Efficient Inversion of Multi-frequency and Multi-source Electromagnetic Data: Final report

BES grant DE-FG02-06ER15819 supported efforts at Oregon State University (OSU) to develop improved inversion methods for 3D subsurface electromagnetic (EM) imaging. Three interrelated activities have been supported by this grant, and its predecessor (DE-FG02-06ER15818): (1) collaboration with a former student of the PI, Dr. Weerachai Siripunvaraporn (now Professor at Mahidol University in Bangkok, Thailand) on developing and refining inversion methods for 3D Magnetotelluric (MT) data . (2) Development at Oregon State University of a new modular system of computer codes for EM inversion, and initial testing and application of this inversion on several large field data sets. (3) Research on more efficient approaches to multi-transmitter EM inverse problems, to optimize use of expensive data sensitivity calculations needed for gradient based inversion schemes. The last of these activities was the main motivation for this research project, but the first two activities were important enabling steps that produced useful products and results in their own right, including freely avaialable software for 3D inversion of EM geophysical data

Efficient Inversion of Mult-frequency and Multi-Source Electromagnetic Data

The project covered by this report focused on development of efficient but robust non-linear inversion algorithms for electromagnetic induction data, in particular for data collected with multiple receivers, and multiple transmitters, a situation extremely common in eophysical EM subsurface imaging methods. A key observation is that for such multi-transmitter problems each step in commonly used linearized iterative limited memory search schemes such as conjugate gradients (CG) requires solution of forward and adjoint EM problems for each of the N frequencies or sources, essentially generating data sensitivities for an N dimensional data-subspace. These multiple sensitivities allow a good approximation to the full Jacobian of the data mapping to be built up in many fewer search steps than would be required by application of textbook optimization methods, which take no account of the multiplicity of forward problems that must be solved for each search step. We have applied this idea to a develop a hybrid inversion scheme that combines features of the iterative limited memory type methods with a Newton-type approach using a partial calculation of the Jacobian. Initial tests on 2D problems show that the new approach produces results essentially identical to a Newton type Occam minimum structure inversion, while running more rapidly than an iterative (fixed regularization parameter) CG style inversion. Memory requirements, while greater than for something like CG, are modest enough that even in 3D the scheme should allow 3D inverse problems to be solved on a common desktop PC, at least for modest (~ 100 sites, 15-20 frequencies) data sets. A secondary focus of the research has been development of a modular system for EM inversion, using an object oriented approach. This system has proven useful for more rapid prototyping of inversion algorithms, in particular allowing initial development and testing to be conducted with two-dimensional example problems, before approaching more computationally cumbersome three-dimensional problems

Hybrid conjugate gradient-Occam algorithms for inversion of multifrequency and multitransmitter EM data

We describe novel hybrid algorithms for inversion of electromagnetic geophysical data, combining the computational and storage efficiency of a conjugate gradient approach with an Occam scheme for regularization and step-length control. The basic algorithm is based on the observation that iterative solution of the symmetric (Gauss-Newton) normal equations with conjugate gradients effectively generates a sequence of sensitivities for different linear combinations of the data, allowing construction of the Jacobian for a projection of the original full data space. The Occam scheme can then be applied to this projected problem, with the tradeoff parameter chosen by assessing fit to the full data set. For EM geophysical problems with multiple transmitters (either multiple frequencies or source geometries) an extension of the basic hybrid algorithm is possible. In this case multiple forward and adjoint solutions (one each for each transmitter) are required for each step in the iterative normal equation solver, and each corresponds to the sensitivity for a separate linear combination of data. From the perspective of the hybrid approach, with conjugate gradients generating an approximation to the full Jacobian, it is advantageous to save all of the component sensitivities, and use these to solve the projected problem in a larger subspace. We illustrate the algorithms on a simple problem, 2-D magnetotelluric inversion, using synthetic data. Both the basic and modified hybrid schemes produce essentially the same result as an Occam inversion based on a full calculation of the Jacobian, and the modified scheme requires significantly fewer steps (relative to the basic hybrid scheme) to converge to an adequate solution to the normal equations. The algorithms are expected to be useful primarily for 3-D inverse problems for which the computational burden is heavily dominated by solution to the forward and adjoint problems.This is the publisher’s final pdf. The published article is copyrighted by the author and published by John Wiley & Sons, Inc. for the Royal Astronomical Society and can be found at: http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1365-246X/.Keywords: Magnetotelluric, Inverse theory, Geomagnetic inductionKeywords: Magnetotelluric, Inverse theory, Geomagnetic inductio

Sampling bias in VGP longitudes

I derive probability densities for virtual geomagnetic pole( VGP) longitudes for a general statistical model of local magnetic field variations. I show that even for very simple statistically homogeneous models of secular variation, the distribution of VGP longitudes is peaked 90° away from the sampling longitude. Thus, when sites are distributed unevenly, a non-uniform overall distribution of VGP longitudies to be expected. Analysis of the recent geomagnetic field indicates that this bias can be significant, particularly when random errors in paleomagnetic data are allowed for. It is possible that some of the deviations from uniformity seen in recent compilations of paleomagnetic reversal and secular variation data are a statistical artifact resulting from the distortion of the VGP transformation, and the non- uniformity of paleomagnetic sampling sites

