GNGTS 2017 - 36° Convegno Nazionale

GNGTS 2017 S essione 3.1 535 (7) where C d and C m are the data and model covariance matrices, respectively; m prior is the prior model, and R par and R perp represent the structural regularization terms that are oriented parallel and perpendicular to the direction of the local geological structure, respectively. These operators are standard 1 st order Tikhonov operators that are locally rotated for each point in the model space. In this paper, we call these terms the structural operators and their derivation is detailed in the following. In Eq. 7, α and β are weight factors that control the emphasis on each regularization term and they can be set using (for example) the cross-validation method. In Eq. 7, d is a column vector that contains the N considered AVA responses extracted along the interpreted top of the reservoir d =[ Rpp 1 , Rpp 2 , … , Rpp N ] T ; G is a block diagonal matrix containing the N forward modelling operators, while m is a N x3 column vector that expresses the petrophysical properties at all the considered spatial locations. The least square solution of equation 7 can be derived as follows: (8) we solve equation 8 iteratively by making use of the conjugate gradient method. To define the structural regularization terms, we use the orientation of the geological features at all the spatial locations considered in the inversion. Unlike Buland et al. (2003) and Bongajum et al. (2013), this information is local, and carries no information about correlation length. The structural orientation can be derived from many sources (i.e. geologic understanding, borehole data, seismic data). In this work, we directly derive this information from the migrated seismic data. To this end, we take the migrated stack image, and we calculate the first order horizontal derivatives ( g x , g y ) along the interpreted top of the investigated reservoir. Given the two vectors g x and g y we define the angle of dip as the angle between the x (horizontal) axis and the vector describing the direction of the minimum gradient of the data. Given the above definition, we calculated the structural dip as follows: (9) where is defined in degrees from the horizontal axis. Note that derivative operation amplifies noise in the seismic data and can result in noisy estimates of the angle. In the case of noisy data, it can be useful to apply spatial smoothing to the seismic data (or else to the estimated gradients, or estimated angles). The smoothing operator needs to be addressed on a case-by-case basis. It is worth noting, that during this process we only use the angle information from the seismic data. We could also consider the magnitude of the gradient field to provide variable model weighting throughout the inversion. However, the magnitude of the gradient is dependent on processing decisions (i.e. gains), and thus it could easily produce biased and erroneous results. Once we have derived an estimate of the structural orientation from the seismic data, we need to construct the operators R par and R perp to enforce smoothness along the angle of local orientation (minimum gradient direction). To this end, we construct gradient operators in a locally rotated frame of reference. To create the rotated regularization operators, we start with the 1 st order Tikhonov regularization operators D x and D y , which are the first order derivatives along the two horizontal x and y directions, respectively. We then use the angle estimated from equation 9 to construct the following rotation operators: (10) Finally, we apply these rotation operators to our 1 st order regularization operators as follows: R par = M cos D x + M sin D y (11.1) R perp = – M sin D x + M cos D y (11.2)