Seismic Uncertainty and Resolution in Time-Lapse Studies
Authors: Harris Peter 
Published: June 2014

Abstract
In time-lapse seismic, the goal is to monitor changes in the reservoir state, that is, changes in fluid, pressure or temperature. This opens the possibility of identifying unswept compartments or monitoring fluid movements. However, this goal can only be achieved if the changes in the reservoir state affect those properties of the rocks which influence seismic wave propagation, ie. their elastic properties and density. These rock properties then have a direct effect on seismically observable quantities such as amplitudes, phase variations, and travel time variations. The accuracy with which we can estimate the seismic variations feeds back into the uncertainty of estimates of the reservoir state. Typically, however, the seismic variations are at a similar level to the noise in the seismic data and therefore accurate estimates are difficult to obtain.
The definition of data quality also requires discussion. We consider that the most important aspects are resolution and noise. Temporal resolution is related to the length of the wavelet, which is inversely proportional to its temporal bandwidth. For example, Kirchhoff migration is a weighted stack along some travel time surface, and it is evident that a high resolution output requires a high spatial bandwidth to include steeply dipping energy on the flanks of the travel time surface, and also a high temporal bandwidth to obtain a sharply focussed stack. Temporal bandwidth may be estimated using the method of Walden and White (1990). It is harder to obtain estimates of spatial bandwidth because such features as curved reflections and faults tend to contain low levels of power over much of the wavenumber spectrum and are therefore relatively difficult to distinguish from noise.
The most general definition of noise is all the energy we cannot use, which includes relatively random energy such as swell noise, and coherent energy such as multiples. In practice, however, this definition is too general and different types of noise must be characterised by different properties. This implies that the definition of signal-to-noise ratio (SNR) is dependent on the type of noise considered. For the most part we use the convention that noise is the spatially unpredictable energy in the data. Spectral coherence (White, 1984) is used to estimate signal and noise spectra, thereby giving estimates of SNR.
Irrespective of the exact definition of noise, there is a trade-off between noise suppression and improved resolution. This may be illustrated by the migration example; a small migration aperture gives a poorly resolved image. For a migration, noise is any energy which is not scattered solely from the image point. Therefore since the SNR is typically poorer on the flanks of a diffraction, increasing the aperture to improve resolution is likely to increase the noise level in the image.

Time shifts
To motivate the study of processing effects we consider the accuracy with which a time shift can be estimated between segments of two seismic traces. The results give useful guidelines as to whether predicted seismic changes due to reservoir production effects may be visible or measurable.