Datum-shift, error-estimation and gross-error detection when using least-squares collocation for geoid determination. by C.C.Tscherning Department of Geophysics, C.C.Tscherning, University of Copenhagen,

Using LSC for gravity field modeling, initial hypothesis: model (the approximation to the anomalous potential) is associated with a geocentric reference system, and that the zero and first order spherical harmonic coefficients are all zero. The data to be used to determine the model must then also refer to this system, or have to be transformed to the system. A well known example is height anomalies determined as the difference between ellipsoidal heights from GPS and normal heights determined from levelling. Collocation Preconditions C.C.Tscherning, University of Copenhagen,

Error due to the fact that the zero levels have been fixed by a convention and not through a physical measurement. We will denote the error for a particular system by N 0. In geodetic practice this does not cause problems, because height differences are the quantity of interest. (In oceanography, however, the absolute heights are important.) Levelling datum error C.C.Tscherning, University of Copenhagen,

The ellipsoidal heights may also suffer from errors due to the fact that most positions are determined differentially, i.e. with respect to a set of reference points. If these reference points are in a non-geocentric system such as NAD83 a conversion to a geocentric system must be done, see e.g. (Smith and Milbert, 1999). This conversion is generally given as a 7- parameter similarity transformation. Position errors: C.C.Tscherning, University of Copenhagen,

The parameters of this kind of transformation are easily estimated if we have two sets of 3-dimensional positions either the Cartesian coordinates or latitude, longitude and ellipsoidal height, using a least-squares adjustment. Datum shift estimation: C.C.Tscherning, University of Copenhagen,

When a purely gravimetric quasi-geoid is compared to a surface constructed from height anomalies derived from GPS and levelling one will often note that the surfaces disagree, DISAGREEMENT: C.C.Tscherning, University of Copenhagen, Ellipsoid Gravimetric geoid GPS/Lev. geoid

Frequently they are related through a bias or tilt. Close agreement by estimating and applying the bias and the tilt. The 3 quantities (height bias, tilt in East and West) corresponds to a 3-parameter datum shift using a translation of the center of the reference ellipsoid (Δx, Δy, Δz), see e.g. Torge (2001). Datum-shilf/Bias-tilt: C.C.Tscherning, University of Copenhagen,

A simple adjustment used for the determination of the parameters will in general be sufficient to obtain an agreement between the gravimetric geoid and the height anomalies. The gravimetric geoid has been used as an interpolator to construct a height reference surface. Parameters through adjustment: C.C.Tscherning, University of Copenhagen,

This procedure does not take into account that the two data types are physically correlated, so that both the gravimetric geoid and the height reference surface may be improved. A 3 or 7-parameter adjustment does not take into account the spatially varying quality of the gravimetric geoid or the spatial distribution of GPS/levelling derived height anomalies. Physical correlation: C.C.Tscherning, University of Copenhagen,

Correlations may be accounted for using LSC LSC can be used to estimate a gravimetric geoid, a corresponding height reference surface and N 0 or datum-shift parameters. The reference surface may be used to convert GPS ellipsoidal heights to normal heights in the used height system. Use Collocation: C.C.Tscherning, University of Copenhagen,

Errors in the data must be taken into account. In many cases are the errors not known, and the data may include gross-errors. The error estimates of leveled heights are generally only known as the error of the height differences relative to set of higher order bench marks. (See format99.txt ). format99.txt Accounting for Errors: C.C.Tscherning, University of Copenhagen,

For ellipsoidal heights determined using GPS the error estimates available are also relative errors. These heights are frequently in error due to erroneous identification of the reference point or the antenna height. GPS errors: C.C.Tscherning, University of Copenhagen,

LSC filters the data. We may as done in geodetic network adjustment inspect the residuals by using LSC for the prediction and comparison with the data used to determine the model. Large difference – possible gross error. Error detection: C.C.Tscherning, University of Copenhagen, Outlier Predicted

This requires that we have unbiased estimates, which we will not have due to the N 0 -problem and the possibly non-geocentric datum. Consequently we have to estimate N 0 and the datum shift parameters simultaneously with the determination of the model. Bias must be removed/estimated: C.C.Tscherning, University of Copenhagen,

We should have a statistically homogeneous data distribution. Unfortunately this is often not true, but there are anyway possibilities for using the residuals for gross-error detection. In the following the theory will be briefly reviewed without proofs. Then the theory is exemplified using the New Mexico test data plus GPS/levelling height anomalies from the area obtained from 99.html html Statistical homogeiety: C.C.Tscherning, University of Copenhagen,

Observations: Parameter is equal to N 0 A i = 1 for all values of i which are associated with height anomalies. Parameter and error-estimation C.C.Tscherning, University of Copenhagen,

Then an estimate of T and of the parameters X are obtained as where W is the a-priori weight matrix for the parameters (Generally the zero matrix). Parameter estimate C.C.Tscherning, University of Copenhagen,

The associated error estimates are with the mean square error of the parameter vector and the mean square error of an estimated quantity. Error-estimates C.C.Tscherning, University of Copenhagen,

