The seven parameters have their own uncertainty

A datum transformation is published as seven numbers. OSGB36 to WGS84 is 446.448, −125.157, 542.060 metres, then 0.1502, 0.2470, 0.8421 arcseconds, then −20.4894 parts per million, and this collection has printed that row, and rows like it, in nine essays without once asking where the digits stop meaning anything.

They stop somewhere. Those seven numbers are the output of a least-squares fit to a set of common points — the same kind of solve the reduction ladder has just taken apart — and the fit produces a covariance matrix alongside the parameters. It is as much a part of the result as they are, in the same way that where a fit leaves residuals is part of the fit. It is published alongside them by the agencies that do the work properly, and nothing downstream carries it.

The widths are an input, said out loud #

Every number in this essay comes from a stated covariance, and none of it is quoted from an agency. That is the same rule the height ladder reached when it stopped citing a geoid and started stating a mass, and it is here for the same reason: quoting one agency’s standard errors would make every figure a measurement of that agency’s network rather than of the propagation.

The widths used are the order of magnitude a national transformation is published at: twenty millimetres on each horizontal translation, fifty on the vertical one, six ten-thousandths of an arcsecond on the two horizontal rotations, twelve on the one about the polar axis, and four parts per billion on the scale.

The three translations are given unequal widths deliberately, and it is not decoration. A network observed largely from one hemisphere constrains the component along the Earth’s axis worst, which is why the z entries are the large ones — and making all seven equal is the special case in which the answer comes out a circle. That case is not the real one and it is not the default here.

What each parameter is worth, and the one that is not what it looks like #

Taking one parameter at a time, with its own width and the other six held exact, separates the seven contributions cleanly, because the transformation is linear in six of them and very nearly linear in the seventh.

The scale parameter’s uncertainty is a height uncertainty and almost nothing else. That is the finding, and it follows from the geometry once it is stated: a scale change is radial, so at the Earth’s surface it displaces a point along the local vertical and leaves its horizontal position where it was. Four parts per billion of 6,371 km is 25 mm, straight up.

The consequence is a reading instruction for a parameter table. A transformation’s scale term is a statement about the vertical that is printed in a table nobody reads for vertical information, and a horizontal survey is almost indifferent to it while a height is not. This collection’s earlier finding that the third coordinate moves too is the same object from the other side: the vertical is the component of a datum shift most likely to be ignored and most affected by its least-discussed parameter.

The rotations are the other reading instruction. Six ten-thousandths of an arcsecond is 2.9 × 10⁻⁹ radians, and 2.9 × 10⁻⁹ radians of the Earth’s radius is 18 millimetres. A rotation’s width is multiplied by the distance from the geocentre, not by anything about the region, so it is a floor: no amount of local work reduces it, and a transformation derived from a national network carries it everywhere the network’s frame is used.

It is an ellipse, and it is a field #

Propagating all seven at once is one sandwich — the local frame times the transformation’s Jacobian times the parameter covariance, and back — and what comes out is a 3 × 3 covariance in metres².

Two things follow that a single quoted accuracy cannot express.

The uncertainty has a direction. At the equator the ellipse is 1.27 to 1, elongated east–west. A survey whose critical dimension runs east–west inherits a different width from one whose critical dimension runs north–south, from the same transformation at the same point.

The uncertainty is a field. Sixty-eight millimetres at the equator, forty-three at 70°, and everything between. A transformation is more certain in some places than others, which is what a datum being fitted to a region means once the fit has a covariance, and where it is most certain is not where the network that produced it was densest — it is where the geometry of the rotation terms happens to be favourable.

Reading the table with the widths in hand #

Put the two halves together and a parameter table becomes readable in a way it was not.

Three readings follow that a table of the parameters alone does not support.

The largest single contributor to horizontal uncertainty is a rotation, not a translation — 21 mm from rz against 20 mm from ty — and it is the parameter quoted to the fewest significant figures, in the smallest units, in the column a reader skims.

