A deflection is the slope of a mass

The plumb line is not the normal computes the whole of the relation between a geoid slope and a deflection of the vertical. One arcsecond is the geoid rising 4.85 millimetres in a kilometre, and it displaces an astronomically observed position by 31 metres of ground. Both numbers are exact and neither is a deflection.

The essay said so at the time:

the plumb line is not the normal computes the exact relation between a geoid slope and a deflection — one arcsecond is 4.85 mm/km and 31 m of ground — and computes no deflection anywhere, because a deflection is the gradient of the geoid and the geoid is a citation under this collection’s own standing decision.

The way out is not to import a geoid model. It is to state a mass.

Why a mass is admissible where a geoid model is not #

The site’s standing decision, taken in the scale phase and recorded in the vertical machinery, is that a published spherical-harmonic geoid model is a citation rather than a measurement: EGM2008 has a truncation degree, and the separation it reports at a point is partly a measurement of that choice, which a reader cannot disentangle.

A stated mass is different in kind. A sphere of this radius at this depth with this density contrast is an input with no provenance to hide, exactly as a cap of 30° radius is. Everything that follows from it is arithmetic:

$$T(s) = \frac{G\,\Delta M}{\sqrt{s^2 + d^2}}$$

the disturbing potential at horizontal distance s from a mass ΔM buried at depth d. Outside a uniform sphere the field is a point mass’s exactly — Newton’s theorem — so the expression covers a real geological body and not merely an idealisation of one.

From it, by differentiation and Bruns’ theorem:

• the geoid rise is N = T/γ;
• the deflection of the vertical is (1/γ)·∂T/∂s, which is an angle this site now computes rather than relates;
• the gravity anomaly is −∂T/∂h.

Three quantities, one function, and two of its derivatives.

The width ratio, which depends on nothing #

The two half-widths have closed forms, and their ratio is the same for every mass at every depth:

$$N \text{ falls to half at } s = \sqrt{3}\,d = 1.732\,d, \qquad \Delta g \text{ at } s = \sqrt{2^{2/3} - 1}\,d = 0.766\,d$$

so the geoid’s feature is √3 / 0.766 = 2.2599 times as wide as the gravity anomaly’s, for a body of any mass at any depth.

The mechanism is the exponent. The potential falls as 1/r, the vertical attraction as 1/r², so the second is more sharply peaked and its half-width is smaller. Everything about the shape of the surface signature is that one fact.

This is the origin of a rule of thumb that appears in every gravity-survey textbook — the half-width rule, that the depth to a source is about the anomaly’s half-width — and the constant here is that rule’s exact value for a sphere.

The deflection’s own width, which completes the set #

Two of the three signals have half-widths in the section above and the third does not, which leaves the comparison incomplete in the direction that matters: the essay’s claim is that a deflection map is an edge-detected gravity map, and an edge detector’s output is sharper than its input rather than broader.

The deflection profile is s/(s² + d²)^(3/2). It is zero at the centre, peaks at d/√2, and falls to half its peak at 0.2047 d on the inside and 1.8964 d on the outside — so its full width at half maximum is 1.6917 d, which has no elementary closed form and is a root of a quartic.

Putting all three on the same footing, as full widths rather than half-widths:

The deflection is a sharp feature: only ten per cent wider than the gravity anomaly and less than half the width of the geoid it is the slope of. That is the quantitative form of the essay’s description, and it is not the intuition a reader brings from the previous section, where the geoid was 2.26 times the gravity anomaly and the deflection sits between them in derivative order. Differentiating horizontally sharpens, and it sharpens nearly as much as differentiating vertically does.

The three shapes also settle which measurement locates a body best. The deflection’s peak sits at 0.7071 d and the gravity anomaly’s half-max at 0.7664 d — within eight per cent of each other, so the two depth estimates use nearly the same feature of the ground. What differs is what has to be measured. A peak position is a maximum and a half-width is a level, so the first needs no calibration and no knowledge of the far field, while the second needs the anomaly’s own amplitude, which needs a regional trend to have been removed first. The deflection therefore gives a depth from a purely geometric observation, and that is the practical argument for surveying deflections in a field where gravity is cheaper to measure.

How much rock an arcsecond takes #

The relation the earlier essay computed says what an arcsecond is worth on the ground. This says what it costs in mass.

