Two parameter sets, one transformation

Look up the transformation between two datums and there is rarely one answer. National agencies, standards bodies and software vendors publish sets of seven parameters for the same pair that differ by tens or hundreds of metres in the translations and by arcseconds in the rotations.

The usual reading is that somebody’s fit is better. Over the region either was fitted to, they are the same transformation, and the reason is not a matter of opinion: a seven-parameter fit over a country does not determine seven independent numbers.

The question, made precise #

A Helmert transformation has three translations, three rotations and a scale. Fitting it to a set of markers means solving normal equations $N\mathbf{p} = \mathbf{b}$, and the interesting object is not the solution but the matrix.

The question “how far can the translation be moved if the other four are re-fitted” has an exact answer. Split the seven into the translations $\mathbf{t}$ and the rest $\mathbf{r}$. For any given $\mathbf{t}$, the $\mathbf{r}$ that best absorbs it is the least-squares solution of the remaining equations, and substituting it back leaves a quadratic form in $\mathbf{t}$ alone — the Schur complement

$$S = N_{tt} - N_{tr}N_{rr}^{-1}N_{rt}$$

whose smallest eigenvector is the translation direction the network can least see. The root-mean-square ground displacement a translation of size $d$ along it produces is $d\sqrt{\lambda_{\min}/3m}$ over $m$ markers, which is a number a surveyor can put beside a tolerance.

Nothing here is fitted or searched. The direction is a property of where the markers are, and for a British network it comes out as almost pure $t_y$ — 0.9986 of it — with a small $t_x$ component and no $t_z$ at all.

The two sets #

Moving the translation 100 metres along that direction and re-fitting gives a second set differing from the first by

Those look nothing alike. A hundred metres and two and a half arcseconds are large numbers in this context: the seven parameters, and what each one does converts each into ground displacement and finds that one arcsecond of rotation is worth about 31 metres at the Earth’s surface, so the rotations here are individually worth 80 metres of ground.

They cancel. Over the region, the two sets agree to 5.6 metres at the worst marker and 3.8 in the mean.

The residual is proportional to the network’s extent #

That cancellation is not a coincidence of Britain’s size, and the law is clean.

The mechanism is the reason the law is linear. A rotation about a distant axis, seen over a small patch, is almost a translation — the difference between them is second order in the patch’s angular size, and the patch’s angular size is its extent over the Earth’s radius. Halve the network and halve what the rotation fails to absorb.

Inverting it gives the useful form: at a tolerance $\tau$ over a network of extent $L$, the translation is free to within roughly $100\,\text{m}\times\tau/r(L)$, where $r$ is the residual from the table. At 2 centimetres over Britain that is 35 centimetres of translational freedom; over a town it is 18 metres.

The refusal, and it is the practical warning #

If the two sets were genuinely the same transformation there would be nothing to warn about. They are not: they agree over the region and diverge away from it, by 193 metres in Australia — more than the hundred-metre translation that generated them, because the compensating rotation is a rotation of the whole Earth.

That is the practical rule this essay exists for. A parameter set fitted in one country and applied in another is not a small extrapolation; it is a different transformation. The number 193 is not a bound either: the far-field disagreement is roughly twice the translation gap, and published sets differ by more than a hundred metres.

The far-field figure is almost independent of the network’s size — 194 metres for every span in the table — which is the same fact from the other side. What the network’s size controls is how well the region hides the difference, not how large the difference is.

What the ambiguity looks like at one point #

A single coordinate transformed by the two sets moves by metres, and metres are the scale at which the whole datum question is usually argued. Putting this beside the shifts the datums themselves carry says how much of the argument is real.

So the ordering is: the datum shift is hundreds of metres, the parameter indeterminacy is metres, the network strain is metres, and the projection is centimetres. A practitioner who has understood the first and not the second or third is in the common case, and the second and third are the same size as each other — which is why a published set and a published grid file disagree by amounts that look like carelessness and are not.

