Implicit extrapolation with the 45 equation Differential equation

Implicit extrapolation with the 45 equation Differential equation: Matrix equation:

3 -D implicit depth migration • Matrix D is blocked tridiagonal – NOT easily invertible – Splitting methods • 3 -D implicit not widely used – Explicit methods

Conclusions • Shown how helical boundary conditions enable implicit 3 -D wavefield extrapolation • Lateral variations in velocity are handled by non-stationary inverse filtering

Conclusions • Demonstrated 3 -D depth migration with 45 wave equation • Helical boundary conditions applicable for full range of implicit migration methods

1979 1991 Envelope

1979 survey 1991 survey

Warping • Different velocity fields – Effects positioning of imaged events – Need residual migration operator • v unknown – Need to estimate operator from the data

Motivation • Cross-equalization pre-requisite for 4 D study – Remove processing/acquisition differences – Remaining differences are due to production • Post-stack vs pre-stack – Pre-stack data not easily available – Post-stack analysis quicker and cheaper

Cross-equalization flow • Survey co-alignment • Global operators – Matched-filtering & gain correction • Local (non-stationary) operators – Matched-filtering & gain correction • Warping

Global corrections • L 2 matched-filter – Design window: 0. 5 s to 2 s depth (reservoir at 3 s) – Bandwidth and phase matching – Bulk time shift • Amplitude scale

Cross-equalization flow • Survey co-alignment • Global operators – Matched-filtering & gain correction • Local (non-stationary) operators – Matched-filtering & gain correction • Warping

Cross-equalization flow • Survey co-alignment • Global operators – Matched-filtering & gain correction • Local (non-stationary) operators – Matched-filtering & gain correction • Warping

Cross-equalization flow • Survey co-alignment • Global operators – Matched-filtering & gain correction • Local (non-stationary) operators – Matched-filtering & gain correction • Warping

Cross-equalization flow • Survey co-alignment • Global operators – Matched-filtering & gain correction • Local (non-stationary) operators – Matched-filtering & gain correction • Warping

Cross-equalization flow • Survey co-alignment • Global operators – Matched-filtering & gain correction • Local (non-stationary) operators – Matched-filtering & gain correction • Warping

Non-stationary corrections • L 2 matched-filter – Separate filters for each trace: 23 points – Design window: 3 traces wide – Residual static/phase/bandwidth differences • Amplitude balancing – Different T. V. gain functions – Incorrect amplitudes from matched-filters

3 -D depth migration results