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Improved Hybrid Geoid Modeling and the FY 2000 Geoid Models

Improved Hybrid Geoid Modeling and the FY 2000 Geoid Models. Dr. Daniel R. Roman. January 16, 2001 9:30 - 10:30. Conference Room 9836. Introduction G99SSS GPSBM’s, alternative geoid height point data Residual values from GPSBM’s - G99SSS Overview of LSC GEOID99. Overview of studies

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Improved Hybrid Geoid Modeling and the FY 2000 Geoid Models

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  1. Improved Hybrid Geoid Modelingand the FY 2000 Geoid Models Dr. Daniel R. Roman January 16, 2001 9:30 - 10:30 Conference Room 9836

  2. Introduction G99SSS GPSBM’s, alternative geoid height point data Residual values from GPSBM’s - G99SSS Overview of LSC GEOID99 Overview of studies Iterative-LSC Multi-Matrix-LSC Summary of Modeling FY2000 Geoid Models Conclusions Future Research OUTLINE

  3. Introduction • GEOID90/GEOID93 used gravimetric data • GEOID96/GEOID99 were hybrids created from gravimetric & other geoid height data • FOCUS: on the approach taken to combine these different data sources and the best solutions for modeling remaining signal • Data are the same as used for the GEOID99 computation to facilitate comparison

  4. The gravimetric geoid model, G99SSS • Derived from more than three million terrestrial, marine and altimeter gravity data • EGM96 gravity removed to create residuals • Gridded at one arcminute to maximize the resolution of the gravity field • Reference datum is ITRF96(1997.0) • Converted to residual geoid height grid with 1D FFT and restored EGM96 geoid values

  5. GPSBM’s, alternative geoid height point data • GPS-derived ellipsoid heights on spirit-leveled Bench Marks (GPSBM’s) give a spot estimate of the geoid height • GPS heights are WRT NAD 83 (ellipsoidal) • Leveling is WRT NAVD 88 (orthometric) • Accuracy of geoid heights is dependent on the quality of the ellipsoid and orthometric point values

  6. Residual values from GPSBM’s - G99SSS • Interpolating G99SSS to the GPSBM locations gives two geoid height estimates • The differences between them should be zero values assuming perfect models, observations, and interpolation algorithms • Any residuals derive from errors in the gravimetric geoid, the GPS-derived ellipsoid heights, and/or the spirit-leveling

  7. Overview of LSC • Simply gridding residuals yields no error analysis - what is signal and what is noise? • Must find broader signal in the residuals that correlates over longer distances • Signal amplitude matches the auto-correlated variance (A0) of the residuals • The character of the correlated signal drop off with distance (D) is defined by A0, the correlation length (L) and a constant ()

  8. Elements of a Correlation Curve it is easier to think in terms of cm than cm2, so use standard deviation instead of the variance signal amplitude (A0) 100% Correlated Signal Power (cm2) if: Dll = L if: Dll = L 50% then: CL = 0.5 A0 then: 0% Correlation (L) length 0 increasing distance => Distance (D) from Reference Point (km)

  9. Overview of LSC (cont.) • Find the best fitting L and A0 values in Mode 1: • Iterate for a posteriori data sigma ( ) in Mode 2: • Use the correlation parameters determined between the 6169 GPSBM points to find the expected correlation at the nodes of the desired grid (s) in Mode 3:

  10. GEOID99 • A national bias of 51.7 cm & trend of 0.15 ppm (azimuth = 327o) were removed from the GPSBM-G99SSS residuals • Best fit parameters of A0 = (18.2 cm)2, L = 400 km & = (4.6 cm)2 were determined for the remaining residual signal • Note the discrepancy between the empirical data (+) and the modeled values (line)

  11. Empirical (+) Versus Modeled (-) Correlation

  12. GEOID99 (cont.) • The conversion surface contains data at 30’ intervals but was regridded to 1’ • G99SSS - conversion surface = GEOID99 • GEOID99 is then compared to GPSBM’s to determine final residual values for analysis • of the 4.6 cm final RMS difference, 2.6 cm is correlated with a 23 km correlation length