The vertical is dominated by two parameters that look horizontal: tz and the scale. A user who cares only about heights and assumes the transformation is a horizontal matter has picked exactly the wrong two to ignore.

And rz contributes to easting and to nothing else, while ry contributes to northing and to nothing else, at this longitude. The seven parameters do not spread their uncertainty evenly over the three coordinates; each of them has a direction, and the directions are geometric rather than statistical.

What a difference between two published sets means #

The ladder already knows that two parameter sets can differ in every number and produce the same transformation, because the seven parameters are correlated and a network can trade a translation against a rotation. This rung adds the other half of the reading.

A difference between two published sets means something only when it is measured against their widths. A difference of two standard errors is a disagreement worth investigating; a difference of a tenth of one is two ways of writing the same fit. Without the widths, a table comparing two agencies’ parameters is a table of numbers with no scale, and the natural response to it — treating the larger differences as more significant — is not available.

What was computed, and how #

The Jacobian is analytic, not numerical. The transformation is linear in the three translations and in the scale, and linear to first order in the three rotations at the magnitudes a datum shift uses, so differencing it would add rounding error to something that has none. The seven columns are the identity for the translations, the position vector for the scale, and the three skew-symmetric generators for the rotations, each with its own unit conversion built in — which is where the arcsecond-to-radian factor enters and is why a rotation’s contribution scales with the geocentric radius.

The rotation into the local east–north–up frame is the standard one, and it is what turns a geocentric covariance into the two numbers a surveyor wants: a horizontal ellipse and a vertical standard deviation. Keeping the answer geocentric would be correct and unreadable.

Nothing here uses a real network’s covariance, and the essay is careful never to say a real transformation is this accurate. What it says is that a transformation with widths of this order carries a positional uncertainty of this shape, and that the shape is the finding. The same discipline applies as in the essay on a coordinate’s written width: the covariance is an input, said out loud, and every number is derived from it.

Where the model stops #

Uncorrelated parameters. The covariance used is diagonal, and a real one is not: the translations and rotations of a fit are strongly correlated, because a small rotation of a regional network looks very like a translation of it, which is exactly the freedom the previous rung measured. Correlations would change the ellipses’ orientations and could shrink them substantially — a correlated pair can cancel where an uncorrelated pair adds. The direction of the effect is not predictable without the actual matrix, which is why the honest position is that these ellipses are for the uncorrelated case and a real one needs the real matrix.

No distortion term. A seven-parameter transformation cannot absorb the internal distortion of an old network, which is why national grids ship a shift grid instead. Where a grid is used, the transformation’s parameter covariance is not the dominant uncertainty and the grid’s own interpolation error is.

One transformation shape. The seven-parameter similarity is what national transformations use and it is what is measured here. A fourteen-parameter transformation with rates of change — which is what a modern reference frame realisation actually is, because the epoch is part of the coordinate — has seven more widths and a covariance that grows with the time since the reference epoch. That growth is the dominant term for a coordinate a decade old, and nothing here computes it.

And the widths are stated rather than measured. Everything scales linearly with them. Halve the assumed rotation widths and the rotation contributions halve; the shape of the field and the direction of every conclusion are unchanged, because they follow from the geometry rather than from the magnitudes.

The generalisation #

A transformation’s parameters are estimates, and a transformed coordinate inherits their covariance whether or not anybody carries it. The habit this argues against is treating a conversion as exact because it is deterministic. The arithmetic is exact; the numbers it is done with are not.

The general form is the delta method, and every field that fits a model and then applies it meets the same question. A calibration curve’s coefficients have a covariance and every reading through it inherits one. An instrument’s correction table has an uncertainty and every corrected measurement carries it. A currency conversion at a published rate is the trivial case, where nobody would think the rate exact.

The reason the geodetic case is worth singling out is that the arithmetic looks so much like a unit conversion. Metres to feet is exact; WGS84 to OSGB36 is a fit, and both are invoked the same way, in the same function call, with the same air of finality. The difference is invisible at the call site and is the whole of the question.