Why this is not the same as a bad fit #

Where a fit leaves residuals measures a different thing that is easy to confuse with this one: a triangulation network is not a rigid body, so no seven-parameter transformation reproduces it exactly, and the best possible fit leaves metres on the table in a pattern.

That is a statement about the data. This is a statement about the parameters: even with perfect data and a perfect rigid motion, the seven numbers describing it are not separately determined by observations over a small region.

The two combine badly in practice. A published set carries both — an irreducible residual from the network’s own strain, and an arbitrary position along the near-null direction chosen by whatever constraint the adjustment used — and neither is recoverable from the seven numbers alone.

What the machinery had to prove about itself #

The eigen-decomposition is written here rather than borrowed, and a routine written for one use is exactly where an error hides. So it is required to reproduce the matrix it decomposed: summing $\lambda_i v_i v_i^{\mathsf{T}}$ over all seven eigenpairs must give back $N$, and it does to $1.2\times10^{-15}$ of the largest eigenvalue.

Measuring against the largest eigenvalue rather than element by element is deliberate. The normal matrix’s entries run from 1 to $4\times10^{13}$, because a translation’s column is a column of ones and a rotation’s is a column of coordinates in metres, so an element-relative residual would be asking for more digits than a double has.

The same care is why the seven parameters are converted to ground displacement before anything is said about their sizes: three metres, three arcseconds and one part per million are three different units, and the seven parameters, and what each one does shows that converting them makes the scale term beat two of the translations. A comparison across mixed units is not a comparison.

That scaling is also why the raw condition number of the matrix — around $10^{16}$ for a country-sized network — is not quoted anywhere in this essay as evidence of anything. It is a units artefact: change the rotations from radians to arcseconds and it moves by ten orders of magnitude. The Schur complement’s eigenvalues are in ground metres per parameter metre and mean something; a condition number computed across mixed units does not.

The same shape, in a survey figure #

A closed traverse has the same structure and the site measures it in the practice field: the misclosure check cannot see a uniform scale error, cannot see a rotation of the whole figure, and cannot see an angle blunder at a station standing next to the closing point.

Those blind directions are the similarity transformations — translation, rotation, scale — which is the same four-parameter group a Helmert fit absorbs. A closed figure and a datum fit are blind to the same things, and neither blindness is a defect: it is what makes the check a check on shape rather than on placement.

The connection is worth naming rather than left as a resemblance. Both are least-squares problems whose design matrix has a near-null space, and in both cases the null directions are exactly the transformations that leave the observations invariant. What differs is only which observations: a traverse observes lengths and angles, which are invariant under the similarity group; a datum fit observes coordinates, which are not — but over a small region they are nearly so, and “nearly” is what this whole essay measures.

What a published set should carry and does not #

Three things would make a seven-parameter set usable outside the region it was fitted to, and none of them is conventionally published.

The region of validity. Almost every published set has one and most citations of it do not carry it. The measurement above says why it is not a formality: outside the region the set is a different transformation by hundreds of metres.

The constraint that fixed the near-null direction. Two adjustments of the same data with different constraints give different sets, and knowing which constraint was used is the only way to compare two published sets meaningfully.

The residual pattern. The strain the seven parameters could not absorb is the difference between the transformation and the truth, and it is what a grid file carries — which is why when a formula is not enough exists and why national agencies distribute tables rather than parameters.

The alternative in practice is to stop treating the seven numbers as a description of anything and treat them as a black box with a stated domain, which is what the better standards do.

The same indeterminacy, in a smaller and more familiar place #

A grid’s false origin is the one-parameter version of all this, and the site has already measured it: a grid has an origin that is not there shows that moving it changes every coordinate in the country and not one distance or bearing.

The difference is that a false origin is exactly indeterminate — it is a convention, and no measurement anywhere can see it — while the near-null direction here is nearly indeterminate, visible only at the level the network’s extent permits. The two are the same phenomenon at two ends of a scale, and the practical consequences differ accordingly: a false origin can be redefined at will and a translation cannot, because at some tolerance it starts to show.

