Airborne gravimetry: An Introduction Madjid ABBASI Surveying Engineering Department, Zanjan University, Zanjan, Iran National Cartographic Center (NCC) 28/09/1385

2 Principle of airborne gravimetry

3 Why airborne gravimetry? Inaccessible regions Medium wavelengths Quasi-regular measurements Cost and time

4 Instrumentation I. GPS multi-receiver configuration II. Gravimetric sensor Strapdown Inertial Navigation System (INS) Gravimeter+stabilized platform III. Radar altimeter IV. INS

A new approach for data reduction in airborne LaCoste & Romberg gravimetry M. ABBASI, J.P.Barriot (1), J. Verdun and H. Duquenne (2) (1) Bureau Gravimétrique International, Toulouse, France (2) Ecole Nationale des Sciences Géographiques, Paris, France Observatoire Midi-Pyrénées

6 L&R Air/Sea gravimetric system Two axis stabilized platform + Gravity sensor unit

7 B Centre of mass of the sensor Proof mass Beam Spring

8 Mathematical modelling Fundamental equation of mobile gravimetry : Gravity sensor unit Inertial system Geocenter Frame of the gravimeter Northwards

9 Differential equation of the gravimeter This equation relies the gravimetric measurements ( S, B and CC ) and the accelerations experienced by the gravimeter. The only unknown in this equation is

10 The differentiation is not a bounded operator The frequency content of each term is different Inconvenients of direct computation is a low frequency signal Power spectrum of spring tensionPower spectrum of cross-coupling effect Line number Frequency (Hz) 200 s 20 s

11 Low pass filtering we know the filtering cut-off frequency just approximately the filter is applied separately to each acquisition line … 10 s 200 s

12 Our new approach: integral equation of the gravimeter We transform the differential equation into an integral equation:

13 Fredholm integral equation of the first kind Intrinsically ill-posed problem Regularization W

14 The least squares solution x ……. the gravity disturbance vector for all the profiles Q x the a priori covariance matrix of the unknowns for all the profiles  regularization matrix Q y ….. the covariance matrix related to the observations Regularization parameter For each profile:

15 Choice of the regularization parameter Helmert method The ‘L’ curve method Both methods give almost the same results.

16 The a priori covariance matrix of the gravity disturbances Correlogram Positive definite function Using the existing geopotential models: Isotropic covariance function

17 Numerical results of the airborne gravimetric project over the Alps Number of data …………………… Mean altitude ……………………… 5200 m Number of North-South lines …..…. 22 Number of East-West lines ……….. 16 Mean distance between the lines … 9.6 km

18 Exponential filter (resolution 8.4km) std = 25 mGal std = 16 mGal Histogramme des écarts aux nœuds avant l’ajustement Histogram of the crossover discrepancies Gravity disturbance obtained from the direct filtering

19 Correlations between the gravity disturbances are taken into account via the regularization matrix. Integral equation method Histogramme des écarts aux nœuds avant l’ajustement std = 3 mGal std = 2.7 mGal Gravity disturbance obtained from the new method Histogram of the crossover discrepancies

20 Standard deviation of the estimated gravity disturbances The standard deviations decrease around the crossover points The a posteriori covariance matrix of the estimated gravity disturbances: and increase near the borders. A criterion for the «precision» of the results

21 Comparison of the results Result of the integral equation method Result of the exponential filter. CHAMBERY. CHAMBERY. CHAMONIX. CHAMONIX

22 Correlation coeffiecient: 0.71

23 Correlation coeffiecient: 0.76

24 Profile of the topography beneath the line L01a The great part of the gravity disturbances at the flight height (particularly short wavelenghts) is due to the topography Profile of the topography beneath the line L08a The integral equation method provides a more detailed gravity field.

25 Future work: validation of the results These results need to be validated. The validation can be done by the upward continuation of the ground based gravity data to the flight height.

26 Conclusion The method based on the integral equation of the gravimeter is a promising method for a better cartography of the gravity field from the airborne data This method provides also the covariance matrix of the estimated gravity disturbances as a precision criterion It is possible to introduce additional constraint equations; for example for crossover points It is possible also to take into account other types of data: ground based data, …

27 Thank you very much

28 Gravity anomaly upward continued to the flight height Perturbation de gravité estimée par différentes méthodes poor coverage of the anomalies specially in the Eastern part of the region upward continuation method: we compared the gravity disturbances with the gravity anomalies! Difference of gravity disturbances with upward continued gravity anomalies Upward continue the ground based gravity anomalies to obtain the corresponding gravity disturbances at the flight height Future work