GEOID PRECISION FROM LIMITED-AREA GRAVIMETRIC SURVEYS

We derive a number of rough theoretical estimates for the precision of a geoid model computed from a local gravimetric survey combined with global reference model information. Example calculations for Finland and Estonia are presented.


Introduction
It is possible to give a number of theoretical estimates for the precision of a gravimetric geoid computed using gravimetric survey data obtained from a bounded area and having a finite density of measurement points. The derivations presented below have in the past appeared in different form and in small pieces in the nonreviewed literature, often in connection with practical geoid determination projects [1][2][3]. These derivations, which are of simple, brute-force nature, are summarized and presented more clearly and systematically in the present article, while we also take the opportunity to correct some errors present in the earlier derivations.
We will consider three main types of geoid error deriving from a gravity survey's limited nature: • The error of omission due to the finite spatial density of the gravity survey.

•
The aliasing error, also due to the finite spatial density of the survey, where error patterns with halfwavelengths shorter than this spacing are misrepresented as geoid error patterns with longer wavelengths.
• The out-of-area error, caused by the absence of gravity survey coverage outside the area of study. Gravity information may be missing completely; or it may be available as long wavelength only information, eg as a spherical harmonic expansion of a reference gravity model. Estimating the out-of-area error requires knowledge of the signal covariance function, either of the full gravity field, or of the part of gravity not described by the reference model.
As input to the expressions we derive we need mainly two quantities: • The mean separation of gravimetric measurement points, and • the average "error of prediction" in the form of a mean error for an arbitrary point somewhere in the terrain, when predicting its gravity anomaly from those of nearby points. This error is in principle computable if we know two quantities defining the gravity anomaly field's signal covariance function: its signal covariance 0 C and its correlation length . is probably more realistic.

The error of omission
If we have gravimetric data given on a grid with spacing d, the Nyquist theorem tells us that the grid points can only represent a low-pass filtered anomaly field with cut-off wavelength 2d, ie degree .
If the mean error of prediction of gravity g ∆ σ is given, which we may assume equal to the RMS of the degree variances above the truncation limit; and if we assume for simplicity that all power is concentrated at , / d R n π = then we can compute (using, as in the sequel, ): This near-trivial derivation appeared first in [1]. Obviously this is only an upper bound; gravity anomaly power will in reality be present in all degrees between d R n / π = and .

∞
For the Finnish gravimetric survey and values quoted above, one obtains for the error of omission in the geoid undulation: . mm 3 ± = σ N For Estonia, using the values quoted above, we similarly obtain .

The aliasing error (1)
If the true gravity anomaly field we are trying to represent is not low-pass filtered in this way, we will have an aliasing error caused by the part of the field above the truncation degree.
If we assume for a moment the grid to be regular with spacing , d and the high-frequency part of the g in order to produce the given total RMS power.
Computing the corresponding geoid error means summing up the squares of these contributions, which are , 1 2 excluded, something to keep in the backs of our minds throughout the below derivation.
We may approximate this sum by an integral: Here, D i u = and D j v = are wave numbers in the two co-ordinate directions. This again is approximated by the integral over a circular disc with a hole in the middle, with bounds would be from D π 4 to slightly less than d 1 ).
The total result then is . ln 2 Compared to the expression (1) found earlier, this will typically be dominant. Note that whereas d is the spacing of the g ∆ data points, D represents the scale at which geoid errors are constrained, eg by GPS/levelling points, or by a global geopotential reference model.

Application to the Finnish gravimetric survey
Cf Table 1, recomputed from an earlier version appearing in [2]. The same values for d and g ∆ σ were used as earlier.

Application to the Estonian gravimetric survey
Cf Table 2, computed using the values referred to earlier. An independent derivation starts from the Stokes integral. In the near-field limit the Stokes kernel is identical to Eq (2).

The out-of-area error
The two above calculations assume an infinite extent of the gravimetric survey data. In reality this data will always be limited in extent, and then error will be generated especially in the border areas due to the lack of gravimetric data on the other side of the border.
We assume that the lacking gravimetric data -or alternatively, the lacking short-wavelength part of the gravimetry not contained in a global reference model used -can be described by the parameters of a Hirvonen signal covariance function l r is the inter-point distance.
In order to determine the error, we apply a discretised Stokes integration. We divide the area to the other side of the border into squares of size , l ie we choose , l y x = ∆ = ∆ the correlation length: Here, i and j are block subscripts in the two geographical directions x and y, and the "blocks" of gravimetric data are of size . l l × For simplicity, we consider a straight line border only, extending to infinity. On the left side of the border, good gravimetric data is available. On the right side, we have a missing component in the gravimetric data which is described by 0 C and . l We assume the l l × size blocks to be statistically independent from each other, with each being characterised by a signal variance of 0 C . Then we obtain using propagation of variances on Eq δ is the distance of the evaluation point from the border. We have here used a plane coordinate system centered upon the evaluation point , 0 See the figure.
Geometry for estimating out-of-area geoid error Clearly the plane approximation breaks down herewe get an infinite total error. Also for the limit 0 → δ the result diverges. Both these effects are non-physical. We eliminate the lower bound problem by requiring it to be at least , l ie we replace δ by max ( ).
This expression describes the prediction accuracy obtained when trying to predict the gravity anomaly

Application to the Finnish gravimetric survey
With this formula we have computed Table 3 for the Finnish gravimetric survey parameters d and .
g ∆ σ A realistic value for the correlation length may well be km 20 = l or a little longer. As seen, improvement of the out-of-area gravity field brought about by the satellite gravity missions, bringing down the shortest wellrepresented half-wavelength from 500 to 100 km, will somewhat improve the quality of the local geoid, though not spectacularly so.
Remember, though, that the values tabulated here only apply close to the border: the error will diminish going in-land (increasing δ ), though not very quickly, as δ appears within the logarithm. For λ ≥ δ (and thus l > δ ), the logarithm will vanish. This will actually happen inland in Finland when the new GOCE geopotential model is available with its small . km 200 = λ Let us remember that the values in Table 3 are valid only in the border zone, and the width of this border zone may be taken as the half-wavelength λ of the global reference model used. Using a better (higherdegree) expansion will narrow down the border zone influenced by these errors, an additional benefit of GOCE that is not obvious from the Table: in the second rightmost column, the italicised value 27,80 represents the error 100 km inland from the border.
Thus, while a clear improvement is to be expected from these missions, it will not certainly do away with the need to obtain a good, dense gravimetric coverage for the problem areas immediately outside Finland's borders. Where terrestrial gravimetry is not available, airborne gravimetry recommends itself.

Application to the Estonian gravimetric survey
Doing the same computation for Estonia, with the earlier quoted parameter values d and , g ∆ σ yields the results listed in Table 4. Here one may guess that the correlation length in Estonia is a bit shorter than in Finland, more like 10 km, producing plausible-looking geoid mean errors for a wide range of ( ) δ λ , combinations.
Otherwise the conclusions are similar to those for Finland. Due to the smallness of Estonia, the values in this table underestimate the true uncertainty of the absolute geoid, which will improve substantially due to the satellite missions. But also here, even more so, good gravimetric survey data immediately across the borders must also be obtained.