Strict bounds on seismic velocity in the spherical earth

Stark, PB, Parker RL, Masters G, Orcutt JA.  1986.  Strict bounds on seismic velocity in the spherical earth. Journal of Geophysical Research-Solid Earth and Planets. 91:13892-13902.

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We address the inverse problem of finding the smallest envelope containing all velocity profiles consistent with a finite set of imprecise τ(p) data from a spherical earth. Traditionally, the problem has been attacked after mapping the data relations into relations on an equivalent flat earth. Of the two contemporary direct methods for finding bounds on velocities in the flat earth consistent with uncertain τ(p) data, a nonlinear (NL) approach descended from the Herglotz-Wiechert inversion and a linear programming (LP) approach, only NL has been used to solve the spherical earth problem. On the basis of the finite collection of τ(p) measurements alone, NL produces an envelope that is too narrow: there are numerous physically acceptable models that satisfy the data and violate the NL bounds, primarily because the NL method requires continuous functions as bounds on τ(p) and thus data must be fabricated between measured values by some sort of interpolation. We use the alternative LP approach, which does not require interpolation, to place optimal bounds on the velocity in the core. The resulting velocity corridor is disappointingly wide, and we therefore seek reasonable physical assumptions about the earth to reduce the range of permissible models. We argue from thermodynamic relations that P wave velocity decreases with distance from the earth's center within the outer core and quite probably within the inner core and lower mantle. We also show that the second derivative of velocity with respect to radius is probably not positive in the core. The first radial derivative constraint is readily incorporated into LP. The second derivative constraint is nonlinear and cannot be implemented exactly with LP; however, geometrical arguments enable us to apply a weak form of the constraint without any additional computation. LP inversions of core τ(p) data using the first radial derivative constraint give new, extremely tight bounds on the P wave velocity in the core. The weak second derivative constraint improves them slightly.