Closing the loop: synthetic seismograms and inversion

The convolutional model (recap from Section 1.2)

At normal incidence, counting only primary reflections, a trace is the earth’s reflectivity series convolved with the wavelet:

• $r(t)$, the reflectivity series: a spike at the two-way time $t_k$ of every boundary, as high as its normal-incidence reflection coefficient $R_k = (Z_2 - Z_1)/(Z_2 + Z_1)$, where $Z = \rho V_{\mathrm{P}}$ is the acoustic impedance, $Z_1$ above the boundary and $Z_2$ below it.
• $w(t)$, the wavelet: the source signature as filtered by the earth, the recording and the processing.
• $s(t)$, the synthetic trace: what the seismic would record at the well if the model were complete.

Each boundary contributes a copy of the wavelet scaled by its coefficient, and the trace is their sum, so the copies from close boundaries overlap and interfere. The model leaves out multiples, transmission losses, the loss of high frequencies with depth and the change of reflectivity with angle; the pitfalls below return to each.

Building a synthetic from a well

1. Impedance. Convert the sonic slowness to velocity, $V_{\mathrm{P}} = 304\,800/\mathrm{DT}$ (m/s, with DT in $\mu$s/ft), and multiply by the density log, $Z = \rho V_{\mathrm{P}}$, at every sample. Normal-incidence reflectivity needs no $V_{\mathrm{S}}$.
2. Reflectivity. $R = (Z_2 - Z_1)/(Z_2 + Z_1)$ between each pair of samples gives a dense series in depth. Edit washouts and other bad log intervals first: their false steps become false reflections.
3. Time from depth. Integrate the sonic, $t(z) = t_0 + \sum 2\,\Delta z/V_{\mathrm{P}}$, from a known time at the top of the log. The integrated sonic almost always runs short of the times a checkshot survey or a VSP measures, by a drift that grows with depth. A sonic tool samples the rock at 10 to 20 kHz, and attenuating rock is faster there than at the 30 Hz of the seismic (dispersion); short-path multiples in thin layering delay the seismic further (Stewart et al., 1984). So the sonic is stretched to match the checkshots at every level, with the drift interpolated between them.
4. Wavelet. Choose or extract one, and say what it is: a Ricker of stated peak frequency, a zero-phase band-pass with stated corner frequencies, or a wavelet estimated from the seismic at the well. State its polarity too. In SEG normal polarity, used throughout this book, an increase of impedance downward gives a positive coefficient, which a zero-phase wavelet shows as a peak.
5. Convolve the reflectivity, now in two-way time, with the wavelet.
6. Tie. Lay the synthetic $s$ beside the recorded trace $d$ at the well and measure the match, usually by the correlation coefficient $\sum s\,d \big/ \sqrt{\sum s^2 \sum d^2}$ over a window. A small bulk shift is normal; a large shift, or one that grows with depth, means the time-depth relation needs fixing, not the seismic.

Exercise, tie the well

1. Start with the default well, a brine sand over a gas sand in shale, with a 30 Hz Ricker and the checkshot correction on. In (e) the brine sand’s top and base reflect +0.03 and −0.03; the gas sand’s top reflects −0.13 and its base +0.13, about four times as strong. In (g) the gas sand is a bright trough over a bright peak, the bright spot of Section 5.5, made by the fluid substitution of Section 5.3. The correlation is 0.97.
2. Choose Sonic alone. The drift in (d) grows to 8.8 ms at 2500 m, so the synthetic’s events arrive early, the deeper the earlier, and the correlation falls to 0.55. Moving the whole synthetic 4 ms later recovers 0.92 but not 0.97: a drift that grows with depth needs a stretch, not a shift.
3. Back on Checkshots, drag the frequency. A Ricker ties best at 28 Hz, 0.97; at 10 Hz the correlation is 0.36 and at 60 Hz 0.43, because a wavelet too broad or too sharp puts lobes where the recorded trace has none. Switch to Band-pass at 30 Hz: 0.99, with the wavelet the trace was made with. In practice the wavelet is estimated from the seismic at the well, not guessed.
4. Choose Two thin gas sands, each 15 m thick and 15 m apart. A bed is resolved when it is thicker than about a quarter of the dominant wavelength, $\lambda_{\mathrm{d}}/4 = V_{\mathrm{P}} T_{\mathrm{d}}/4$, with $T_{\mathrm{d}}$ the wavelet’s trough-to-trough period, $\sqrt{6}/(\pi f)$ for a Ricker (Section 1.7). In the gas sand that limit falls to 15 m at a 37 Hz Ricker. Above it (g) shows a trough and a peak at each sand’s top and base; below it the four reflections overlap and interfere: the composite is brightest at about 37 Hz, where the sands are at the tuning thickness, and dims as the frequency falls, until at 15 Hz the two sands read as one trough over one peak.
5. Choose Shale on limestone. Limestone at 5500 m/s and 2.65 g/cm³ has twice the impedance of the shale, so its top reflects +0.34, the strongest event in the four wells, and its base −0.33.
6. Choose Salt over a gas sand. The top of the salt reflects +0.18, but the strongest event is its base, −0.26, where fast salt sits on soft gas sand, and the base of the gas sand reflects +0.14. Nothing in this model dims the sand below the salt, because the convolutional model has no transmission losses and no ray bending. Below real salt those are exactly the problems: the salt’s high velocity and irregular shape bend and spread the rays, and only migration with an accurate salt model repairs the image (Section 3.5).

