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Our main research areas are seismic modeling, wave equation migration velocity analysis (WEMVA) and full waveform inversion (FWI). Below we give short introductions to our main research areas; seismic modeling, wave equation migration velocity analysis (WEMVA) and full waveform inversion (FWI).

Seismic modeling is the process of modeling (simulating) how seismic waves propagate through the earth using a computer. The results from the simulations are used to understand how seismic waves behave and propagate and a typical practical application is designing and optimising new seismic surveys. Also seismic modelling is an integral part of seismic software used to obtain images and mechanical parameters of the subsurface. In order to actually perform seismic modelling, software based on a modeling algorithm is needed in addition to computer hardware.

The starting point for a modeling algorithm is mathematical equations which describe seismic waves. Two standard equations are the acoustic wave equation (AWE) and the elastic wave equation (EWE). The major difference between these two equations is that the latter includes shear waves in addition to pressure waves which the former equation describes. In that sense, the latter equation is physically more correct since it includes more true physics. On the other hand, the EWE is more complex than the AWE, and thus needs more computer resources.

The mathematical equation must be translated into a “language” which the computer understand. There exist several useful methods to do this, where the two most popular maybe are the finite difference method and the finite element method. In addition to a wave equation understandable by a computer, a model of the earth subsurface is also needed in the form of mechanical parameters. These parameters are used as input data to the modelling software.

It is impossible to simulate wave propagation phenomena which are completely identical to the real ones observed in nature. This fact is due to the principle that when performing modelling several approximations are necessary; i.e the mathematical equations involved in the modeling are not able to describe all physical phenomena occurring in a real wave propagation. In addition, when transforming the equation to computer language several approximations are used. However, the most important wave phenomena are described by the equation and the software, and are not too far away from the one observed in nature.

Consider a two-dimensional medium with a box which is placed in the middle of the model, see the image below. The density, P- and S-wave velocities are used as physical model in the simulation. Below is an image of the P-wave velocities. The topology of the density and S-wave velocity are the same as the P-wave velocity.

Now, a seismic wave is starting to propagate from the left in the model. The wave propagation at different times after the start are given in the images below.

The ultimate objective of seismic surveys is to compute mechanical parameters like bulk modulus, shear modulus and density of the subsurface using seismic data observed at the surface. If we imagine that the subsurface is divided into small regular cells of size, say, 10 meters in all three directions, the seismic inverse problem then consists of estimating an average mechanical quantity for each cell. Usually seismic surveys covers an area of several thousand square kilometers to a depth of about 5-10 kilometer, implying that a very large number of unknown parameters need to be estimated.

Wave equation migration velocity analysis is one of the approaches used to estimate primarily the bulk modulus, or equivalently the wave velocity for pressure-waves. The main idea for performing this is to create an image of the subsurface by focusing the reflected seismic waves. The focusing is entirely artificial and performed by software on a computer system. It turns out that the sharpness of the image depends on the wave velocity. By systematically changing the wave velocity until a sharp image is found, one obtains an estimate of the correct wave velocity. Focusing is mainly sensitive to the average of the seismic velocities over many cells, which implies that the estimated velocities are a smoothed version of the true velocity as can be seen on the example below.

Below is the result of estimating the velocity model from synthetic data using Wave equation Migration Velocity Analysis (WEMVA) based on reverse-time migration (right). The true velocity model is displayed on the left.

The image below shows the result of Full-Waveform Inversion (FWI) using the WEMVA velocity model as an initial guess.

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