Those are ordinary geological objects. A salt dome of that size is unremarkable; so is an ore body, a granite pluton or a sedimentary basin of the same mass deficit with the sign reversed. The measurement says that deflections of the order of an arcsecond are what the shallow crust produces routinely, which is exactly what surveyors find: deflections of a few arcseconds are typical, ten or twenty occur near mountains, and the record values are around a minute.

And an arcsecond is 31 metres of ground. That is the number that made the deflection a practical problem rather than a curiosity, because before satellite positioning a national network’s orientation came from astronomical observations at a few points, and each of those points had a deflection nobody could measure.

Why the deflection peaks off to one side #

The figure puts the largest deflection 5.7 kilometres from a body buried 8 kilometres down, which is d/√2, and the position is worth deriving because it is the whole geometry of the thing in one line.

The deflection is proportional to s/(s² + d²)^{3/2}. At s = 0 it is zero by symmetry; far away it falls as 1/s²; so it has a maximum in between, and differentiating gives s = d/√2 exactly.

Two consequences follow immediately. A surveyor standing directly over an anomalous body sees no deflection from it at all — the mass pulls straight down, which is where the plumb line already points. And the largest deflections are found on the flanks of a feature, at a distance comparable with its depth, which is why deflection is an edge-finding measurement and gravity is a mass-finding one.

The same relation runs backwards: observing where a deflection peaks locates the depth of what causes it, without knowing the mass. That is the deflection’s version of the half-width rule, and it is sharper, because a peak position is easier to measure than a curve’s width.

One more consequence of the deflection’s shape is worth having, because it is what a survey planner needs rather than what an interpreter needs. The profile has a zero at the centre and a zero at infinity, so a deflection survey run over a body of unknown depth can miss it entirely by being either too tight or too coarse: stations within about a fifth of the depth of the centre see almost nothing, and so do stations beyond about twice it. The whole signal lives in a ring, and the ring’s inner and outer radii are 0.20 d and 1.90 d — a span of less than a decade in distance.

That is a narrow window compared with a gravity survey, whose signal is largest exactly where the deflection’s is zero and which therefore cannot be missed by standing in the wrong place. It is the price of the sharpening. An edge detector reports nothing in the interior of a region and nothing outside it, and a sparse survey over an unknown body is very likely to be sampling one or the other.

A gravity anomaly does not imply a deflection #

The relation invites a wrong inference and the machinery is required to refuse it.

An anomalous mass shows up in gravity, so a place with a large gravity anomaly ought to have a large deflection. It need not. A laterally uniform sheet — the Bouguer slab, Δg = 2πGρt — produces a gravity anomaly of 112 milligal for a kilometre of ordinary crust and a deflection of exactly zero, because the plumb line is pulled equally in every horizontal direction and the horizontal components cancel.

Meanwhile the buried sphere above produces 27 milligal and 2.21 arcseconds. So a body with a quarter of the slab’s anomaly bends the plumb line measurably and the slab does not bend it at all.

The reason is that the deflection is a horizontal derivative. What produces one is not mass but the lateral variation of mass — which is the same statement as the width ratio above, and the reason a deflection map looks like an edge-detected version of a gravity map.

What the deflection does to a survey, and to a coordinate #

Three effects, in increasing order of how easy they are to forget.

An astronomical latitude is not a geodetic one. A star sight measures the direction of the plumb line, so it returns astronomic latitude; a coordinate on an ellipsoid is referred to the ellipsoid’s normal. The difference is the meridian component of the deflection, and it is 31 metres per arcsecond on the ground. Every pre-satellite national network was oriented by such observations, so a deflection at the origin point is a rotation of the whole network.

A levelled height is referred to a surface that leans. Levelling follows the equipotential surface, and the equipotential surface’s normal is the plumb line, so a level’s line of sight is perpendicular to the plumb line rather than to the ellipsoid — which is one of the reasons a levelled height is not a distance.

A total station is plumbed. An instrument set up over a mark is levelled by its own bubble, which aligns it with the plumb line. Its vertical angles are therefore measured from the local vertical, and reducing them to a geodetic reference frame requires the deflection — which in practice is either ignored, because the effect on a short line is small, or taken from a geoid model.

