Crustal constraint through complete model space screening for diverse geophysical datasets facilitated by emulation

Deep crustal constraint is often carried out using deterministic inverse methods, sometimes using seismic refraction, gravity and electromagnetic datasets in a complementary or “joint” scheme. With increasingly powerful parallel computer systems it is now possible to apply joint inversion schemes to derive an optimum model from diverse input data. These methods are highly effective where the uncertainty in the system is small. However, given the complex nature of these schemes it is often difficult to discern the uniqueness of the output model given the noise in the data, and the application of necessary regularization and weighting in the inversion process means that the extent of user prejudice pertaining to the final result may be unclear. We can rigorously address the subject of uncertainty using standard statistical tools but these methods also become less feasible if the prior model space is large or the forward simulations are computationally expensive. We present a simple Monte Carlo scheme to screen model space in a fully joint fashion, in which we replace the forward simulation with a fast and uncertainty-calibrated mathematical function, or emulator. This emulator is used as a proxy to run the very large number of models necessary to fully explore the plausible model space. We develop the method using a simple synthetic dataset then demonstrate its use on a joint data set comprising first-arrival seismic refraction, MT and scalar gravity data over a diapiric salt body. This study demonstrates both the value of a forward Monte Carlo approach (as distinct from a search-based or conventional inverse approach) in incorporating all kinds of uncertainty in the modelling process, exploring the entire model space, and shows the potential value of applying emulator technology throughout geophysics. Though the target here is relatively shallow, the methodology can be readily extended to address the whole crust

Cube law, condition factor and weight-length relationships: history, meta-analysis and recommendations

This study presents a historical review, a meta-analysis, and recommendations for users about weight–length relationships, condition factors and relative weight equations. The historical review traces the developments of the respective concepts. The meta-analysis explores 3929 weight–length relationships of the type W = aLb for 1773 species of fishes. It shows that 82% of the variance in a plot of log a over b can be explained by allometric versus isometric growth patterns and by different body shapes of the respective species. Across species median b = 3.03 is significantly larger than 3.0, thus indicating a tendency towards slightly positive-allometric growth (increase in relative body thickness or plumpness) in most fishes. The expected range of 2.5 < b < 3.5 is confirmed. Mean estimates of b outside this range are often based on only one or two weight–length relationships per species. However, true cases of strong allometric growth do exist and three examples are given. Within species, a plot of log a vs b can be used to detect outliers in weight–length relationships. An equation to calculate mean condition factors from weight–length relationships is given as Kmean = 100aLb−3. Relative weight Wrm = 100W/(amLbm) can be used for comparing the condition of individuals across populations, where am is the geometric mean of a and bm is the mean of b across all available weight–length relationships for a given species. Twelve recommendations for proper use and presentation of weight–length relationships, condition factors and relative weight are given

Verification of velocity-resistivity relationships derived from structural joint inversion with borehole data

We present results of three-dimensional joint inversion of seismic, magnetotelluric, and gravity data over a marine salt dome. Such structures are difficult to image with a single method, and our results demonstrate how combining different techniques can yield improved results. More importantly, we examine the reliability of velocity-conductivity relationships derived from structure-coupled joint inversion approaches. Comparison with a seismic reflection section shows that our models match the upper limit of the salt. Furthermore, velocity and resistivity logs from a borehole drilled into the salt dome's flank match, within error, those recovered by the inversion. The good match suggests that the difference in length scale does not have a significant effect in this case. This provides a strong incentive to incorporate borehole data into the joint inversion in the future and substantiates approaches that use the relationships derived from joint inversion models for lithological classification

A framework for 3-D joint inversion of MT, gravity and seismic refraction data

We present a 3-D joint inversion framework for seismic, magnetotelluric (MT) and scalar and tensorial gravity data. Using large-scale optimization methods, parallel forward solvers and a flexible implementation in terms of model parametrization allows us to investigate different coupling approaches for the various physical parameters involved in the joint inversion. Here we compare two different coupling approaches, direct parameter coupling where we calculate conductivities and densities from seismic slownesses and cross-gradient coupling, where each model cell has an independent value for each physical property and a structural similarity is enforced through a term in the objective function. For both types of approaches we see an improvement of the inversion results over single inversions when the inverted data sets are generated from compatible models. As expected the direct coupling approach results in a stronger interaction between the data sets and in this case better results compared to the cross-gradient coupling. In contrast, when the inverted MT data is generated from a model that violates the parameter relationship in some regions but conforms with the cross-gradient assumptions, we obtain good results with the cross-gradient approach, while the direct coupling approach results in spurious features. This makes the cross-gradient approach the first choice for regions were a direct relationship between the physical parameters is unclear