NS31C-1578
Statistical Parameter Clustering for Inverse Problems
Many geophysical and other inverse problems are ill-posed, meaning that they yield nonunique solutions, either because there are not enough data to constrain the model parameters or because the analytic model function is itself noninvertible. The key issue with such problems is that they have nontrivial null spaces, making necessary the use of constraint/regularization techniques that in effect pare down or eliminate null spaces, thereby forcing well-posedness. As Portniaguine (1999) and others have observed for some time, the major issue with many constraint/regularization schemes (e.g., Tikhonov Regularization, Damped Least Squares, etc.) is that they restrict the range of variability of the solution. Large parameter contrasts and sharp spatial gradients are essentially penalized, often forcing solutions to be blurred or distorted with respect to the true geological distribution of material properties. In this study, a novel constraint method called Statistical Parameter Clustering (SPC) is developed and demonstrated. This constraint technique attempts to avoid restrictions on model-parameter contrasts and spatial gradients. SPC achieves this end by invoking Occam's Razor, which advises that the ideal description of a phenomenon is that which uses the fewest possible assumptions. In this context, the �phenomena' are observed data and the �assumptions' are model parameters. In other words, the best solution to an inverse problem is assumed to be the one that uses the fewest parameters to predict the data. Statistical Parameter Clustering enforces this assumption by constraining the inverse problem to seek out models whose parameters are statistically tightly multimodal. The smaller the variance is about each �mode' of the parameter distribution, the more the solution obeys Occam's Razor. To actualize this constraint, a two-step methodology is adopted: at each iteration of a nonlinear inversion, the current model is clustered using a suitable clustering algorithm; second, the clustering assignments are used to create a constraint functional that measures the sum of the variances of parameters about the means of their respective clusters. This cluster-variance functional is then simultaneously minimized with a least squares objective, effecting the so-called Statistical Parameter Clustering constraint. A synthetic and a real-world example are offered to demonstrate and elucidate this inversion constraint method.
NS31C-1579
Difference full waveform inversion for time-lapse cross-hole radar imaging
Cross borehole radar is widely used not only for geological structure investigation but also for hydrology, soil science, and environmental science to make use of the advantage in resolution in space and in sensitivity for water content and solute in soil. Recently, time-lapse crosshole measurements are conducted for monitoring soil water dynamics or tracer movement in the subsurface. We propose an algorithm of difference waveform inversion for time-lapse cross-hole imaging for monitoring subsurface dynamical phenomena like soil water flow in vadose zone, solute movement as tracer in groundwater, or the occurrence of change in soil property due to piping, etc. Full waveform inversion in time domain originally utilizes the gradient method for updating an assumed initial model. Our approach is to employ this method to interpret differences in waveforms of time-lapse radar measurements. This approach has the following advantages: 1. No need to pick the exact travel time and wave amplitude 2. High spatial resolution and high sensitivity to small fluctuation in physical or chemical property of soil 3. The ability to estimate for the change in permittivity and conductivity of soil may be improved.
NS31C-1580
Identifying sources of crosshole radar tomography uncertainty and its impact on the resulting images
Data and model uncertainty in crosshole radar measurements tend to corrupt the final tomograms and physical property estimation. These uncertainties can arise due to errors in the data acquisition and the inability to model the physics adequately. Specifically the sources and magnitudes of errors in crosshole field data are investigated and quantified. The affects of these uncertainties are demonstrated through a series of data error distribution plots. Although these crosshole data uncertainties, such as timing and positioning errors, have been described previously, this study aims to quantify thoroughly and display these using a comprehensive series of plots. These plots may be used a diagnostic check in the data processing flow or incorporated as errors in the tomogram inversion. In addition, errors due to inaccurate physics have a significant effect in the imaging problem. The second contribution will focus on using first order scattering information in the data to image the velocity anomalies. Both synthetic and field examples will be used to demonstrate the validity of our approach.
NS31C-1581
Deterministic approaches to understanding image uncertainty: Point spread functions and region of data influence
Geophysical inversion is inherently ill-posed and non-unique. Regularization of ill-posed geophysical inverse problems can lead to a stable solution, but often complicates the interpretation of the solution because of the bias introduced by prior information. If the end goal of our data analysis involves decisions based on our solution, it is of great importance that we quantify how much information came from data and how much came from prior information. For linear inverse problems this procedure is well established; but for nonlinear problems the procedure is far more complicated. Here we compare two different approaches applicable to nonlinear problems: region of data influence (RDI) index and a resolution spread computed using point spread functions (PSF). The RDI method is a fully nonlinear approach, while the PSF analysis is a linearized approach. We outline the mathematical relationships between the two approaches and then demonstrate the use of both approaches on synthetic and field data examples. From a practical point of view, if two different approaches indicate similar interpretation on post-inversion images the confidence in the interpretation is enhanced. Using these example problems, we demonstrate the additional insights gained and how this analysis can improve our interpretations, ultimately helping us to make better informed decisions.
NS31C-1582
Resolution of GPS data from the 2004 Mw6.0 Parkfield Earthquake
The long-awaited 2004 Mw6.0 Parkfield Earthquake provides a unique opportunity to probe the resolution limits of source inversions due to the large amount of near-field seismic stations. This earthquake was well recorded by a dense network of strong-motion seismographs and GPS 1-Hz receivers. We investigate the spatial resolution of the GPS data, which provides a constraint on the static field. Even for a well-recorded earthquake such as Parkfield, static GPS inversions are poorly resolved at depth and near the edges of the fault. We demonstrate how in underdetermined inversions such as this, it is possible to obtain structure in poorly resolved areas that is not real. Furthermore, bootstrapping is unlikely to give correct error bounds in these regions. As such, much of the structure shown in GPS inversions of Parkfield is highly uncertain.
NS31C-1583
Constraining 2.5D Inversion of Electrical Resistivity Tomography (ERT) Field Data With Resistivity Cone Penetrometry (RCPT) Data
We investigate the potential for using resistivity cone penetrometry (RCPT) to constrain 2.5D electrical resistivity tomography (ERT) inversion. ERT data, consisting of several parallel 2D lines, RCPT and EM-31 data were collected at a site on the East Yorkshire coast, U.K. The site is well-characterised, with sand/gravel bodies in clay till. The ERT data were collated and a 2.5D inversion was carried out with a homogenous reference model. The final model did not adequately characterise the sand/gravel bodies; both underestimating their resistivity and overestimating their size and depth. In previous work, we used reference models generated from the RCPT and EM-31 data in inversions of the 2D ERT lines and showed that the least squares misfit between the final model and the reference model used in that inversion could be used as a proxy for the misfit of the final model to the true geoelectrical structure. Using the misfit, we correctly identified the best final models and final models that were worse than that produced by a blind inversion (using a homogenous reference model). We now extend this approach into 2.5D by using 3D reference models in 2.5D inversions of the ERT data. A method for constructing 3D geoelectrical reference models based on sparse RCPT and EM-31 data was developed for this study. We discuss the implications for the characterisation of sites in 2.5D and compare the 2D and 2.5D approaches.