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

Document Type : Research

Author

Assistant Professor, Faculty of Civil Engineering, Shahrood University of Technology, Shahrood, Iran

Abstract

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.

Keywords

Main Subjects


به‌نبیان، ب. و مشهدی حسینعلی، م.، 1398، استفاده از مدل کوواریانس ناهمسانگرد به‌منظور محاسبه تغییر شکل پوسته با استفاده از کولوکیشن کمترین‌مربعات، مطالعه موردی: شبه‌جزیره کنای. نشریه علمی علوم و فنون نقشه برداری، 6(4)، 143–159.
گلی، م.، 1398، بررسی تراکم ایستگاه‌‌های شبکه چندمنظوره ژئودزی سازمان نقشه‌برداری در تعیین ژئوئید: مطالعه موردی منطقه شمال-غرب کشور. نشریه علمی پژوهشی علوم و فنون نقشه‌برداری. ۸ (۴)، ۳۱-۳۹.
Ardalan, A. A., 1999, High Resolution Regional Geoid Computation in the World Geodetic Datum 2000, based up on collocation of linearized observational functionals of the type GPS, gravity potential and gravity intensity, Ph.D. thesis, University of Stuttgart.
Alberts, B. and Klees, R., 2004, A comparison of methods for the inversion of airborne gravity data. Journal of Geodesy, 78, 1, 55–65.
Barzaghi, R., Borghi, B., Keller, K., Forsberg, R., Giori, I., Lorreti, F., Olsen, A.V. and Srenseng, L., 2009, Airborne gravity tests in the Italian area to improve the geoid model of Italy, Geophysical Prospecting, 57(4), 625-632.
Cooper, G., 2004, The stable downward continuation of potential field data. Exploration Geophysics, 35, 4, 260–265.
Darbeheshti, N., 2009, Modification of the Least-Squares Collocation Method for Non-Stationary Gravity Field Modelling. Curtin University of Technology. PhD thesis.
Fedi, M. and Florio, G., 2002, A stable downward continuation by using the ISVD method. Geophysical Journal International, 151, 1, 146–156.
Forsberg, R., 1987, A new covariance model for inertial gravimetry and gradiometry. Journal of Geophysical Research: Solid Earth, 92(B2), 1305–1310.
Goli, M., Foroughi, I. and Novak, P., 2018, On estimation of stopping criteria for iterative solutions of gravity downward continuation. Canadian Journal of Earth Sciences.
Goli, M., Foroughi, I. and Novák, P., 2019, The effect of the noise, spatial distribution, and interpolation of ground gravity data on uncertainties of estimated geoidal heights. Studia Geophysica et Geodaetica, 63(1), 35–54.
Goli, M. and Najafi-Alamdari, M., 2011, Planar, spherical and ellipsoidal approximations of Poisson’s integral in near zone, Journal of Geodetic Science, 17-24.
Goli, M., Najafi-Alamdari, M. and Vaníček, P., 2011, Numerical behaviour of the downward continuation of gravity anomalies. Studia Geophysica et Geodaetica, 55, 191–202.
Hirt, C., Bucha, B., Yang, M. and Kuhn, M., 2019, A numerical study of residual terrain modelling (RTM) techniques and the harmonic correction using ultra-high-degree spectral gravity modelling. Journal of Geodesy, 93(9), 1469–1486.
Hofmann-Wellenhof, B. and Moritz, H., 2006, Physical geodesy. Springer Science and Business Media.
Hsiao, Y. S. and Hwang, C., 2010, Topography-assisted downward continuation of airborne gravity: An application for geoid determination in Taiwan. Terrestrial, Atmospheric and Oceanic Sciences., 21, 627-637.
Hwang, C., Hsiao, Y.-S., Shih, H.-C., Yang, M., Chen, K.-H., Forsberg, R. and Olesen, A. V., 2007, Geodetic and geophysical results from a Taiwan airborne gravity survey: Data reduction and accuracy assessment. Journal of Geophysical Research: Solid Earth, 112(B4).
Martinec, Z., 1996, Stability investigations of a discrete downward continuation problem for geoid determination in the Canadian Rocky Mountains. Journal of Geodesy, 70, 805–828.
Martinec, Z. and Grafarend, E. W., 1997, Construction of Green’s function to the external Dirichlet boundary-value problem for the Laplace equation on an ellipsoid of revolution. Journal of Geodesy, 71, 562–570.
Novák, P. and Heck, B., 2002, Downward continuation and geoid determination based on band-limited airborne gravity data. Journal of Geodesy, 76, 269–278.
Pilkington, M. and Boulanger, O., 2017, Potential field continuation between arbitrary surfaces—Comparing methods. Geophysics, 82(3), J9–J25.
Rexer, M., Hirt, C. and Pail, R., 2017, High-resolution global forward modelling: a degree-5480 global ellipsoidal topographic potential model. EGU General Assembly Conference Vienna, Austria.
Saleh, J., Li, X., Wang, Y. M., Roman, D. R. and Smith, D. A., 2013, Error analysis of the NGS’ surface gravity database. Journal of Geodesy, 87(3), 203-221.
Saadat, A., Safari, A. and Needell, D., 2018, IRG2016: RBF-based regional geoid model of Iran. Studia Geophysica et Geodaetica, 1–28.
Sansò, F., 2013, The local modelling of the gravity field by collocation. In: Sansò F, Sideris MG (eds) Geoid determination: theory and methods. Springer, Heidelberg.
Tscherning, C. C. and Rapp, R. H., 1974, Closed covariance expressions for gravity anomalies, geoid undulations, and deflections of the vertical implied by anomaly degree variance models. Report 208, department of geodetic sciences, Ohio State University.
Tziavos, I. N., Andritsanos, V. D., Forsberg, R. and Olesen, A. V., 2005, Numerical investigation of downward continuation methods for airborne gravity data, in Gravity, Geoid and Space Missions, C. Jekeli, L. Bastos, and J. Fernandes (eds.); 119–124.
Wang, Y. M., Roman, D. R. and Saleh, J., 2008, Analytical Downward and Upward Continuation Based on the Method of Domain Decomposition and Local Functions, in VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy (P. Xu, J. Liu, and A. Dermanis (eds.); 356–360.
Wang, Y. M., Li, X., Ahlgren, K. and Krcmaric, J., 2020, Colorado geoid modeling at the US National Geodetic Survey. Journal of Geodesy, 94,10, 106.
Willberg, M., Zingerle, P. and Pail, R., 2019, Residual least-squares collocation: use of covariance matrices from high-resolution global geopotential models. Journal of Geodesy, 93(9), 1739-1757.
Xu, S., Yang, J., Yang, C., Xiao, P., Chen, S. and Guo, Z., 2007, The iteration method for downward continuation of a potential field from a horizontal plane. Geophysical Prospecting, 55, 6, 883–889.
Vaníček, P., Sun, W., Ong, P., Martinec, Z., Najafi, M., Vajda, P. and Ter Horst, B., 1996, Downward continuation of Helmert’s gravity. Journal of Geodesy, 71, 21–34.
Zhang, C., Lü, Q., Yan, J. and Qi, G., 2018, Numerical Solutions of the Mean-Value Theorem: New Methods for Downward Continuation of Potential Fields. Geophysical Research Letters, 45, 8, 3461–3470.
Zhao, Q., Xu, X., Forsberg, R. and Strykowski, G., 2018, Improvement of Downward Continuation Values of Airborne Gravity Data in Taiwan. In Remote Sensing 10, 12.