Inversion: the reverse direction

The synthetic is forward modelling: rock properties to reflectivity to trace. Seismic inversion runs the other way, from the trace to the impedance. Its simplest form, recursive inversion (Lindseth, 1979), inverts the definition of the reflection coefficient: given the impedance above a boundary and the coefficient at it,

so impedance is the running sum of the reflectivity, exponentiated. With complete reflectivity this would be exact. A seismic trace is not reflectivity, though: it is reflectivity seen through a band-limited wavelet. Once the wavelet is removed, or the trace simply divided by the wavelet’s gain in its passband, the scale a tie measures, what remains is the reflectivity within the seismic band, here 7.5 to 60 Hz. Summing it recovers the changes of impedance inside that band and nothing else. The frequencies below the band, which carry the compaction trend and the average impedance of any bed thicker than a few tens of milliseconds, are missing, so the result is relative impedance.

The missing low frequencies must come from a model: well logs interpolated along interpreted horizons, or velocities from processing or tomography, low-pass filtered and merged with the inverted band (Russell, 1988; Latimer et al., 2000). Only then is the result absolute impedance, a property of the rock that can be compared from well to well and converted to porosity or lithology.

Plate (h) shows three cases. With Nothing, started from the log’s value at its top, the inverted impedance follows every step of the log but wanders away from it, 18% low by the base and 12% rms over the well. A trend, the least-squares straight line through the log, removes that growing error (7.2% rms) but not the error in the gas sand, which reads 17% high: the sand’s own low frequencies lie below the band and are no part of a straight line. The well, the log itself below 15 Hz, brings the inversion to 2.5% rms, close to the 2.4% that the log’s detail beyond the trace’s band, above its 45 to 60 Hz taper, costs any inversion of this trace. That match is made by construction, since at the well the model is the answer; between wells the model is interpolated, and its errors become the inversion’s. In the limestone well a trend leaves the limestone 37% low.

Pre-stack inversion extends this to angle stacks and recovers P-impedance, S-impedance and density (or $V_{\mathrm{P}}$, $V_{\mathrm{S}}$ and $\rho$) separately, which feed the fluid and lithology analysis of Sections 5.3 to 5.5. It needs what post-stack inversion needs, wavelets from well ties (one for each angle range), a low-frequency model and amplitudes preserved through processing, and it is the most complete and most demanding product of quantitative interpretation. Part 7 builds on both.

Common pitfalls

• The wavelet. A synthetic is only as good as its wavelet. One of the wrong frequency or phase gives a synthetic that looks roughly right and ties poorly in detail, and every inversion built on it inherits the error. Estimate it from the seismic at the well and refine it as the tie improves.
• Time from depth. Integrated sonic alone drifts from the seismic times by milliseconds per few hundred metres, more in slow, attenuating rock. Correct it with checkshots or a VSP before judging the tie. Stretching and squeezing a synthetic to force a match without them can produce a high correlation and a wrong tie.
• The logs. Washouts, invasion and cycle skips put false steps in the sonic and density, and every false step is a false reflection. Edit the logs first.
• What the model leaves out. Multiples, transmission losses, the loss of high frequencies with depth and noise are in the recorded seismic and not in a primaries-only synthetic. A tie that is good through the target and poor elsewhere is often acceptable; say so.
• Angle and anisotropy. A stack mixes angles. Where reflectivity changes with angle (Section 5.4), a normal-incidence synthetic ties a near-angle stack better than a full stack. Anisotropic shales, faster horizontally than vertically, add to the drift.
• Phase. Recorded wavelets often carry residual phase from acquisition and processing. A zero-phase synthetic against a phase-rotated wavelet leaves an asymmetric misfit that a phase rotation removes and a time shift does not (Section 2.2).

Closing Part 5: the workflow

The six sections of Part 5 make one chain:

1. Section 5.1, acoustic impedance and the reflection coefficient: $Z = \rho V_{\mathrm{P}}$ and $R = (Z_2 - Z_1)/(Z_2 + Z_1)$, the least a boundary must be described by.
2. Section 5.2, moduli and $V_{\mathrm{P}}/V_{\mathrm{S}}$: $K$ and $\mu$ set the velocities, $V_{\mathrm{P}}/V_{\mathrm{S}}$ separates lithology and fluid, and Castagna’s mudrock line is the brine-saturated reference.
3. Section 5.3, Gassmann fluid substitution: how the velocities change when the pore fluid changes, which tells brine, oil and gas apart in the same rock.
4. Section 5.4, AVO: the Aki-Richards approximation gives the intercept $R_0$ and gradient $G$ of the angle-dependent reflection coefficient; $G$ is the more sensitive to fluid.
5. Section 5.5, the $(R_0, G)$ crossplot and the AVO classes: a background trend and class regions, with anomalies off the trend as candidate prospects.
6. Section 5.6, the synthetic and the tie calibrate the seismic to the wells, and inversion turns calibrated seismic back into rock properties, absolute only with a low-frequency model.

With the structural toolkit of Parts 2 and 3 and the attributes of Part 6, this is the working core of quantitative interpretation: reservoir-property volumes, fluid maps and prospect rankings all pass through this chain. What remains is calibration to a particular basin and judgement about a particular prospect.