Comparison of least squares collocation and Poisson's integral methods in downward continuation of airborne gravity data

Terrestrial gravimetry in large countries such as Iran with mountainous areas is time consuming and costly. Airborne gravimetry can be used to fill the data gravity gaps. Airborne gravity data are contaminated with different kinds of systematic and random errors that should be evaluated before use. In this study, the downward continued airborne gravity data is compared with existing terrestrial gravity data for detecting probable biases and measurement error. For this purpose, the efficiencies of the two least squares colocation and Poisson's integral methods are compared.
Collocation is an optimal linear prediction method in which the base functions are directly related to the covariance functions. The covariance function can be derived from empirical covariance fitting. This method can be utilized for downward continuation (DWC) of gravity data with arbitrary distribution. Often the homogeneous and isotropic covariance functions are used in collocation. However, in reality the statistical parameters of gravity data change with location and azimuth. This is the main drawback of collocation with stationary covariance function. Based on the Dirichlet’s boundary values problem for harmonic functions, the downward continuation of airborne gravity data from the flight altitude to the geoid/ellipsoid surface is given by inverse of Poisson’s integral. Similar to collocation, this method can be utilized for DWC of gravity data with arbitrary distribution. Poisson’s integral as inverse problem is unstable in continuous form. However, for discrete data, the instability depends of the amplitude of high frequency components in the gravity observation such as error measurements.
Numerical computations for this study were performed in the Colorado region and northern parts of New Mexico that is bounded by . In this region, 524,381 airborne data are available in 106 flight lines. The along track sampling is 1 Hz (about 128 meters) and the cross distance between lines is about 10 km. To reduce the edge effect, the final test area is reduced to  which includes 5494 ground gravity points. To improve the efficiency of the computations, the sampling interval is decreased to  Hz (about 2 km).
We first demonstrate the applications of the DWC methods using simulated gravity data. Short wavelength of gravity disturbance related to degree 360-2190, was generated using experimental global gravity model 'refB' at the two true positions of airborne and ground data. Two (white) noise 1 and 2 mGal was added to airborne data. Using these simulated observations, the two aforementioned methods were employed to determine the terrestrial disturbances. The comparison of computed and simulated terrestrial disturbances show that the accuracy of the Poisson method for both noise levels is about 30% better than the collocation.
For real data, the residual gravity data is computed by subtracting the long wavelengths up to degree 360 and corresponding residual topographical effect (RTM) from the real gravity observation. RTM is derived from the harmonic model (dV_ELL_Earth2014_5480) of spherical harmonic degrees between 360-5480. This model provides spherical harmonics of gravitational potential of upper crust. According to previous studies, the level noise of airborne gravity of Colorado is about 2.0 mGal. By introducing this noise into collocation, the problem becomes stable. In Poisson method, the iterative 'lsqr' method is used to solve the system of linear equations. To achieve stable solution, the iterations was terminate using discrepancy principal rule.
The residual anomaly gravity at Earth's surface can be computed directly using collocation. But in the Poisson method, computation is performed in two steps: 1 the airborne gravity disturbances are downward continued to a  grid on the reference ellipsoid, 2- the terrestrial gravity disturbance is computed by upward continuation from ellipsoid disturbances. Despite of simulated data, the accuracy of the two methods is the same in terms of standard deviation of the differences. The mean and the standard values of difference is about 2mGal and 8mGal, respectively. According to a study by Saleh et al. (2013), the bias of in parts of Colorado reaches more than 2mGal. Therefore, due to the bias of terrestrial data, the estimated bias in airborne data cannot be confirmed.