Rescaling of incorrect source strength using Marchenko Redatuming

Abstract

The iterative Marchenko scheme is a recent development in the field of Geophysics and a way to retrieve Green's functions at any point in the subsurface, called a focal point. To achieve this, only the reflection response measured at the Earth's surface and an estimation of the first arrival at the focal point are required. If the amplitude of the reflection response is scaled incorrectly, the method will suffer from artifacts in the estimation of the Green's function. The amplitudes of the reflection response are unknown when the source strength of the recording is unknown. To correct for source strength, a correction factor is used. The correction factor can be retrieved by using a function that has its minimum at the required correction factor, a so-called cost function. Additionally a scaling factor can be determined, which is used to ensure that the final Green's function has the correct amplitudes. Three cost functions are proposed. The first cost function minimizes the upgoing Green's function and only works if no reflectors are present below the focal point. The second cost function minimizes the reflection of a truncated medium that has no reflectors above the focal point. The second cost function can handle a focal point with reflectors below it. However, it is very computationally expensive, especially in 2D and 3D. Therefore the third cost function is introduced. The third cost function is more efficient than the second one and is based on the minimization of the upgoing Green's function, with a source and receiver at the focal point. It is less accurate, retrieving only very close approximations of the required correction factor in case reflectors are present below the focal point. The minimization of the cost functions fails if there is an overlap in time of physical events and artifacts. In case physical events overlap with each other, none of the cost functions works perfectly.