An operation defined on a raster has no surface in its statement. Step to the steepest downhill neighbour mentions cells and elevations and nothing else, so the sheet the cells live on is not a term in the algorithm and does not appear in any discussion of its accuracy. Every other source of error — the eight-direction quantisation, the pits, the resolution, the vertical noise — is in the statement, and each of those is discussed thoroughly. The projection is the one input the algorithm cannot see, which is why it is the one nobody argues about.

Where the ladder goes next #

Three rungs establish what a map does to a field: the direction of its gradient, the readings taken off its contours, and the trajectories of its descent. Each has a different answer and each answer names a different kind of map as the right one.

What none of them touches is a field with an error bar — a surface estimated from observations rather than stated by a formula. A gradient of a noisy field is noisier than the field by a factor that depends on the sampling, and a divide computed from one moves whether or not a projection is involved. That is the ladder’s natural next rung and it belongs beside a coordinate that is a number with a width rather than here.