What geodesy adds is that the propagation is geometric — the Jacobian is a rotation and a lever arm, so the answer’s shape can be reasoned about before it is computed — and that the discipline already has the vocabulary for the result. An error ellipse is not a new object here. It is the object this collection drew in its first figure, arriving from a third direction — after a projection’s own distortion and a position’s measured covariance, now a transformation’s parameters.

Who found it, and when #

Friedrich Robert Helmert gave the seven-parameter similarity transformation its modern form in his 1880 Die mathematischen und physikalischen Theorien der höheren Geodäsie, and the transformation carries his name. He was also, in the same work, one of the first to insist that a geodetic result be published with its precision — the covariance of a fit was not an afterthought for him but the second half of the answer.

The practice fell away in the middle of the twentieth century for a practical reason. Transformations were published as tables for hand computation, the widths would have doubled the table, and survey precision was well above them. It returned with satellite geodesy, and the modern frame definitions — the ITRF realisations — publish parameters, rates of change, and full covariance matrices as a matter of course.

What has not returned is the carrying. The exchange formats that move coordinates between systems have a field for the transformation’s identifier and no field for its covariance, so the information exists, is published, and is discarded at the first hop. That is not a scientific gap; it is a plumbing one, and it is why a coordinate arriving from a transformation is usually treated as exact by the software that receives it.

What the registry does carry, and what it is not #

The plumbing complaint needs qualifying, because the exchange formats are not quite empty: a published transformation usually carries an accuracy figure, a single number in metres. It is worth being exact about what that number can and cannot do.

It is a scalar summary of a field. The uncertainty a transformation induces in a coordinate varies with position — it grows with distance from the centroid of the stations the fit was made over, and it is an ellipse rather than a circle at each place. One number is that field’s value somewhere, or its average, or its worst case, and which of the three is rarely stated.

So it is right at one place and wrong everywhere else, which is precisely the defect a screen map’s printed scale has, arriving in a different quantity. The failure mode is the same: a portable single number outlives the assumption that the thing it summarises is roughly constant, and nothing in the number reports the failure.

It is also the wrong shape for combining. A user who wants the total uncertainty of a coordinate — the observation, the adjustment, the transformation — needs a covariance to add, and cannot construct one from a scalar. So the transformation’s contribution is either dropped or treated as an isotropic circle, and the second is a guess dressed as a calculation.

What it is good for is triage, and that is not nothing. An accuracy figure of a metre and one of a centimetre are different kinds of transformation, and the number sorts them instantly. It answers is this good enough for my job in the common case where the answer is obvious, and fails exactly where a careful user would want it.

There is one more thing the scalar cannot express and it matters more than the shape. The uncertainty of a transformation is highly correlated between nearby points — two coordinates a kilometre apart, transformed by the same parameters, are displaced almost identically — so the difference between them is far better determined than either. A user computing a distance, a bearing or an area from transformed coordinates inherits almost none of the transformation’s uncertainty, and a user computing an absolute position inherits all of it.

A scalar accuracy figure cannot make that distinction, and it is the distinction that decides whether the figure matters. Applied naively it tells a surveyor that their setting-out is uncertain by a metre when the relative geometry is good to millimetres, which is the same over-statement a coordinate list makes about its own stations and for the same reason: an absolute figure quoted where a relative one is needed.

Which sharpens the recommendation. The six extra numbers would let a user propagate properly; a note that the transformation’s error is nearly common-mode over short distances would let them do the right thing without any numbers at all. The second costs a sentence and is not written anywhere.

The honest description is that the field carries a magnitude and discards a shape. Publishing the covariance would cost six more numbers in a record that already holds seven, and the information exists in every office that fitted one.

Where the ladder goes next #

This rung gives a transformation a width. The bodies ladder alongside it has a different debt: it solved a conformal map on a triangulated surface and reported an areal spread of 3.07, and recorded that the dependence of that number on where the body was pinned and where it was cut had never been measured.