The CRS stack can be considered as a generalized velocity analysis tool that provides kinematic information in the form of stacking attributes. Whenever the CRS operator fits well enough the traveltimes of the reflection events on the pre-stack data, the cor- responding kinematic information have a well-established physical meaning and can be used to derive velocity information.
4.7.1
Stacking velocities
Stacking velocities can be expressed in term of CRS attributes α and RN IP through
Equation 4.37. Therefore, their estimation requires to extract from the corresponding attribute sections the optimized α and RN IP values related to genuine reflection events.
This can be done picking on the CRS-stacked section the reflection events from which the velocity information want to be derived. The high S/N ratio and the coherence section help in distinguishing them. The required α and RN IP values are then derived
from each picked (x0, t0) pair.
4.7.2
Interval velocities
Interval velocities may be evaluated from the CRS attributes through the so-called NIP- wave tomography, which is an inversion method to determine a smooth depth velocity model from the CRS attributes [Duveneck, 2004]. This method is based on the NIP- wave theorem [Hubral and Krey,1980], which states that up to the second order the rays connecting sources and receivers on the measurement surface with a common reflection point are geometrically identical to the rays of a hypothetical emerging wave generated by a point source placed at the corresponding CRP (Figure4.14). If the CRP coincides with a normal incident point, i.e. the reflection point associated to a ZO ray, then such a wave is called NIP-wave. Accordingly, a velocity model can be considered consistent with the data if all NIP-waves focus at zero traveltime after their back-propagation. This implies that the RN IP values become zero at t0/2 = 0.
The input for the NIP-wave tomography inversion process consists on a number of sparse picked points in the CRS stacked section related to genuine reflection events. From each picked point (x0, t0) the CRS attributes RN IP and α, i.e. the radii of curvature and the
emergence angle of the hypothetical NIP-wave, are extracted from the corresponding CRS attribute sections. Actually, for practical purposes the inversion rather than RN IP
NIP NIP
CRP
Figure 4.14: Ray trajectories associated with a CRP and NIP points. (a) Ray segments of specular rays reflect at a CRP in the subsurface. (b) Geometrically, ray trajectories associated with a hypothetical wave due to a point source at the NIP coincide with the CRP ray segments. (c) In a consistent velocity model, the NIP-wave focuses at the NIP at zero traveltime, when they are propagated back into the subsurface.
the horizontal component of the slowness vector, respectively:
MN IP = cos2α v0RN IP (4.45) p = sin β0 v0 (4.46)
The input data vector is then: ˜
dobs(t0/2, MN IP, p, x)i for i = 1, . . . , N (4.47)
where N is a number of picked data points.
In principle, once the input data ˜dobs are determined it is possible to find the correct
velocity model simply using the focusing properties of NIP-waves for all picked data points. However, these must be expected to be affected by noise or measurement errors which may lead to a destabilization of the inversion process. Accordingly, prior to the inversion process they must be checked for the presence of outliers, multiples and other noise and if necessary edited. In addition, since the picked NIPs data are extracted from a time image section, their true subsurface locations (x0, t0) as well as the associated
local dip θN IP (see Figure 4.15) are initially unknown. Consequently, these quantities
need to be considered as additional model parameters to be determined together with velocity distribution. The velocity model is described in terms of B-Spline coefficients for each point of a grid with nx and nz nodes in the horizontal and vertical directions
respectively: v(x, z) = nx X j=1 nz X k=1 vjkβj(x)βk(z) (4.48)
being βj(x) and βk(z) the B-spline basis functions. Therefore, the aim of the inversion
process is to find a model vector ˜m:
˜
m[(x, z, θN IP)ni=1, vjk] (4.49)
NIP α ϑ x (x,z,v(x,z)) MNIP
Figure 4.15: Data and model components for the NIP-wave tomographic inversion.
vector ˜dobs and the forward-modelled data vector ˜dm = f ( ˜m), having defined with f a
nonlinear operator symbolizing the dynamic ray tracing:
S( ˜m) = 1 2 ˜ d − ˜dm 2 D (4.50)
This is an ill-posed problem as the data space does not contain sufficient information to uniquely determine all model parameters. Its regularisation is obtained by imposing the velocity model of having minimum second derivatives, namely limiting the roughness of the velocity distribution. This is a reasonable constraint, since the smooth model is the simplest one that explains the data. To apply the mentioned constrains, a term depending on the B-spline coefficients is added to the cost function:
S( ˜m) = 1 2 ˜ d − ˜dm 2 D + c (vjk) (4.51)
where c (vjk) is a measure for the model smoothness that has to be provided during the
inversion process.
The inversion algorithm starts from an initial velocity model which can be described by a constant gradient model V (z) = v0 + g z, where v0 is the near-surface velocity, g
the velocity gradient and z the depth. Any other available a priori velocity information can be used to constrain the initial velocity model. After that, kinematic ray tracing in the downward direction for each of the picked data points in the starting velocity model is performed. This yields the initial elements of vector ˜m[(x, z, θN IP), vjk]. Using
these values, dynamic ray tracing in the upward direction is then performed until the rays reach the measurement surface to obtain the elements of model vector ˜dm = f ( ˜m).
Then the cost function is calculated using Equation4.51 to describe the misfit between picked and modelled data. The linear system of equation is solved by using the least- square inversion method and the model update vector is obtained. The current model
˜
m is updated, and the forward dynamic ray tracing is repeated using the new model. If the cost function increases, the model update vector decreases, and the cost function is recalculated; otherwise, a next iteration is started. The inversion process is stopped when the data misfit reaches the specified minimum, or when the given maximum num- ber of iterations is reached, or when the minimum of the cost function is found. The
characteristic decrease of the model update vector during the inversion has the effect of the determination of the long-wavelength features during the first iterations, while more and more details can be resolved in further iterations.