Inversion basics: from reflectivity to impedance

The forward model

A post-stack trace is modelled by the convolutional model of Section 1.2: the reflectivity series convolved with the wavelet, plus noise,

where $s(t)$ is the recorded trace, $r(t)$ the reflectivity, $w(t)$ the wavelet, $*$ convolution and $n(t)$ noise. At normal incidence each reflection coefficient comes from the acoustic impedance $Z = \rho V_{\mathrm{P}}$ on either side of a boundary (Section 1.1; much of the QI literature writes $I_{\mathrm{P}}$):

So reflectivity is half the change in $\ln Z$ from one sample to the next: the trace carries the derivative of log-impedance, seen through the wavelet. The forward chain is

Given the impedance log at a well and the wavelet, you can predict the trace there, up to the noise. Inversion runs the chain backwards: from $s$ to $Z$.

Reading Figure 7.3

The figure takes the well of Figure 5.6, the brine sand over the gas sand by default, places its impedance in two-way time in 2 ms samples, and makes the trace recorded at it with the same zero-phase 7.5 to 60 Hz band-pass and 15 % band-limited noise. Plate (a) is the true impedance and the low-frequency model; (b) the recorded trace $d$, the trace $p$ the inverted impedance predicts, and their difference, the residual; (c) the true reflectivity and the inversion’s; (d) the inverted impedance against the true; (e) the spectrum of the error against the band the trace carries. The table scores each inversion by its rms difference from the true impedance, its correlation with it, and the residual against the noise.

• Integrate the trace. Choose Recursive with nothing for the low frequencies. Reading the trace as reflectivity and integrating it recovers the steps in impedance but not the level they sit on: the result is about 11 % rms from the true impedance, correlates only about 0.48 with it and ends about 16 % low at the base. In (e) the error piles up below 7.5 Hz, where the trace holds nothing. That is relative impedance.
• Fit the trace. Choose Model-based. With a straight trend for the low frequencies the inversion leaves a residual of about 18 % of the trace against 15 % noise and reads the gas sand about 21 % high, against the 17 % of Figure 5.6, which scores its recursive result against the sonic log (where dispersion leaves the gas sand about 2 % stiffer beside the shale than at seismic frequency) and adds the whole relative trace, less its mean, to the trend, where this figure takes everything below the band’s low ramp from the trend: the sand’s level lies below the band, and the trend puts it there. Choose The well and the sand comes within 3 %, because at the well the model is the log itself.
• Assume a blocky earth. Switch to the limestone and Sparse-spike with nothing for the low frequencies. Asking for as few reflections as possible restores most of the 150 m limestone’s level from below the band (correlation about 0.98), which integration cannot. Merge a straight trend underneath and the correlation falls to about 0.56: a wrong model overrides a right answer.
• Regularise to the noise. With the well’s low frequencies, sweep the regularisation. Too little and the residual falls far below the noise, which means the inversion is fitting noise; too much and it climbs above it. The residual meets the noise between $\varepsilon$ = 0.04 and 0.05, the choice of the discrepancy principle. A residual of that size is not therefore the noise: set the noise to zero and at 0.04 it hardly changes, because this regularisation gives up first the low end of the band, where the trace is least sensitive to $\ln Z$.
• Rotate the wavelet. A 60° phase error drops the correlation from 0.96 to about 0.87 and nearly doubles the rms difference, from 2.4 % to 4.1 %, while the residual rises only from 14 % to about 17 %. A small residual does not prove a right wavelet.

Two controls test the data rather than the method. Moving the band centre $f$ moves the whole band, $f/4$ to $2f$: a lower band hands more of the impedance to the low-frequency model, a higher one resolves thinner beds. Setting the noise to zero is the solvers’ own test: with the well’s low frequencies and little regularisation the model-based fit leaves a residual of about 0.1 % of the trace.

Why the low-frequency model carries the answer

A wavelet is a band-pass filter. A marine wavelet typically carries useful energy from a few hertz to somewhere between 40 and 80 Hz, depending on depth, attenuation and acquisition; the recorded trace in Figure 7.3 carries 7.5 to 60 Hz. Whatever lies outside the band never reached the data, so no inversion of the trace alone can recover:

• The mean level (0 Hz): the absolute impedance. Without it you know the shape of $Z(t)$ but not where it sits.
• The frequencies below the band: the compaction trend and the average impedance of thick beds. A 44 ms gas sand keeps much of its level below 7.5 Hz, which is why a model that misplaces it moves the inverted sand with it.
• The frequencies above the band: detail thinner than the band resolves, which an inversion can only restore by assuming something about the earth, as sparse-spike inversion assumes blocks.

