Modified Gravity and Cosmology

A computationally efficient relocatable system for generalized inverse (GI) modeling of barotropic ocean tides is described. The GI penalty functional is minimized using a representer method, which requires repeated solution of the forward and adjoint linearized shallow water equations (SWEs). To make representer computations efficient, the SWEs are solved in the frequency domain by factoring the coefficient matrix for a finite-difference discretization of the second-order wave equation in elevation. Once this matrix is factored representers can be calculated rapidly. By retaining the first-order SWE system (defined in terms of both elevations and currents) in the definition of the discretized GI penalty functional, complete generality in the choice of dynamical error covariances is retained. This allows rational assumptions about errors in the SWE, with soft momentum balance constraints (e.g., to account for inaccurate parameterization of dissipation), but holds mass conservation constraints. Wh...

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Efficient Inverse Modeling of Barotropic Ocean Tides

Two models, a simple cooling model and the plate model, have been advanced to account for the variation in depth and heat flow with increasing age of the ocean floor. The simple cooling model predicts a linear relation between depth and t½, and heat flow and 1/t½, where t is the age of the ocean floor. We show that the same t½ dependence is implicit in the solutions for the plate model for sufficiently young ocean floor. For larger ages these relations break down, and depth and heat flow decay exponentially to constant values. The two forms of the solution are developed to provide a simple method of inverting the data to give the model parameters. The empirical depth versus age relation for the North Pacific and North Atlantic has been extended out to 160 m.y. B.P. The depth initially increases as t½, but between 60 and 80 m.y. B.P. the variation of depth with age departs from this simple relation. For older ocean floor the depth decays exponentially with age toward a constant asymptotic value. Such characteristics would be produced by a thermal structure close to that of the plate model. Inverting the data gives a plate thickness of 125±10 km, a bottom boundary temperature of 1350°±275°C, and a thermal expansion coefficient of (3.2±1.1) × 10−5°C−1. Between 0 and 70 m.y. B.P. the depth can be represented by the relation d(t) = 2500 + 350t½ m, with t in m.y. B.P., and for regions older than 20 m.y. B.P. by the relation d(t) = 6400 - 3200 exp (−t/62.8) m. The heat flow data were treated in a similar, but less extensive manner. Although the data are compatible with the same model that accounts for the topography, their scatter prevents their use in the same quantitative fashion. Our analysis shows that the heat flow only responds to the bottom boundary at approximately twice the age at which the depth does. Within the scatter of the data, from 0 to 120 m.y. B.P., the heat flow pan be represented by the relation q(t) = 11.3/t½ μcal cm−2s−1. The previously accepted view that the heat flow observations approach a constant asymptotic value in the old ocean basins needs to be tested more stringently. The above results imply that a mechanism is required to supply heat at the base of the plate.

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An analysis of the variation of ocean floor bathymetry and heat flow with age

The inversion of electromagnetic sounding data does not yield a unique solution, but inevitably a single model to interpret the observations is sought. We recommend that this model be as simple, or smooth, as possible, in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models.To obtain smooth models, the nonlinear forward problem is linearized about a starting model in the usual way, but it is then solved explicitly for the desired model rather than for a model correction. By parameterizing the model in terms of its first or second derivative with depth, the minimum norm solution yields the smoothest possible model.Rather than fitting the experimental data as well as possible (which maximizes the roughness of the model), the smoothest model which fits the data to within an expected tolerance is sought. A practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models, resulting in a stable and rapidly convergent algorithm. The inversion of both magnetotelluric and Schlumberger sounding field data, and a joint magnetotelluric-resistivity inversion, demonstrate the method and show it to have practical application.

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Occam's inversion; a practical algorithm for generating smooth models from electromagnetic sounding data

J+=j

Summary It is shown how a series of Fourier transforms can be used to calculate the magnetic or gravitational anomaly caused by an uneven, non-uniform layer of material. Modern methods for finding Fourier transforms numerically are very fast and make this approach attractive in situations where large quantities of observations are available.

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The Rapid Calculation of Potential Anomalies

PrefaceCh. 1Mathematical PrecursorCh. 2Linear Problems with Exact DataCh. 3Linear Problems with Uncertain DataCh. 4Resolution and InferenceCh. 5Nonlinear ProblemsAppendix A: The Dilogarithm FunctionAppendix B: Table for 1-norm MisfitsReferencesIndex

Geophysical Inverse Theory

Individual aseismic areas move as rigid plates on the surface of a sphere. Application of the Mercator projection to slip vectors shows that the paving stone theory of world tectonics is correct and applies to about a quarter of the Earth's surface.

The North Pacific: an Example of Tectonics on a Sphere

Much of our knowledge of the Earth's interior is perforce based on the interpretation of measurements made at the surface, rather than direct sampling of the material in the interior. In the past few years there have been great advances in the mathematical aspects of this problem, and the topic has come to be called geo­ physical inverse theory. To apply these ideas, there must be a valid mathematical model of the physics of the system under study, so that one would be able to calculate the values of observations made on an exactly known structure: the calculation of the behavior of a specified system is the solution of the "forward" or "direct" problem. Frequently it is the forward problem that presents a difficult challenge to the theoretical geophysicist. Illustrations include the mechanism of earthquake rupture or the generation of the Earth's magnetic field; in problems like these, inverse theory is normally quite inappropriate. When the forward problem has been completely solved, there are of course unknown parameters in the mathematical model representing physical properties of the Earth such as Lame parameters, density or electrical conductivity. The goal of inverse theory is to determine the parameters from the observations or, in the face of the inevitable limitations of actual measurement, to find out as much as possible about them. The quality that distinguishes inverse theory from the parameter estimation problem of statistics (Bard 1974, Rao 1973) is that the unknowns are junctions, not merely a handful of real numbers. This' means that the solution contains in principle an infinite number of variables, and therefore with real data the problem is as under­ determined as it can be. Naturally, there are geophysical problems containing a relatively small number of free parameters: for example, in describing the relative instantaneous motion of N lithospheric plates, we find that the assumption of internal rigidity reduces the number of unknowns to 3N 3 for the N 1 relative angular velocity vectors (McKenzie & Parker 1974). Sometimes, however, unknown structures are conceived in terms of small numbers of homogeneous layers for reasons of computational simplicity rather than on any convincing geophysical or

Robert L. Parker

Papers

Occam's inversion; a practical algorithm for generating smooth models from electromagnetic sounding data

The Rapid Calculation of Potential Anomalies

Geophysical Inverse Theory

The North Pacific: an Example of Tectonics on a Sphere

Understanding Inverse Theory