Delineation of Basement Surface Relief... 35 JKAU: Earth Sci., vol. 16, pp: 39-49 (2005 A.D. / 1426 A.H.) 35 Delineation of Basement Surface Relief from its Magnetic Anomaly Using Hybrid Numerical Algorithm MOHAMED GOBASHY 1 and MAHA ABD EL-AZEEM 2 1 Cairo University, Faculty of Science, Dept. of Geophysics, Giza, and 2 National Research Institute of Astronomy and Geophsyics, Helwan, Egypt Received: 15/11/2003 Revised: 39/10/2004 Accepted: 18/12/2004 ABSTRACT. Magnetic field mapping is a very powerful geophysical tool for recognizing basement relief. In the present work, the problem of finding the depth to the basement surface is transformed into the problem of minimizing a constraint objective function ψ (z j ) of the L 1 norm instead of the traditional L 2 in the unknown depths (z j ). The as- sumed subsurface model is consisting of a number of vertical prisms, homogeneously magnetized and of fixed magnetic susceptibility κ. The Finite Difference-Quasi Newton (FD-Newton) algorithm with ac- tive set strategy (ASS) is used as a hybrid technique to minimize the constraint objective function ψ (z j ). Both linear and nonlinear bound constraints are implemented as a priori information to reduce ambigu- ity and improve the solution. For simplicity, the problem can be solved as an even determined; however, it is directly solved as an over-determined with little modification The method is found to be stable, robust and convergent even for large size models and when im- plementing inherent noise in the synthetic examples. Further, the present method is applied to a field example from the gulf of Suez, Egypt. The calculated depths show a good agreement with results of drilled wells in that area. Introduction Parameter estimation is a common target in processing geophysical data both in linear and non-linear problems. The formulation of such problems may be clas- sified into two main categories. The first category of formulation is dealing with the source geometry, while fixing its physical property (density, magnetic sus-