  13. Empirical (+) Versus Modeled (-) Correlation

  14. Iterative-LSC (lower then upper) Minimum Curvature (MC) of GPSBM-G99SSS residuals MC of GEOID99 LSC point estimates Single-pass LSC with corr. length = 33 km Iterative-LSC (left then right) MC of GPSBM-GEOID99 residuals Weighted-LSC of GPSBM-G99SSS res. Weighted-LSC of GPSBM-GEOID99 res. Multi-Matrix-LSC Overview of studies

  15. Iterative-LSC • A0 = (15.0 cm)2, L = 550 km, & = (5.2 cm)2 were chosen for best fit of the broader signal in the GPSBM-G99SSS residuals (lower hump) • The resulting grid, the national trend & bias, and a conversion from ITRF96 to NAD 83 are all used to create a conversion surface • The conversion surface is removed from G99SSS to create the intermediate geoid

  16. Empirical (+) Versus Modeled (-) Correlation

  17. Iterative-LSC (cont.) • Revised residuals are generated by removing interpolated values from the intermediate geoid from the GPSBM’s (GPSBM’s - inter. geoid = rev. residuals) • L=33 km, A0=(3.0 cm)2 and = (2.3 cm)2 were selected to best fit these residuals • Note the uncorrelated signal component • Resulting grid = 2nd conversion surface

  18. Empirical (+) Versus Modeled (-) Correlation

  19. Iterative-LSC (cont.) • The second conversion surface is removed from the intermediate hybrid geoid to create the final hybrid geoid model • Heights from this model are removed from the GPSBM’s for final residuals • of the 3.3 cm final RMS difference, 2.4 cm is correlated with a 14 km correlation length

  20. Empirical (+) Versus Modeled (-) Correlation

  21. Multi-Matrix-LSC • The combination of two or more correlation matrices that best model all the signal in the GPSBM-G99SSS residuals (both humps) • Matrices: • Adding 2 positive definite matrices yields a positive definite matrix • The combined matrix is used in the LSC solution

  22. Multi-Matrix-LSC (cont.) • Correlation length and amplitude for each matrix are varied to find the overall best fit • 1st matrix: A0 = (14.0 cm)2 and L = 650 km • 2nd matrix: A0 = (11.6 cm)2 and L = 100 km • The resulting grid, national trend & bias, and ITRF96 conversion are combined into a conversion surface

  23. Empirical (+) Versus Modeled (-) Correlation

  24. Multi-Matrix-LSC (cont.) • This conversion surface is removed from G99SSS to create the final hybrid geoid • Heights from this model are removed from the GPSBM’s for final residuals • of the 3.0 cm final RMS difference, 1.7 cm is correlated with an 8 km correlation length

  25. Empirical (+) Versus Modeled (-) Correlation

  26. Summary of Modeling Studies • Two approaches gave improved results in modeling GPSBM-G99SSS residual signals • The iterative-LSC process models broader signal with a single matrix, generating an intermediate geoid and revised residuals that are modeled with another single matrix • Multi-Matrix-LSC uses multiple matrices in a single pass to best fit the initial residuals

  27. Summary of Relevant Statistics

  28. FY 2000 Geoid Models • Two hybrid geoids were created using FY 2000 GPSBM data and G99SSS • The first, XUSHG2000A, was generated using the same methods as for GEOID99 (single-pass and single-matrix) • The other, XUSHG2000B, was generated using Iterative-LSC (multi-pass and single-matrix)

  29. FY 2000 GPSBM Data • Pulled on September 15, 2000 • 7775 total points = 254 rejected + 7521 kept • Of the 7521 retained points, 1358 were new covering more regions than FY 1999 • More FBN/CBN values with increased accuracies for ellipsoid heights (12 states)

  30. XUSHG2000A • Model: single-pass, single-matrix • Correlation Parameters: A0 = (17.7 cm)2, L = 400 km and = (4.5 cm)2 • Comparison with FY 2000 GPSBM’s: of the final 4.5 cm RMS difference, 2.7 cm correlated with a 22 km correlation length

  31. Empirical (+) Versus Modeled (-) Correlation

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