Bootstrapping Nonlinear Least Squares Estimates in the Kalman Filter Model.

1986 ◽  
Author(s):  
David S. Stoffer
Keyword(s):  
Kalman Filter ◽  
Least Squares ◽  
Nonlinear Least Squares ◽  
Filter Model ◽  
Least Squares Estimates

A LEAST‐SQUARES METHOD OF INTERPRETING MAGNETIC ANOMALIES CAUSED BY TWO‐DIMENSIONAL STRUCTURES

Geophysics ◽  
1969 ◽  
Vol 34 (1) ◽  
pp. 65-74 ◽  
Author(s):  
William W. Johnson
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Magnetic Anomalies ◽  
Descent Method ◽  
The Body ◽  
Steepest Descent Method ◽  
Two Dimensional ◽  
Gauss Method ◽  
Least Squares Estimates

The equations relating the magnetic anomalies to the shape and susceptibility of a body are nonlinear with respect to the coordinates describing the shape. Therefore, iterative procedures must be used to obtain least‐squares estimates of the body coordinates. One method in general use for obtaining nonlinear least‐squares estimates is the Gauss method. This method often fails when the initial values for the structures and susceptibilities do not adequately account for the magnetic anomalies. Another method known as the steepest descent method generally converges to a solution; however, a large number of iterations are required. A method suggested by Marquardt (1963) incorporates the best features of the previous methods. In this paper the Marquardt method is applied to the interpretation of magnetic anomalies. For this purpose the two‐dimensional formulas derived by Talwani and Heirtzler (1964) are used to relate the geometry of a body to the resulting magnetic anomalies. The procedure efficiently controls the amount of change made to an interpreted structure at each iteration, assuring rapid convergence to a solution which satisfies the observed data better in the least‐squares sense than does the initial solution. The method is applied to representative problems.

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