Improved Covariance Modeling of Gravimetric, GPS, and Leveling Data

in High-Resolution Hybrid Geoids

Daniel R. Roman, Ph.D.

Research Geodesist

AbstractHybrid geoids are created from gravimetric geoids and GPS-derived ellipsoidal heights at spirit-leveled Bench Marks (GPSBM's). Modeling of the residuals between the GPSBM's and gravimetric values was previously accomplished nationally using single values for correlation length and signal amplitude in Least Squares Collocation (LSC). The most recent high resolution hybrid geoid (GEOID99) converts heights between the NAD 83 and NAVD 88 datums at about 2.5 cm RMS accuracy, which represents the remaining correlated signal in the residuals. While this signal is lower in power compared to the initial residuals (21 cm RMS) and the features implied by it are generally narrow, these features can have great lateral extent (100's to 1000's of km). Hence it is desirable to further reduce this correlated signal, and more elaborate LSC modeling techniques were explored to do this. The two most promising are the of progressively reduced correlation lengths & signal amplitudes and also combination of multiple covariance matrices in a single pass. The results of these tests demonstrate that the correlated signal of the residuals can be reduced to 1.5 - 2.0 cm RMS. These results are discussed in relation to error sources deriving from the observed heights above the NAVD 88 and NAD 83 datums (the GPSBM's) as well as those from the gravimetric geoid model.

Outline

• Introduction

• Unresolved issues after GEOID99

• Alternative LSC modeling

• Separation of error sources

• Conclusions

100%

50%

Correlation (L) length

Cor

rela

ted

Si g

nal

Pow

er (

cm2 )

Distance (D) from Reference Point (km)

Elements of a Correlation Curve

0%

signal amplitude (A0)

0

if: Dll = L

then: CL = 0.5 A0

increasing distance =>

sll llC=

if: Dll = L

then: s =

=

0 5 0

0 7 1

0

0

.

.

A

A

it is easier to thinkin terms of cm than cm2, so use standarddeviation instead ofthe variance

C A ell

DLll

=- æèç

öø÷

0

2

k

k = @12 1 2ln .

Empirical (+) versus Gaussian Function (line) for GPSBM-G99SSS

Empirical (+) versus Gaussian Function (line) for GPSBM-GEOID99

Empirical Error Statistics for GEOID96 (100 km range)

Empirical Error Statistics for GEOID96 (1000 km range)

First empirical covariance function for iterative-LSC

Second empirical covariance function for iterative-LSC

Empirical covariance function for MM-LSC

Empirical covariance function for Gaussian-Sinusoidal combination function

Error Sources

• GPS Obs.• Short/Int. • Statewide

adjustments (HARNs)

• CORS• National

adjustment

• Gravimetric Geoid• Faye anomalies• DEM resolution and

accuracy• Remove-and-

Restore (EGM96)• 1D FFT solution• New DEM/gravity• Combined data &

Fourier solution

• Leveling (BM)• Long/Int. • Quality of

initial gravity• The effect is

greatest in the mountains

• Propagation• GPS/Leveling

Summary & Outlook

• More complex models of the Gaussian function better emulate GPSBM residuals

• Further near term improvements will derive from readjusting and improving input data

• Long term improvements require revising the entire approach taken to generate the underlying gravimetric geoid