What was computed, and how #

Newton’s constant is the only physical constant this file does not derive. Everything else — the mass from the radius and the density contrast, the potential, the two derivatives — follows from it and from the stated geometry.

The deflection is computed twice. Once as the analytic derivative of the potential, and once by differencing the geoid height numerically and passing the slope through the site’s own deflectionFromSlope, which has been in the vertical machinery since the practice phase and has never had a geoid to apply to. The two agree to 2 × 10⁻⁸ relative, which checks the new machinery against the old relation and the old relation against the new machinery.

The half-widths are measured by bisection on the profile and compared with the closed forms, which agree to the tolerance the bisection is run to.

The refusal is a zero mass. With no anomalous body stated, the expressions return exactly zero for the geoid rise and exactly zero for the deflection — because there is nothing there. A routine that returned a small non-zero deflection for no mass would be reporting its own arithmetic, and every number above would be that arithmetic plus a signal.

What this does not turn into a geoid #

The essential limit is worth stating early rather than in the closing section, because it is the difference between this essay and a geodesy paper.

A real geoid is the sum of the fields of every mass anomaly in the Earth, and computing it means knowing the density everywhere — which nobody does. What is computed here is the field of one stated body, exactly, which gives the mechanism, the scaling laws and the sizes, and gives no deflection at any real place.

That is the same bargain the whole site makes with coastlines and datasets: compute the mechanism from a stated input, cite the number that needs a survey behind it, and never blur the two.

Where the model stops #

One body, spherical, in a half-space. A real anomaly has a shape, and the field of an irregular body differs from a sphere’s at distances comparable with its size. The point-mass expression is exact outside a sphere and approximate outside anything else, which is why the figures keep the observation point outside the body by construction and refuse a geometry where the sphere reaches the surface.

The surrounding Earth is flat and homogeneous. The expressions here are the flat-Earth forms, appropriate over the few tens of kilometres these features span. Over a region large enough for curvature to matter, the geoid’s own definition brings in the normal field, which the site computes but has not composed with this.

Terrain is not topography. A mountain is a mass above the reference surface rather than a density contrast below it, and its deflection has an extra term — the attraction of the visible terrain — which is what the classical deflection of the vertical computations spend most of their effort on. Nothing here computes one.

No real deflection anywhere. The essay computes what a stated mass does. Turning that into a deflection at a place needs the mass distribution under that place, which is a survey.

Isostasy is not modelled, and it dominates. A mountain range is not simply a mass above the reference surface: it floats on a root of lighter crust whose mass deficit largely cancels the topography’s excess, which is why Bouguer’s Chimborazo deflection came out several times smaller than the visible mountain predicted. Any calculation of a real deflection from topography alone is therefore wrong by a large factor and in a known direction, and the correction is a model of the compensation rather than a measurement of it.

All three of the widths above scale with the depth and with nothing else, which is what makes them a measurement of the source rather than of the survey. Nothing in them depends on the mass, the density contrast or the radius of the body, so a shape read off a profile is a depth and an amplitude read off the same profile is a mass — two independent readings from one curve.

Who found it, and when #

Bruns’ theorem — that the geoid height is the disturbing potential over gravity — is from 1878, and it is the identity that makes the whole of physical geodesy a problem about one scalar function.

The deflection of the vertical is older as an observation than as a theory: Bouguer measured the attraction of Chimborazo in 1738 and found the plumb line deflected by far less than the mountain’s visible mass predicted, which is the first evidence of isostasy, though nobody called it that for a century. Maskelyne’s 1774 experiment on Schiehallion is the famous one — a deflection of about 11 arcseconds, used to weigh the Earth.

The half-width rule and the whole apparatus of interpreting anomaly widths as depths belong to twentieth-century exploration geophysics, where the buried sphere is the standard first model for exactly the reason it is used here: it is the shape whose field is a closed form.

Where the ladder goes next #

The height ladder is now able to compute what it could previously only relate, and the last recorded gap in the datums field is closed.

What is left open is in a different field. The flexion ladder measures the second derivative of a projection at a point and over the whole sphere; it has never measured it over a region, which is the only unit anybody chooses a projection for. And the first thing that measurement finds is that one of the site’s own projections has no second derivative at all.