Frequency

Selecting the best starting frequency is an important element of any FWI scheme. Warner et al., (2013) stated that it is the signal-to-noise ratio, rather than the amplitude of the low frequencies that is of interest when choosing the lowest inversion frequency, even if the absolute amplitudes are lower than the amplitudes at higher frequencies. Ideally phase plots, that is, plots of the phase of the data at a single frequency, after Fourier transformations, should be analysed when choosing the starting frequency. Coherent structure for example, in common-receiver gathers, on such plots indicates source-generated signal.

The lowest useable frequency in the Tommeliten field data is 3 Hz. This is not clear from analysis of the shot gathers with different high-cut filters applied in the time domain (Figure 4.6), or from the Fourier amplitude spectra for the field data (Figure 4.7). These suggest that the inversions should begin at 4 Hz or higher. Figure 4.8, Figure 4.9 and Figure 4.10 show phase plots at 2 Hz, 3 Hz and 4 Hz for both the muted and unmuted field data. At 4 Hz, the smooth concentric circular structure indicates a good signal-to-noise ratio. At 3 Hz, there is a lower signal-to-noise ratio than at 4 Hz, but the energy is still coherent. For all offsets of the muted data, concentric circles represent the coherency. For the unmuted data, at medium and large offsets, the coherency pattern is the same, and at the smallest offsets, the coherency is represented by a cross-like pattern with four-fold symmetry in the direction of the receiver and source lines due to the dominance of Scholte waves. At 2 Hz, the data is dominated by noise at all offsets, with little or no coherent energy except the Scholte waves.

Scholte waves are boundary waves associated with the interface between a fluid and a solid medium. These couple strongly with the source in shallow water at very low frequencies, and are localized near the sea floor. The cross pattern at low frequencies is due to the azimuthally variable effects of the source-receiver array, which are partially effective in suppressing the Scholte waves at low frequencies due to their low velocities (Warner et al., 2013). The mutually perpendicular source-receiver arrays with approximately the same dimensions cause four-fold symmetry in the horizontal plane.

Figure 4.6 A randomly selected shot gather with different high-cut filters applied: (a) 6-9 Hz and (b) 3-5 Hz.

Figure 4.7 Amplitude spectra for a high-cut (12-15 Hz) filtered shot gather.

Source Y Position (km)

12

Source X Position (km) 0

0 10 0 10 Source X Position (km)

(a) (b)

Source Y Position (km)

12

0

phase (degrees)

-180 +180

Figure 4.9 Phase plots for at 3 Hz for (a) the unmuted field data and (b) the muted field data from a randomly selected receiver gather located at the center of the concentric circles, and highlighted by the yellow circle.

Figure 4.10 Phase plots for at 4 Hz for (a) the unmuted field data and (b) the muted field data from a randomly selected receiver gather located at the center of the concentric circles, and highlighted by the yellow circle.

The maximum useable frequency is controlled by the lowest velocity and stability of the forward modeling algorithm. Ideally there should be 4 to 5 samples per wavelength (accuracy of modeling algorithm).

The geology of the study area suggests that the lowest velocities are expected in the gas media or in the water. The velocity of water is 1480 m/s and the velocity of the gas sediments encountered by wells drilled in the area is ~1500 m/s. Thus the maximum useable frequency is ~ 5.9 - 7.4 Hz with the selected 50 m grid spacing.

Initial inversion tests were run using a multi-scale approach with frequencies increasing from 3 to 7 Hz (3, 3.9, 5.1, 5.85 and 7 Hz). At 7 Hz, a grid spacing of 50 m provides more than 4.2 grid points per wavelength in the water layer, and more than 5 grid points per wavelength everywhere that the velocity is greater than 1750 m/s.

Table 4.4 Summary of the parameters chosen for the inversion.

water velocity (m/s) 1480

minimum frequency (Hz) 3.0

grid spacing (m) 50

grid points per wavelength 4 5 maximum frequency (Hz) 7.4 5.9

maximum velocity (m/s) 6250

time step (ms) 4.0

4.12 Start Model and Anisotropic Model

FWI uses a linearised least squares technique to minimize the misfit between observed and predicted data. In addition to sufficiently long offsets and low frequencies, FWI also requires a sufficiently accurate, low wave number, initial velocity model that predicts the travel time to within half a period of the dominant wavelength at the lowest inversion frequency (Pratt, 1999) to avoid the cycle-skipped local minima and to assure convergence to the global minimum. Inversion clearly works if the starting model is in the vicinity of the global minimum, however it may fail if the starting model is too far away from the true model.

