Example: a CMP from the North Sea

Figure 1 shows a common midpoint gather from the Mobil AVO data set Keys and Foster (1998). This part of the North Sea covers relatively flat lying sediments to a depth of 2 s, where an unconformity introduces older, more deformed rock which is nonetheless still for the most part flat lying. Therefore layered modeling seems reasonable for this data, at least to perhaps 3 s and as a first approximation. The work reported here views whatever converted wave energy is present in the data as noise, so acoustic modeling is reasonable. Finally, as the range of offsets in this data is modest (2.5 km maximum), the hyperbolic moveout approximation seemed likely to be adequate, at least when combined with an agressive mute as displayed in Figure 1.

 

cmpfig Figure 1 CMP from Mobil AVO data.

Strong surface related multiple energy is characteristic of this region. The main preprocessing steps were hyperbolic Radon filtering and bandpass filtering. The Radon filter suppressed but did not entirely remove coherent noise: it seemed reasonable to hope that primary energy dominates the filtered data, as is required by the dual regularization strategy. The bandpass filter ensured that the data were not spatially aliased, so that the differential semblance could be computed accurately.

Minimization of J₀[v;d] produced the RMS velocity displayed in Figure 2, which exhibits a characteristic feature of the differential semblance function: when faced with contradictory moveout (as for example in the interval 1.8-2.4 s), it averages the apparent velocities to come as close as possible to flattening all events. The moveout corrected data (G[v]d in the notation of the last section) displayed in Figure 3 shows a mixture of overcorrected and undercorrected events. The minimization process (via a quasi-Newton algorithm) required approximately 12 s on an SGI Origin2000 processor.

 

sigdisplay0 Figure 2 $V_{\rm RMS}$ from velocity inversion, $\sigma = 0.0$, overplotted on velocity spectrum. Note that the estimated RMS velocity navigates between peaks.

 

siginvdata0 Figure 3 Image gather ( = NMO corrected CMP) using $V_{\rm RMS}$ from velocity inversion with $\sigma = 0.0$. Note residual curvature in all events.

We used the output of the J₀[v;d] minimization as the initial guess for minimization of J_0.5[v;d]; the latter required approximately 3 min on the Origin. Figure 4 shows that $\sigma=0.5$ was not a bad guess at the level of coherent noise: the automatic velocity analysis has now essentially ignored slow (multiple reflection) and smaller fast (steeply dipping or out of plane) phases and placed its estimate of RMS velocity squarely in the main corridor of apparent primary reflection phases, as one also sees in the the conventional image gather (= moveout corrected data G[v]d, Figure 5). Dual regularization also produces an inverted reflectivity ( $r[v;\sigma]$ in the last section, Figure 6) or denoised data, which is superior to the conventional image gather as a basis for further processing.

 

sigdisplay5 Figure 4 $V_{\rm RMS}$ from velocity inversion, $\sigma=0.5$, overplotted on velocity spectrum. Note that the estimated RMS velocity picks apparent primary phases.

 

siginvdata5 Figure 5 Image gather ( = NMO corrected CMP) using $V_{\rm RMS}$ from velocity inversion with $\sigma=0.5$. Primary reflections are essentially flat.

 

sigestrefl5 Figure 6 Inverted reflectivity, $\sigma=0.5$, essentially a cleaned up version of the data, with non-primary phases suppressed.