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

2. OUTLINE 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

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

6. 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

9. 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

11. 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 (A 0 ) of the residuals The character of the correlated signal drop off with distance (D) is defined by A 0, the correlation length (L) and a constant (  ) Overview of LSC

12. 100% 50% Correlation (L) length Correlated Signal Power (cm 2 ) Distance (D) from Reference Point (km) Elements of a Correlation Curve 0% signal amplitude (A 0 ) 0 if: D ll = L then: C L = 0.5 A 0 increasing distance => if: D ll = L then: it is easier to think in terms of cm than cm 2, so use standard deviation instead of the variance

13. Overview of LSC (cont.) Find the best fitting  L and A 0 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:

14. A national bias of 51.7 cm & trend of 0.15 ppm (azimuth = 327 o ) were removed from the GPSBM-G99SSS residuals Best fit parameters of A 0 = (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) GEOID99

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

18. 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

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

22. Overview of studies 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

23. A 0 = (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 Iterative-LSC

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

26. 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, A 0 =(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

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

30. 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

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

34. 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

35. Multi-Matrix-LSC (cont.) Correlation length and amplitude for each matrix are varied to find the overall best fit 1 st matrix: A 0 = (14.0 cm) 2 and  L = 650 km 2 nd matrix: A 0 = (11.6 cm) 2 and  L = 100 km The resulting grid, national trend & bias, and ITRF96 conversion are combined into a conversion surface

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

38. 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

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

42. 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

43. Summary of Relevant Statistics

44. 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)

45. 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)

48. XUSHG2000A Model: single-pass, single-matrix Correlation Parameters: A 0 = (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

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

51. XUSHG2000B Model: multi-pass, single-matrix Parameters first iteration: A 0 = (14.5 cm) 2,  L = 550 km and = (5.0 cm) 2 About the same as for Iterative-LSC(99) Parameters second iteration: A 0 = (2.0 cm) 2,  L = 50 km and = (3.3 cm) 2 Truncated to minimize uncertainties in the short wavelengths of the residuals

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

54. XUSHG2000B (cont.) The two conversion surfaces are removed from G99SSS to create XUSHG2000B Heights from this model are removed from the GPSBM’s for final residuals of the final 3.5 cm RMS difference, 1.9 cm correlated with an 11 km correlation length

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

57. Statistical Comparison of FY 1999 and 2000 Hybrid Geoids

58. Conclusions for Modeling It is possible to incorporate more of the residual signal between gravimetric and GPSBM data into a hybrid geoid model Iterative-LSC modeled increasingly shorter correlation lengths, which also aided in analyzing signals in different bandwidths Multi-Matrix-LSC modeled the most of the signal in one pass with the best results

59. Conclusions for FY 2000 Geoids FY 1999 and 2000 GPSBM data have been culled to about the same level XUSHG2000A is very similar to GEOID99 Use of iterative-LSC permits selective inclusion of residual signal XUSHG2000B does incorporate more of the correlated residual signal than the FY 1999 iterative-LSC model.

60. Future Research Source(s) of residual values Focus will be on relationship with nodes in Basic Net A of NAVD 88 Better understanding of sources will aid in determining how much of the signal to use in future hybrid models