GEOCOL estimate a gravity field model and components of a 7-parameter similarity transformation datum-shift If parameters are (  x,  y,  z) we have (in spherical approximation) Datum shift and N 0 estimation. C.C.Tscherning, University of Copenhagen,

Height system bias may be determined. The determination of a datum-shift requires data which covers a large area, see If the area is not large, this will be reflected in large error estimates. This will be illustrated using 2920 gravity data and 20 height anomalies from the New Mexico test area. N 0 estimation C.C.Tscherning, University of Copenhagen,

Height anomalies obtained from ellipsoidal heights, h, determined by GPS and orthometric heights, H, in the North Americal Vertical Datum 1988 (NAVD88) converted to normal heights, N*. Ellipsoidal heights were available both in the continental North American Datum 1983 (NAD83) and the global ITRF94 datum. The contribution from EGM96 and residual topography was subtracted. Height-anomalies C.C.Tscherning, University of Copenhagen,

Smoothing C.C.Tscherning, University of Copenhagen,

Predict reduced height anomalies from reduced gravity anomalies, c.dat c.dat See job-file Exercise D1. C.C.Tscherning, University of Copenhagen,

Prediction result Good agreement But large bias ! C.C.Tscherning, University of Copenhagen,

Add 20 residual height anomalies to the gravity anomalies and determine a new model together with a 3-parameter translational datum-shift or a height bias shift, respectively, see Exercise D2: Datum-shift C.C.Tscherning, University of Copenhagen,

Results of datum shift determination C.C.Tscherning, University of Copenhagen,

Large error-estimates. We can not expect better results using data from such a small area. Formally the error of the estimated quasi-geoid was improved from 0.07 m to 0.02 m when only N 0 was determined, see but it should be interpreted as the improvement of the height anomaly differences. Results C.C.Tscherning, University of Copenhagen,

The detection of possible gross-errors using the differences between observed values and values calculated using LSC may applied Using GEOCOL we may compute the estimates of the data used as input as well as the error-estimates. These a-posteriori error-estimates will generally be smaller than the a-priori ("input") error estimates. The estimate of the error of the difference will then be the Gaussian sum of the a-priori error and the a-posteriori error. Verification of error-estimates and gross-error search. C.C.Tscherning, University of Copenhagen,

LSC will perform a filtering, so that if closely correlated data differ from a value of a specific measurement, a residual larger than the error estimate of the difference will occur. If this is the case, we have found a "suspected gross error". A threshold may be defined, equal to e.g. 3 times the error of the difference. Now different factors play a role: we may have under or over-estimated the a-priori error. Suspected gross-error C.C.Tscherning, University of Copenhagen,

If the original data (supposed to be distributed in a regular manner in space) follow a normal distribution (may be verified from a histogram), then the ratio between the absolute value of the difference observed minus predicted and the a- posteriori error estimate should follow a t- distribution. If it does not do this, something is wrong. Error-estimates C.C.Tscherning, University of Copenhagen,

GEOCOL will as output bin the residuals divided by the a-posteriori error in 50 % intervals. The percentages of values in each bin may then be used as an indicator of the quality of the a-priori error-estimate. Error-distribution C.C.Tscherning, University of Copenhagen,

Unfortunately the error-estimates does not take into regard that the error will be small in areas with a smooth gravity field and large where the gravity field varies much. The inspection of residuals is therefore quite problematic, and should only be used with great caution. Smooth and non-smooth areas C.C.Tscherning, University of Copenhagen,

Only residuals 5 or more times larger than the error of the difference should be regarded as gross-errors. These values should not necessarily be deleted, but they should have assigned a much larger a- priori error, and the model should be determined again. Change of weight C.C.Tscherning, University of Copenhagen,

Errors Number of suspected gross-errors as a function of a-priori data noise standard deviation and percentage of residuals divided by the a-posteriori error in 0.5 bins. Suspected gross errors are defined as measurements where the residual is 4.0 times larger than the a-posteriori error (mgal). C.C.Tscherning, University of Copenhagen,

It seem like the residuals for a-priori-error 1.0 mgal is most close to a t-distribution. The error estimates of the height anomalies did also increase from 0.02 m to 0.05 m, which seems realistic. There are however too many (56) suspected gross- errors. One would normally expect below 1 %. The values of the differences observed minus predicted which are more than 6 mgal are: T-distribution ? C.C.Tscherning, University of Copenhagen,

Suspected Gross-errors C.C.Tscherning, University of Copenhagen,

LSC may be used for datum shift determination or height system bias estimation by combining gravity data and height anomalies. However, in order to determine reliable estimates of a datum-shift a large area is needed. This limits the applicability of the method. Conclusion C.C.Tscherning, University of Copenhagen,

It is required that we know the error standard deviations of the data and that gross-errors have been deleted or down-weighted. The validity of the a-priori error estimates as well as the occurrence of outliers may be investigated by comparing the a-posteriori determined differences between observed and predicted values with the error estimate of the difference obtained using LSC. Conclusion (II) C.C.Tscherning, University of Copenhagen,

It Conclusion (III) C.C.Tscherning, University of Copenhagen,