What the field says here #

A datum transformation is presented as a fact about two datums. It is a fact about two datums and a set of markers, and the third of those is invisible in the published form.

That is the same shape as the field’s opening claim. What a coordinate refers to says a latitude and longitude are a claim about a model with a shape, a position, a set of marked stones and a date. This says the transformation between two such models carries the marked stones too — not as a residual, which everybody knows about, but in the parameters themselves, which nobody treats as network-dependent.

What a practitioner should actually do #

Four rules follow from the arithmetic, and none of them requires understanding the arithmetic.

Use the set the agency publishes for the region, and do not mix. Two sets that agree over Britain do not agree in Australia, and neither is the “better” one to carry abroad.

Do not compare two published sets parameter by parameter. The differences carry no information about accuracy; a set can differ by a hundred metres of translation and produce the same coordinates.

Compare them on coordinates, over the region. Transform a spread of points with each and look at the difference — which is what the figure at the top of this essay does and is the only comparison that means anything.

Expect a grid file to beat both. The residual a seven-parameter fit cannot absorb is a metre or two of network strain in a pattern, and only a table can carry a pattern. When a formula is not enough measures what the table needs and finds that the spacing is set by the ripple rather than by the shift.

Between them those four say the same thing in four ways: the seven numbers are an interface to a transformation and not a description of one.

What an interface, not a description means for a user #

The closing formulation is the practical content of the whole rung, and it has consequences a user can act on today rather than a proposal for whoever maintains a registry.

Treat a parameter set as an opaque token. Its job is to be fed to an implementation that produces coordinates; it is not a description of how two datums are related, and reading meaning out of individual parameters is reading the solver’s choices. A translation of 375 metres is not a statement that one ellipsoid’s centre is 375 metres from the other’s in that direction — it is one component of a point on a near-null line, and a different point on the same line describes the same transformation with a different number there.

Never interpolate or average two sets. Both operations act on the parameters directly, both are sensitive to exactly the undetermined combination, and the average of two valid parameter sets can be a transformation that matches neither. If a blend between two regions is wanted, blend the coordinates they produce.

Never difference two sets to measure a change. A revised transformation published five years later may differ in every parameter and produce the same coordinates, or agree in six and differ substantially. The difference that means something is between the outputs at stated points.

And do not read precision from the digits. A parameter quoted to four decimal places is quoted that way because the solver printed it; the combination it belongs to may be determined to a part in a thousand of what those digits suggest, and the near-null direction is determined much worse than that.

and none of them requires knowing which combination is undetermined, which is what makes the reading usable by somebody who has not run the fit. There is one thing the token reading does not forbid, and it is worth saying so, because the prohibitions above could be read as the parameters are meaningless. They are not. A parameter set applied to coordinates gives an answer that is right to the accuracy of the fit, everywhere the fit was made — which is the whole of what it was published to do. What is unavailable is everything else: the interpretation, the comparison, the interpolation and the precision. The numbers are exactly as useful as they ever were and no more informative than the coordinates they produce.

All four follow from the same fact — that the map from parameter sets to transformations is not one to one — and all four are things a user does routinely because the seven numbers look like a description. The token reading forbids each of them, and it costs nothing to adopt.

It is the same indeterminacy because it is the same fit with more unknowns.

Where this ladder goes #

The datum ladder has established what a coordinate refers to, why a datum is fitted to a region, what each of the seven parameters does, when a formula is not enough, where a fit leaves residuals, and that the epoch is part of the coordinate.

This adds the observation that the parameters are not seven numbers but a point on a near-null line, and that where on that line a published set sits is a fact about somebody’s adjustment rather than about the Earth.

What is left on this ladder is the modern replacement for the whole apparatus: a time-dependent transformation between reference frames, in which the seven become fourteen with rates attached, and the indeterminacy measured here appears again with a rate of its own.