Abstract. Nonlinear inverse problems usually have no analytical solution and may be solved by Monte Carlo methods that create a set of samples, representative of the a posterJori distribution. We show how neural networks can be trained on these samples to give a continuous approximation to the inverse relation in a compact and computationally efficient form. We examine the strengths and weaknesses of this approach and use it to determine the full a posterJori distribution of crustal thickness from surface wave velocities. The solution to this inverse problem shows significant asymmetry and large uncertainties due to trade-off with shear velocity structure around the Moho. We produce maps of maximum likelihood crustal thickness across Eurasia which are in agreement with current knowledge about the crust; thus we provide an independent confirmation of these models. In this application, characterized by repeated inversion of similar data, the neural network algorithm proves to be very efficient.

S U M M A R YThe possibility that parts of the Earth's continental lower crust can be described with stochastic geological models has been suggested for some time. Recent studies of deeper well logs also indicate a possible stochastic structure at mid-crustal levels. This motivates a closer examination of the relation between the statistics of reflection wavefields and that of the lower crust. Such a relation can put important constraints on possible lower crustal models. This study follows up earlier efforts to quantify the statistics of both stochastic lower crustal models and the reflected wavefield. Since modelling of the seismic response of stochastic (von Karman) fields implies the usage of the impedance contrast field of the latter, we wish to compare the second-order statistics of both types of fields (velocity and impedance). This study concludes that the vertical derivative operator on a von Karman velocity field, implicitly present in the impedance contrast field, alters the second-order horizontal von Karman statistics of the velocity fields in a profound way. Wavefield effects, undoubtedly present in observed seismic data, which have earlier been proposed as possible causes for the aforementioned change, seem to play a secondary role. The vertical derivative operation, inherent in the impedance contrast field, reduces the estimated horizontal scale length and Hurst number by a factor of 2-22 and 1-3, respectively. Original vertical scale length and Hurst number of the velocity fields have a (quasi-)linear influence on this underestimation. Horizontal scale lengths and Hurst numbers were also estimated from the seismic response (Primary Reflectivity Section) of the von Karman fields. The values obtained are close to those obtained from the causative impedance contrast fields, and are similarly underestimated. This suggests a dominant role for the vertical derivative operator in the underestimation of horizontal scale length and Hurst number. This attempt to quantify the relation between the horizontal spatial statistics of von Karman fields and the estimates derived from their seismic response, may be useful in upscaling the latter.

In an area of complicated structure a stacked record section is likely to be characterized by a low signal‐to‐noise ratio, even after substantial velocity analysis and other processing. The interpreter identifies signals showing phase coherence across many traces at physically allowable velocities and compiles them into a line drawing. We have developed a nonlinear filter designed to mimic this process, which passes only signals showing spatial coherence and having slowness within an allowed range. In this algorithm, called the “SSD filter,” overlapping M-trace windows are converted into a p-τ representation, obtained by multiplying the stack along the relevant slant line by the smoothed semblance. The results from all windows are composited in the p-τ domain, then retransformed into x-t. The principal tunable parameter is the width M of the correlation window, adjusted to provide an output which agrees well with the event picks made by an experienced interpreter on a test panel of data. The method was developed to enable production of cleaned‐up sections from the often noisy stacks produced in deep crustal seismic studies. An example from the COCORP Southern Appalachian profile illustrates how removal of incoherent noise from the stacked section substantially enhances the quality of the migrated section.

No abstract

No abstract