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Parker, RL.  1974.  Best bounds on density and depth from gravity data. Geophysics. 39:644-649.   10.1190/1.1440454   AbstractWebsite

Gravity data cannot usually be inverted to yield unique structures from incomplete data; however, there is a smallest density compatible with the data or, if the density is known, a deepest depth of burial. A general theory is derived which gives the greatest lower bound on density or the least upper bound on depth. These bounds are discovered by consideration of a class of “ideal” bodies which achieve the extreme values of depth or density. The theory is illustrated with several examples which are solved by analytic methods. New maximum depth rules derived by this theory are, unlike some earlier rules of this type, optimal for the data they treat.

Stark, PB, Parker RL.  1995.  Bounded-variable least-squares: an algorithm and applications. Computational Statistics. 10:129-141. AbstractWebsite

The Fortran subroutine BVLS (bounded variable least-squares) solves linear least-squares problems with upper and lower bounds on the variables, using an active set strategy. The unconstrained least-squares problems for each candidate set of free variables are solved using the QR decomposition. BVLS has a ''warm-start'' feature permitting some of the variables to be initialized at their upper or lower bounds, which speeds the solution of a sequence of related problems. Such sequences of problems arise, for example, when BVLS is used to find bounds on linear functionals of a model constrained to satisfy, in an approximate l(p)-norm sense, a set of linear equality constraints in addition to upper and lower bounds. We show how to use BVLS to solve that problem when p = 1, 2, or infinity, and to solve minimum l(1) and l(infinity) fitting problems. FORTRAN 77 code implementing BVLS is available from the statlib gopher at Carnegie Mellon University.

Huestis, SP, Parker RL.  1977.  Bounding the thickness of the oceanic magnetized layer. Journal of Geophysical Research. 82:5293-5303.   10.1029/JB082i033p05293   AbstractWebsite

We present a theory for placing a lower bound on the thickness of the oceanic magnetized layer using magnetic anomaly observations and estimates of the intensity of magnetization; the theory makes only a minimum number of assumptions regarding the spatial distribution of the magnetization. The principle of the method is based upon the fact that thin layers imply high magnetizations. We show how to calculate the source distribution that has minimum intensity yet fits the data and is confined to a given thickness layer; because the minimum intensity must be a monotonically decreasing function of layer thickness, it follows that an upper bound on the intensity allows us to obtain a lower limit on the thickness. The practical calculations are performed by using linear programing. The method is applied to two sets of near-bottom magnetic profiles, one on the Galápagos Spreading Center at 86°W and the other set on the Pacific-Antarctic Ridge at 51°S. In the first area we conclude that the magnetic layer must be at least 450 m thick, and in the other a crossing of the Jaramillo event indicates that the magnetic layer is probably more than 1000 m in thickness.