SPATIAL UNDERSTANDING OF MATRIX INVERSION FOR INVERSE FORCE ESTIMATION

A basic approach to study a NVH problem is to break down the system in three basic elements – source, path and receiver. While the receiver (response) and the transfer path can be measured, it is difficult to measure the source (forces) acting on the system. It becomes necessary to predict these forces to know how they influence the responses. This requires inverting the transfer path. Singular Value Decomposition (SVD) method is used to decompose the transfer path matrix into its principle components which is required for the inversion. The usual approach to force prediction requires rejecting the small singular values obtained during SVD by setting a threshold, as these small values dominate the inverse matrix. This assumption of the threshold may be subjected to rejecting important singular values severely affecting force prediction. The new approach discussed in this report looks at the column space of the transfer path matrix which is the basis for the predicted response. The response participation is an indication of how the small singular values influence the force participation. The ability to accurately reconstruct the response vector is important to establish a confidence in force vector prediction. The goal of this report is to suggest a solution that is mathematically feasible, physically meaningful, and numerically more efficient through examples. This understanding adds new insight to the effects of current code and how to apply algorithms and understanding to new codes.

EVALUATION OF THE EVOLVING STRESS FIELD OF THE YELLOWSTONE VOLCANIC PLATEAU FROM 1988 TO 2010 FROM EARTHQUAKE FIRST MOTIONS INVERSION

Within the Yellowstone National Park, Wyoming, the silicic Yellowstone volcanic field is one of the most active volcanic systems all over the world. Although the last rhyolite eruption occurred around 70,000 years ago, Yellowstone is still believed to be volcanically active, due to high hydrothermal and seismic activity. The earthquake data used in this study cover the period of time between 1988 and 2010. Earthquake relocations and a set of 369 well-constrained, double-couple, focal mechanism solutions were computed. Events were grouped according to location and time to investigate trends in faulting. The majority of the events has oblique, normal-faulting solutions. The overall direction of extension throughout the 0.64 Ma Yellowstone caldera looks nearly ENE, consistently with the direction of alignments of volcanic vents within the caldera, but detailed study revealed spatial and temporal variations. Stress-field solutions for different areas and time periods were calculated from earthquake focal mechanism inversion. A well-resolved rotation of σ3 was found, from NNE-SSW near the Hebgen Lake fault zone, to ENE-WSW near Norris Junction. In particular, the σ3 direction changed throughout the years in the Norris Junction area, from being ENE-WSW, as calculated in the study by Waite and Smith (2004), to NNE-SSW, while the other σ3 directions are mostly unchanged over time. The Yellowstone caldera was subject to periods of net uplift and subsidence over the past century, explained in previous studies as caused by expanding or contracting sills, at different depths. Based on the models used to explain these deformation periods, we investigated the relationship between variability in aseismic deformation and seismic activity and faulting styles. Focal mechanisms and P and T axes were divided into temporal and depth intervals, in order to identify spatial or temporal trends in deformation. The presence of “chocolate tablet” structures, with composite dilational faults, was identified in many stages of the deformation history both in the Norris Geyser Basin area and inside the caldera. Strike-slip component movement was found in a depth interval below a contracting sill, indicating the movement of magma towards the caldera.

AMPLITUDE-VERSUS-ANGLE ANALYSIS AND WIDE-ANGLE-INVERSION OF CROSSWELL SEISMIC DATA IN A CARBONATE RESERVOIR

Crosswell data set contains a range of angles limited only by the geometry of the source and receiver configuration, the separation of the boreholes and the depth to the target. However, the wide angles reflections present in crosswell imaging result in amplitude-versus-angle (AVA) features not usually observed in surface data. These features include reflections from angles that are near critical and beyond critical for many of the interfaces; some of these reflections are visible only for a small range of angles, presumably near their critical angle. High-resolution crosswell seismic surveys were conducted over a Silurian (Niagaran) reef at two fields in northern Michigan, Springdale and Coldspring. The Springdale wells extended to much greater depths than the reef, and imaging was conducted from above and from beneath the reef. Combining the results from images obtained from above with those from beneath provides additional information, by exhibiting ranges of angles that are different for the two images, especially for reflectors at shallow depths, and second, by providing additional constraints on the solutions for Zoeppritz equations. Inversion of seismic data for impedance has become a standard part of the workflow for quantitative reservoir characterization. Inversion of crosswell data using either deterministic or geostatistical methods can lead to poor results with phase change beyond the critical angle, however, the simultaneous pre-stack inversion of partial angle stacks may be best conducted with restrictions to angles less than critical. Deterministic inversion is designed to yield only a single model of elastic properties (best-fit), while the geostatistical inversion produces multiple models (realizations) of elastic properties, lithology and reservoir properties. Geostatistical inversion produces results with far more detail than deterministic inversion. The magnitude of difference in details between both types of inversion becomes increasingly pronounced for thinner reservoirs, particularly those beyond the vertical resolution of the seismic. For any interface imaged from above and from beneath, the results AVA characters must result from identical contrasts in elastic properties in the two sets of images, albeit in reverse order. An inversion approach to handle both datasets simultaneously, at pre-critical angles, is demonstrated in this work. The main exploration problem for carbonate reefs is determining the porosity distribution. Images of elastic properties, obtained from deterministic and geostatistical simultaneous inversion of a high-resolution crosswell seismic survey were used to obtain the internal structure and reservoir properties (porosity) of Niagaran Michigan reef. The images obtained are the best of any Niagaran pinnacle reef to date.

VALIDATION OF A 'DISPLACEMENT TOMOGRAPHY' INVERSION METHOD FOR MODELING SHEET INTRUSIONS

The study of volcano deformation data can provide information on magma processes and help assess the potential for future eruptions. In employing inverse deformation modeling on these data, we attempt to characterize the geometry, location and volume/pressure change of a deformation source. Techniques currently used to model sheet intrusions (e.g., dikes and sills) often require significant a priori assumptions about source geometry and can require testing a large number of parameters. Moreover, surface deformations are a non-linear function of the source geometry and location. This requires the use of Monte Carlo inversion techniques which leads to long computation times. Recently, ‘displacement tomography’ models have been used to characterize magma reservoirs by inverting source deformation data for volume changes using a grid of point sources in the subsurface. The computations involved in these models are less intensive as no assumptions are made on the source geometry and location, and the relationship between the point sources and the surface deformation is linear. In this project, seeking a less computationally intensive technique for fracture sources, we tested if this displacement tomography method for reservoirs could be used for sheet intrusions. We began by simulating the opening of three synthetic dikes of known geometry and location using an established deformation model for fracture sources. We then sought to reproduce the displacements and volume changes undergone by the fractures using the sources employed in the tomography methodology. Results of this validation indicate the volumetric point sources are not appropriate for locating fracture sources, however they may provide useful qualitative information on volume changes occurring in the surrounding rock, and therefore indirectly indicate the source location.