COMPUTATIONAL METHODOLOGY
The optimum calculation of 30' mean free-air gravity anomalies would be based on a
infinite number of point gravity anomalies in a specified cell. This definition cannot be
realized because the density and distribution of point gravity anomalies varies by
geographic region and elevation over the Earth's surface. DMA has a worldwide
requirement to acquire point or mean gravity information at required spacing but many
regions still need additional coverage or densification. The computation of 30' mean
free-air anomalies by DMA is based on Least-Squares Collocation (LSC), which is a
technique that combines heterogeneous data types to optimally estimate gravimetric
quantities and their errors. DMA has applied LSC using the Forsberg Covariance Model
(Forsberg, 1987) to estimate the 30' mean gravity anomalies directly using the PGA file.
The Forsberg Covariance Model contains simple, closed formulas for quantities related
to the Earth's anomalous potential using a planar approximation. The power spectral
decay of the self-consistent Forsberg model closely approximates Kaula's rule and three
parameters (D, T, CO) characterize the correlation and power of gravity anomalies in a
local area. The three parameters are defined as: D is the high frequency attenuation factor,
T is the low frequency attenuation factor, and CO is the variability of the gravity field.
The computational process at DMA is to select the most accurate gravity data at
appropriate spacing from the PGA file and then reduce the anomaly data for the effects of
terrain (high frequency effects), if necessary, and long wavelength effects. After these
reductions, analytical covariance functions are closely fitted to empirical functions based
on the three Forsberg Model parameters. The local covariance parameters (D, T, CO) are
then used in a LSC algorithm that utilizes the Forsberg closed expressions for gravimetric
quantities, integral formulas for the mean representation of the gravimetric quantities, and
Cholesky decomposition to efficiently and accurately calculate the mean gravity
anomalies from available PGA data in a specified cell.
There are two techniques to estimate 30' free-air gravity anomalies. Section 1 below
describes the Bouguer anomaly methodology and Section 2 describes the use of free-air
anomalies in the computations. Greenland and the coastlines of all continental areas were
computed from Molodensky free-air gravity anomalies as defined in equation (1). The
free-air gravity anomaly estimation technique is used along the coastlines to incorporate
all the ship borne free-air anomalies in the water. For all interior continental areas and
islands Bouguer anomalies were used in the computations. The Bouguer anomalies are
regionally correlated with elevation and change much smoother with position compared
to free-air gravity anomalies. In high mountain areas the Bouguer anomaly can easily be
highly negative by hundreds of mgals. Since the Bouguer anomaly provides a much
smoother anomaly it provides excellent input to the estimation process of LSC. The main
difference in the two methods to compute mean anomalies relates to the terrain reductions
performed.
The use of an accurate long-wavelength geopotential spherical harmonic model is
critical to the proper reduction of the free-air and Bouguer anomalies. For this project, the
use of the JGM-2 (to degree 70) model (Nerem et al., 1994) augmented by the OSU-91A
(degree 71 to 360) model was selected as the most accurate geopotential model currently
available.
1. Methodology for 30' Mean Free-Air Anomaly Computation/rom Point Bouguer
Gravity Anomalies
To perform LSC (Moritz, 1980) using Bouguer anomalies, the following formulas are
used to predict the 30' mean anomalies and their associated errors:
85
The optimum calculation of 30' mean free-air gravity anomalies would be based on a
infinite number of point gravity anomalies in a specified cell. This definition cannot be
realized because the density and distribution of point gravity anomalies varies by
geographic region and elevation over the Earth's surface. DMA has a worldwide
requirement to acquire point or mean gravity information at required spacing but many
regions still need additional coverage or densification. The computation of 30' mean
free-air anomalies by DMA is based on Least-Squares Collocation (LSC), which is a
technique that combines heterogeneous data types to optimally estimate gravimetric
quantities and their errors. DMA has applied LSC using the Forsberg Covariance Model
(Forsberg, 1987) to estimate the 30' mean gravity anomalies directly using the PGA file.
The Forsberg Covariance Model contains simple, closed formulas for quantities related
to the Earth's anomalous potential using a planar approximation. The power spectral
decay of the self-consistent Forsberg model closely approximates Kaula's rule and three
parameters (D, T, CO) characterize the correlation and power of gravity anomalies in a
local area. The three parameters are defined as: D is the high frequency attenuation factor,
T is the low frequency attenuation factor, and CO is the variability of the gravity field.
The computational process at DMA is to select the most accurate gravity data at
appropriate spacing from the PGA file and then reduce the anomaly data for the effects of
terrain (high frequency effects), if necessary, and long wavelength effects. After these
reductions, analytical covariance functions are closely fitted to empirical functions based
on the three Forsberg Model parameters. The local covariance parameters (D, T, CO) are
then used in a LSC algorithm that utilizes the Forsberg closed expressions for gravimetric
quantities, integral formulas for the mean representation of the gravimetric quantities, and
Cholesky decomposition to efficiently and accurately calculate the mean gravity
anomalies from available PGA data in a specified cell.
There are two techniques to estimate 30' free-air gravity anomalies. Section 1 below
describes the Bouguer anomaly methodology and Section 2 describes the use of free-air
anomalies in the computations. Greenland and the coastlines of all continental areas were
computed from Molodensky free-air gravity anomalies as defined in equation (1). The
free-air gravity anomaly estimation technique is used along the coastlines to incorporate
all the ship borne free-air anomalies in the water. For all interior continental areas and
islands Bouguer anomalies were used in the computations. The Bouguer anomalies are
regionally correlated with elevation and change much smoother with position compared
to free-air gravity anomalies. In high mountain areas the Bouguer anomaly can easily be
highly negative by hundreds of mgals. Since the Bouguer anomaly provides a much
smoother anomaly it provides excellent input to the estimation process of LSC. The main
difference in the two methods to compute mean anomalies relates to the terrain reductions
performed.
The use of an accurate long-wavelength geopotential spherical harmonic model is
critical to the proper reduction of the free-air and Bouguer anomalies. For this project, the
use of the JGM-2 (to degree 70) model (Nerem et al., 1994) augmented by the OSU-91A
(degree 71 to 360) model was selected as the most accurate geopotential model currently
available.
1. Methodology for 30' Mean Free-Air Anomaly Computation/rom Point Bouguer
Gravity Anomalies
To perform LSC (Moritz, 1980) using Bouguer anomalies, the following formulas are
used to predict the 30' mean anomalies and their associated errors:
85