The flux of tidal energy across latitude 60°S

How and where the ocean tides dissipate their energy are longstanding questions with both oceanographic and astronomical implications. Two decades ago, Doake suggested that flexing of Antarctic ice shelves by the underlying ocean tide is an important energy sink, perhaps accounting for over half the global dissipation rate. Observational constraints on Antarctic dissipation have been scarce. Here two new and complementary ocean-tide models, both derived from Topex/Poseidon satellite altimeter measurements are used to determine the flux of tidal energy across 60øS toward the Antarctic coastline. Our results show relatively small fluxes and they therefore rule out Doake's suggestion: Antarctica is an insignificant sink in the global tidal energy budget.Copyrighted by American Geophysical Union

Estimates of M₂ tidal energy dissipation from TOPEX/Poseidon altimeter data

Most of the tidal energy dissipation in the ocean occurs in shallow seas, as had long been recognized. However, recent work has suggested that a significant fraction of the dissipation, perhaps 1 TW or more, occurs in the deep ocean. This paper builds further evidence for that conclusion. More than 6 years of data from the TOPEX/Poseidon satellite altimeter are used to map the tidal dissipation rate throughout the world ocean. The dissipation rate is estimated as a balance between the rate of working by tidal forces and the energy flux divergence, computed using currents derived by least squares fitting of the altimeter data and the shallow water equations. Such calculations require dynamical assumptions, in particular about the nature of dissipation. To assess sensitivity of dissipation estimates to input assumptions, a large suite of tidal inversions based on a wide range of drag parameterizations and employing both real and synthetic altimeter data are compared. These experiments and Monte Carlo error fields from a generalized inverse model are used to establish error uncertainties for the dissipation estimates. Owing to the tight constraints on tidal elevation fields provide by the altimeter, area integrals of the energy balance are remarkably insensitive to required dynamical estimates. Tidal energy dissipation is estimated for all major shallow seas (excluding individual polar seas) and compared with previous model and data-based estimates. Dissipation in the open ocean is significantly enhanced around major bathymetric features, in a manner consistent with simple theories for the generation of baroclinic tides.Copyrighted by American Geophysical Union

Time-Variable Refraction of the Internal Tide at the Hawaiian Ridge

The interaction of the dominant semidiurnal M₂ internal tide with the large-scale subtidal flow is examined in an ocean model by propagating the tide through an ensemble of background fields in a domain centered on the Hawaiian Ridge. The background fields are taken from the Simple Ocean Data Assimilation (SODA) ocean analysis, at 2-month intervals from 1992 through 2001. Tides are computed with the Primitive Equation Z-coordinate Harmonic Analysis of Tides (PEZ-HAT) model by 14-day integrations using SODA initial conditions and M₂ tidal forcing. Variability of the tide is found to occur primarily as the result of propagation through the nonstationary background fields, rather than via generation site variability. Generation of incoherent tidal variability is mapped and shown to occur mostly in association with waves generated at French Frigate Shoals scattering near the Musicians Seamounts to the north of the ridge. The phase-coherent internal tide loses energy at a domain-average rate of 2mWm⁻² by scattering into the non-stationary tide. Because of the interference of waves from multiple generation sites, variability of the internal tide is spatially inhomogeneous and values of the scattering rate 10 times larger occur in localized areas. It is estimated that 20% of the baroclinic tidal energy flux is lost by adiabatic scattering (refraction) within 250 km of the ridge, a value regarded as a lower bound because of the smoothed nature of the SODA fields used in this study

Empirical orthogonal function analysis of magnetic observatory data : further evidence for non-axisymmetric magnetospheric sources for satellite induction studies

Although satellite electromagnetic induction studies have usually assumed a symmetric magnetospheric ring current source, there is growing evidence for significant source asymmetry. Here we apply empirical orthogonal function methods to mid-latitude night-side hourly mean geomagnetic observatory data to search for evidence of nonzonal low-frequency source fields. The dominant spatial mode of variability in residuals, obtained by subtracting symmetric ring current and ionospheric fields of the CM4 comprehensive model, has a substantial Y₂¯¹ quadrupole component and is highly correlated with Dst. This pattern of temporal variability, which implies enhanced ring current densities in the dusk sector, persists even when peak storm-time data are omitted. The observed asymmetry agrees with that inferred previously by Balasis et al. (2004), from the local time dependence of biases in satellite induction transfer functions. Temporal correlation of the leading mode with Dst, and consistency of its spatial structure with recent empirical ring current models, suggest a magnetospheric origin

Semi-diurnal and diurnal tidal dissipation from TOPEX/Poseidon altimetry

Tidal energy dissipation is estimated for eight semi‐diurnal and diurnal constituents using a global inverse solution constrained by TOPEX/Poseidon altimeter data. Very similar spatial patterns are obtained for all semi‐diurnal constituents, with about one third of the total dissipation occurring in the deep ocean over rough topography. Maps for diurnal constituents are also similar amongst themselves, but quite different from the semi‐diurnal results. For diurnals a smaller fraction of dissipation, roughly 10%, occurs in the deep ocean. Much of the difference can be explained by the very different spatial pattern of diurnal and semi‐diurnal tidal currents. The lack of free internal waves at frequencies poleward of 30° at diurnal frequencies also probably plays a role, limiting the effectiveness of baroclinic conversion as an energy sink for barotropic diurnal tides