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Seismic surface waves present a characteristic dispersion behavior in which different frequencies travel at different speeds. This characteristic dispersion curve depends on the shear and compressional wave velocity structure of the crust and upper mantle. These physical parameters can be estimated using some inversion method that can be 1D, 2D, or even 3D. 1D inversions are very common for their relative simplicity and speed. However, the dispersion is usually measured between two stations.

So my question is, where do we "place" the resulting 1D estimated model? Is it the average of the crust between the stations (I did the math for this once long ago and it seems that it is not, but I'm not sure)? Or is there no way to place the estimate at a single point?

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This is a very good question, not just important to seismic inversion, but also modeling in general.

Lets set this problem up differently. Lets say point's A and D are nodes.

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Each node represents a system of equations, and these equations are only calculated on these points. Therefore, the model can only exist on the points in which they are calculated. Since B and C are dimensionally between A and D, a smoothing factor must be imposed to graphically represent them. So while there might be a line in between A and D, the model only exists on points A and D. To find the model at point B and C, you would have to build them in your model, creating a higher resolution mesh. This is perhaps why it is so important to have a dense seismic array when doing seismic studies.

So in your case, the model is dependent on information on information at two stations, making each node TWO STATIONS. This would mean that the it is impossible to give an exact geographic location to where the inversion is, just a generalized idea of between these two stations.

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