An anisotropic velocity model generated for PSDM was used for the inversion. It was provided by the processing contractor, and was obtained using a combination of migration, well velocities and travel-time tomography. Stacking velocities from an earlier surface streamer dataset were refined by picking the residual move-out on the pre-stack time-migrated PZ-summed volume. This latter was stacked, and the match of the stacked data to the sonic velocities at the four well positions was used for an initial estimate of anisotropy. Residual move-out picked on depth-migrated common image gathers was used to refine both the velocity and anisotropic model. Reflection travel-time tomography was carried out in a layer-stripping mode with well ties used for depth calibration. The anisotropic models were further constrained using the geometry reflectors or stratigraphy from the migrated data.

The reflectors in the depth migrated volume are severely deteriorated in the gas-bearing sediments, with the edges of such sediments highlighted by bright reflectors as the gas creeps along the sands only causing an increased acoustic impedance in the sand-silt sediments. This poor reflectivity region was used to outline a low velocity region using generic velocities from the well in the starting model representative of the gas in the overburden sediments. This method commonly used for PSDM when reflection tomography is inadequate, is largely based on the user’s interpretation. FWI seeks to improve such techniques.

The contractor’s model provided is relatively simple and smooth. It is broadly one-dimensional outside the gas cloud down to 3 km in the clastic section with an underlying broad, low-relief, anti-formal high-velocity section in the chalks (Figure 4.11). Within the upper section there is a low-velocity blocky structure representative of gas.

The model measures 9.0 x 9.9 km with a grid spacing of 25 m in the x and y directions and 20 m in the z direction. This velocity model was resampled all three directions such that the grid spacing was 50 m, and smoothed in the x and y directions by a horizontal wavelength of 300 m. Smoothing was minimal, except in areas where the original contractor’s model had sharp boundaries. The starting velocity model used in the inversion should be smooth and void of any structure sharper than

depth and all three dimensions (Warner et al., 2013). Inversion tightens up the velocity model, and adds fine structure and details. Any erroneous structures in the starting model may be grossly exaggerated by the inversion. Practically, the seabed is the only structure known with certainty at the start on any inversion. For the Tommeliten dataset, the top chalk appears as a sharp boundary at all four well positions. Its depth is accurately known at the well positions, but its 3D structure and lateral continuity is unknown. Thus, despite the knowledge that this will most likely be a sharp boundary, it does not appear as a sharp interface in the starting velocity model.

The smoothed model was then extrapolated such that for a given direction, the last slice of the model was repeated. This slice was then smoothed using an average value of the velocity for the nine closes cells. The smoothed was weighted by some value. This was again repeated in the same direction until the model was extended by the required amount. The weighting factor was linearly reduced with the outward extension of the model. Velocity cut-off values were used for the smoothing. This technique was used to extend the model in all directions such that final model measured 16.5 x 13.5 x 4.0 km. This technique helped to ensure sharp boundaries were not reintroduced into the model.

The contractor model contained velocities gradually increasing from water velocity, 1480 m/s, at the top of the model (0 m) to 1700 m/s at the water sediment interface (Figure 4.12). For migration purposes this is sufficient, but initial testing suggested that the seabed boundary was too smooth, and it was thus modified to have a sharp water-sediment interface.

The anisotropic model is a vertical transverse isotropic (VTI) model, as there is no strong evidence to suggest azimuthal p-wave anisotropy, lateral changes in anisotropy or a tilted axis of symmetry. Horizontal and vertical velocities are intrinsically different for VTI media and are related by Thomsen’s anisotropic parameters, delta (δ) and epsilon (ε). Both models are relatively one-dimensional down to 3 km with a low-relief anticlinal geometry at depth that mimics the stratigraphy.

Notably the anisotropy inside and outside the gas cloud is the same due to limited control on the anisotropy of the gas. Due to limited variations in these models laterally, a one-dimensional profile was used for the inversions (Figure 4.13). Both epsilon and delta models contain relatively high values, with delta values ranging from 2 to 8%, and epsilon values consistently above 10%, and as high as 20% at some depths.

Figure 4.11 3D image of the starting velocity model. The low-velocity blocky blob at the center represents the gas anomaly. At depth simple anticlinal feature of increasing velocity with depth is shown. This image is coincident with Figure 4.47.

Figure 4.12 Velocity-depth profile for the top 100 m of the PSDM velocity model (red) and the modified

model with a sharp seabed (blue).

Figure 4.13 1D anisotropic parameters used for the FWI.