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

Datum-shift, error-estimation and gross-error detection when using least-squares collocation for geoid determination.byC.C.TscherningDepartment of Geophysics,C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Collocation PreconditionsUsing 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. C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Levelling datum errorError 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 N0. In geodetic practice this does not cause problems, because height differences are the quantity of interest. (In oceanography, however, the absolute heights are important.) C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Position errors: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. C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Datum shift estimation: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. C.C.Tscherning, University of Copenhagen, 2005-01-13 *

DISAGREEMENT: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, C.C.Tscherning, University of Copenhagen, 2005-01-13 *EllipsoidGravimetric geoidGPS/Lev. geoid

Datum-shilf/Bias-tilt: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 (x, y, z), see e.g. Torge (2001).

C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Parameters through adjustment: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.C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Physical correlation: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.

C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Use Collocation:Correlations may be accounted for using LSC

LSC can be used to estimate a gravimetric geoid, a corresponding height reference surface and N0 or datum-shift parameters.

The reference surface may be used to convert GPS ellipsoidal heights to normal heights in the used height system.C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Accounting for Errors: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 http://www.ngs.noaa.gov/GEOID/GPSonBM99/format99.txt ). C.C.Tscherning, University of Copenhagen, 2005-01-13 *

GPS errors: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. C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Error detection: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.C.C.Tscherning, University of Copenhagen, 2005-01-13 *OutlierPredicted

Bias must be removed/estimated:This requires that we have unbiased estimates, which we will not have due to the N0-problem and the possibly non-geocentric datum.

Consequently we have to estimate N0 and the datum shift parameters simultaneously with the determination of the model. C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Statistical homogeiety: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 http://www.ngs.noaa.gov/GEOID/GPSonBM99/gpsbm99.html .C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Parameter and error-estimationObservations:

Parameter is equal to N0 Ai = 1 for all values of i which are associated with height anomalies . C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Parameter estimateThen 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).

C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Error-estimatesThe associated error estimates are with

the mean square error of the parameter vector

and the mean square error of an estimated quantity .

C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Datum shift and N0 estimation.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)

C.C.Tscherning, University of Copenhagen, 2005-01-13 *

N0 estimation

Height system bias may be determined.The determination of a datum-shift requires data which covers a large area, see http://cct.gfy.ku.dk/geoidschool/figure1.jpg

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.

C.C.Tscherning, University of Copenhagen, 2005-01-13 *

Height-anomaliesHeight 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*. http://cct.gfy.ku.dk/geoidschool/nmzeta83.datEllipsoidal 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. http://cct.gfy.ku.dk/geoidschool/nmzeta.rdC.C.Tscherning, University of Copenhagen, 2005-01-13 *

SmoothingC.C.Tscherning, University of Copenhagen, 2005-01-13 *

Exercise D1.Predict reduced height anomalies http://cct.gfy.ku.dk/geoidschool/nmzeta.rdfrom reduced gravity anomalies, http://cct.gfy.ku.dk/geoidschool/nmfa_egm96_tc.dat

See job-file http://cct.gfy.ku.dk/geoidschool/nmlscfa3.jo