Incorporating Attenuation In Model Building And Imaging
Authors: Farmer Paul , Bloor Robert , Luo Zhijiang , Chen Guoquan , Dolliazal Mohamed , Bai Jianyong , Wang Chao , Zhou Zheng-Zheng , Yingst David 
Published: May 2013

Abstract
Seismic waves propagating in the earth are affected by attenuation. Attenuation is the progressive loss of energy with each cycle of the wave. The resulting seismic data has a frequency and time dependent loss of amplitude and a phase rotation of the wavelet. Traditionally attenuation has been compensated during the preprocessing, however this approach has a number of limitations. It is difficult to relate the recorded seismic trace to a location in the subsurface and hence except in the case of a constant attenuation factor the amount of attenuation correction to apply is difficult to identify. Correcting for attenuation during the model building and imaging stages of processing creates the opportunity to relate attenuation factors to the subsurface model and hence have geologic constraints on the attenuation factors and permits the application of a attenuation correction associated with propagation of the energy through the subsurface. Additionally the attenuation corrections introduce phase shifts into the data that can create biases in the velocity model if they are not rigoroulsy handled in the model building process. Ideally the incorporation of attenuation is handled within all aspects of the model building and imaging. Our experience indicates that incorporating attenuation in this manner results in a more rapid convergence to a more accurate model, especially in the shallow section. Improvements in the shallow section will impact the ability to accurately image and interpret high contrast boundaries in the subsurface such as a salt interface and improvement to the imaging and positiong of such boundaries results in improved imaging of the deeper data to varying degrees.

Statement of Theory and Definitions
Attenuation of the seismic wave at a location in the subsurface is defined by the Q factor of that particular part of the subsurface. The Q factor is defined as the 2 p times the total energy stored divided by the energy lost in a single cycle. Higher frequencies are reduced in amplitude more for the same traveltime than lower frequencies, so as we progress deeper into our data the dataset becomes progressively reduced in bandwidth and lacking in high frequency components. Consequently the amplitude at a given time can be related to the initial amplitude as follows:A(t)?A_0 e^(-?t/2Q)
where A_0 is the initial amplitude, ? is the angular frequency and t is the traveltime.
In addition to the loss of energy and reduction in amplitude as the wave propagates there is an apparent frequency dependence of the velocity. The apparent velocity variation or phase rotation is rigorously handled in the methods we use to propagate energy (Bai et al, 2012, Blanch et al., 1995). For a general understanding there are several approximate models that enable a quantification of the magnitude of this affect (Futterman 1962, Azimi et.al. 1968). Higher frequencies propagate faster with relatively rapid changes in the velocity at low frequenices