A new mathematical model for Airborne LaCoste-Romberg Gravimetry

1. A new mathematical model for Airborne LaCoste-RombergGravimetry M. ABBASI , J.P. BARRIOT , J. VERDUN and H. DUQUENNE Bureau Gravimétrique International (BGI), UMR 5562, 14, Av. E. Belin, Observatoire Midi-Pyrénées, 31400, Toulouse, France madjid.abbasi@obs-mip.fr , Jean-Pierre.Barriot@cnes.fr Ecole Nationale des Sciences Géographiques, 77455 Marne la Vallée Cedex 2, France verdun@ensg.ign.fr , Henri.Duquenne@ensg.ign.fr 1 1 2 2 Observatoire Midi-Pyrénées 1 2 Abstract 5 Least squares solution 3 Differential equation of the beam motion The problem of separation of the perturbing accelerations of the airplane and the gravity signal at flight height is the main problem of airborne gravimetry. The method generally used is the frequency domain filtering of the measured signal. This is based on the assumption that the airplane-induced accelerations have a high frequency nature, whereas the gravity acceleration is a low frequency signal. The cut-off frequency is usually not higher than 0.005 Hz. This corresponds to 12 km at a mean speed of 60 m/s, meaning that the gravity accelerations with shorter wavelength are overfiltered. Such a low frequency makes the filter design extremely delicate as well. We have developed a new filtering method based on integral equation modeling and least squares estimation. By taking into account the covariance matrices of the observations and the a priori unknowns, as well as the mathematical formulation of the gravimeter as a spring-damper system, we obtain results with an improved content on the high frequency band. The method is applied to the data acquired over the French Alps. Introducing the prior information and leads to the biased least squares estimate given by The accelerations sensed by the sensor cause the variations of the beam position (B(t)) and the spring tension (S(t)). The gravimeter measures them as well as a cross coupling effect (CC(t)). A second order differential equation relates all these accelerations and measurements. (5) (2) where and are the cofactor matrices and and are two unknown variance factors. Since the matrix A is a square matrix we cannot use the popular ‘degrees of freedom’ method for evaluating the variance factors. One of the alternatives is the Helmert method, which is what we used. The a priori covariance matrix of the unknowns is constructed from the existing geopotential models like EGM96. For the matrix the covariance propagation law is applied to the individual terms in the r.h.s of Eqn. (3). This requires some precautions because the covariance matrices depend directly on the estimated accuracy of different observations, notably on the GPS derived coordinates of the flight trajectory. The covariance matrix of the estimated parameters is estimated with the following relation: • ….………. initial spring tension (at the airport) • ………….. absolute gravity at the airport • ……..….. vertical acceleration of the airplane • ……..…… Eötvös acceleration • ………….. vertical acceleration compensated by the platform/dampers system • ………….. gravity disturbance • ………….. normal gravity • .…..….. instrument constants 1 Lacoste-Romberg air/sea gravimeter • The only unknown in this equation is the gravity g(t). So, one can compute each term of the equation separately to obtain g(t). But this is not the best solution because: • the spectral contents of the different terms are situated at different frequencies, • differentiation from the time series is needed for some terms. This is an improperly posed problem which causes the random errors to be amplified. • The method generally used for this problem is the low pass filtering of all the terms in Eqn. (2). Since the cut-off frequency is very small (in the order of 0.005 Hz), filter design becomes very difficult and hence some distortions are unavoidable. On the other hand, this kind of filtering cuts off all the frequency content of the gravity with wavelengths smaller than say 7 Km. (6) The Lacoste-Romberg air/sea gravimeter is a highly-damped spring-type gravimeter which is stabilized by a platform. The air-shock dampers absorb the high frequency vertical shocks of the plane. This means that the vertical acceleration sensed by the plane is not the same as the one sensed by the sensor. The plat-form remains the gravimeter’s base plate (more or less) in the level surface. The zero-length spring and the highly-damped measurement system cause the extreme sensitivity of the sensor. So it becomes also very sensitive to systematic effects. This gives a direct estimate of the accuracy of the gravity disturbances derived through airborne gravimetry measurements. 6 Results and conclusion This method was applied to real data acquired during an airborne gravimetry project over the French Alps in 1998. Vertical acceleration Eötvös acceleration Gravity acceleration Beam position (B(t)) Spring tension (S(t)) Cross coupling effect (CC(t)) M 4 Corresponding integral equation We propose to transform the differential equation (2) to the following integral equation Fig. 1 2 The gravimeter’s transfer function The principle that [2] uses for studying the high frequency vibrations of the gravimetric system, is the same as we used for our mathematical modeling. By taking the gravimetric system as two spring-damper systems, one for the sensor and one for the plat-form/dampers, we find a coupled system of second order differential equations. Each equation possesses a transfer function of the form (3) (1) in which is the frequency, is the damping factor and is the natural frequency of the spring/damper system (Fig. 2). These parameters are different for each system. • Since the direct filtering is applied over the individual flight lines, we can distinguish them in the final result. This problem is less apparent in our new method. • Filtering parameters depend sometimes on the upward continued ground based data. With this kind of manipulation, airborne gravimetry becomes a data densification method. The new method uses only the data derived during the flight campaign so that the ground based data can serve as an independent control. • The traditional accuracy criteria comes from the cross-over analysis. The discrepancies in these points are a function of the filtering degree. So, it cannot be an internal criterion for the accuracy of airborne gravimetry. This is a Fredholm integral equation of the first kind. Green's function W is a non-symmetric and split-type kernel (its first derivative with respect to is not continuous). The Fredholm integral equations of the first kind are intrinsically ill-posed. Even supposing a solution exists, a slight perturbation of f(t) may give rise to an arbitrary large variation in the solution. There are two principal methods to solve this type of problem: singular value decomposition and regularization. We use the second method. After discretisation, Eqn. (3) takes the following form: (4) Fig. 2 Fig. 3 where x is the unknown vector containing ‘s, fis the vector of the discretized right hand side f(t) in Eqn. (3) and comes from the observational errors and the discretization error. A is a n by n square matrix in which n depends on the number of observations and also on the preselectioned discretization step. It seems that smaller steps lead to more accurate numerical results, but at the same time to an increase of the observational errors. So, one of the tasks in the numerical evaluation of these integrals is to find a compromise between the variance and the bias to minimize the total (mean square) error. It must be mentioned that any gap or irregularity in the sampling sequence causes the computing procedure to be more complicated and adds errors to the vector . [2] gives the transfer functions for high frequency vibrations but for the low frequency effects it remains to be determined in laboratories. [1] determined the transfer function of the sensor system using the gravimeter’s measurements. (Fig. 2). But for the plat-form/dampers system we used an empirical natural frequency and damping factor. Our tests with real data show that this natural frequency must be very small; in the order of 0.004 Hz. 7 References [1] Verdun, J. R. Bayer, E.E. Klingelé, Airborne gravimetry measurments over montainous areas by using a Lacoste & Romberg air-sea gravimeter, In GEOPHYSICS, VOL. 67, No. 3, P. 807-816, May-June 2002. [2] Aliod, D. E. Mann, M. Holliday, Lcoste & Romberge Air-Sea Gravity Meter Vibration Tests, In http://www.lacosteromberg.com/techdocsfr.htm, March, 2003.