The low-frequency model (also called the background or trend model) supplies the band below the data. It is built, not inverted:

• Compute $Z = \rho V_{\mathrm{P}}$ at every well that reaches the interval, and tie each log to the seismic in time (Section 5.6).
• Low-pass each log so it holds only what the seismic lacks, below the low edge of the seismic band (in Figure 7.3, the complement of the 7.5 to 15 Hz ramp).
• Interpolate between the wells along the mapped horizons of Parts 2 and 3, since the impedance trend follows the stratigraphy.
• The result is a volume of low-frequency impedance that fills the band the seismic cannot.

Merged with the band-limited result, it gives an absolute impedance estimate, and it is only as good as the well control behind it. At a well the model is the log, so an inversion always looks right there; between distant wells the interpolation decides the level of every thick bed. In a frontier basin with one well, the low-frequency model is often the largest single source of error.

Three families of inversion, and a fourth

All post-stack inversions solve the same forward model; they differ in what they assume and how they solve it. Figure 7.3 runs the first three on one trace (Russell and Hampson 1991 compare them the same way).

• Recursive inversion (Lindseth 1979) reads the trace, scaled by the wavelet’s gain, as reflectivity and integrates it down the trace, $Z_k = Z_{k-1}\,(1 + r_k)/(1 - r_k)$, from a starting value. It is fast and simple. Because the trace is band-limited, the result is relative impedance; the wavelet’s shape is never removed; and noise, and any amplitude error, are integrated along with the signal, so errors accumulate downward. It survives as a quick look and as the integration step inside other methods.
• Model-based inversion (Cooke and Schneider 1983) starts from the low-frequency model and updates it so that the trace it predicts through the wavelet matches the recorded one, by least squares, with a regularisation term that keeps it close to the model where the data say nothing. Outside the band the answer is the model; inside it, the data. It is stable and widely used, and depends as much on the wavelet as on the model.
• Sparse-spike inversion (Oldenburg, Scheuer and Levy 1983) looks for the reflectivity with the fewest, largest spikes (the smallest sum of absolute values) that fits the trace, then integrates it. Its blocky answers extend the spectrum beyond the band, which is right where the earth really is blocky and wrong where impedance changes gradually; production versions constrain it with a low-frequency model.
• Bayesian or stochastic inversion treats impedance as a random field with a rock-physics prior and returns not one model but a set of realisations consistent with the data, the prior and the wells. It is slower, and the honest way to answer how confident you can be in an inverted value.

In a real project several engines are often run on the same data and compared: agreement builds confidence, disagreement flags a problem to resolve.

Pre-stack and post-stack inversion

The forward model above is post-stack: one trace per location, stacked over all offsets. At normal incidence the reflection coefficient depends only on the contrast in $Z$, so post-stack inversion recovers one property, acoustic impedance. That is often enough to separate lithologies, but rarely fluids, since fluid sensitivity lives largely in $V_{\mathrm{P}}/V_{\mathrm{S}}$ and density.

Pre-stack inversion uses the angle stacks (near, mid, far). Each responds to a different weighting of $\Delta V_{\mathrm{P}}/V_{\mathrm{P}}$, $\Delta V_{\mathrm{S}}/V_{\mathrm{S}}$ and $\Delta\rho/\rho$ (the Aki-Richards approximation of Section 5.4), so inverting them together returns acoustic impedance, shear impedance (or $V_{\mathrm{P}}/V_{\mathrm{S}}$) and, with long offsets and good data, density. With $V_{\mathrm{P}}/V_{\mathrm{S}}$ a gas sand can be told from a brine sand of the same acoustic impedance.

• Post-stack acoustic inversion: cheaper and faster, sensitive to lithology. Used for regional screening, lithology mapping and feasibility work.
• Pre-stack elastic inversion: needs conditioned angle gathers and a wavelet for each angle stack, but can discriminate fluids. It is the usual basis of quantitative interpretation and supplies the inputs to Section 7.4 (reading inversion products) and Section 7.5 (facies classification).

The rest of Part 7 works mostly with pre-stack elastic products, because they carry the information that property prediction needs.

Wavelet estimation

The wavelet is inferred from the data, not measured. At each well:

• Compute the reflectivity of the well’s impedance log, tied to the seismic in time (Section 5.6).
• Take the seismic trace at the well.
• Find the wavelet $w(t)$ for which $w * r_{\mathrm{well}} \approx s_{\mathrm{well}}$: the well’s reflectivity, convolved with it, reproduces the recorded trace.
• Check that it is plausible: compact, band-limited in the expected range, and close to zero or constant phase.

Wavelets from several wells are compared and averaged, with outliers rejected; the wavelet varies slowly across a survey and can be interpolated. In pre-stack inversion each angle stack gets its own wavelet, since the far stacks are usually lower in frequency than the near.

A poor wavelet is among the commonest causes of a poor inversion, and Figure 7.3 shows why it is hard to see: a constant phase error changes the wavelet’s shape but not its spectrum, so the inversion can still fit the trace nearly as well. The usual failures:

• Misalignment of the log and the seismic at the well. The extraction absorbs the mismatch as a time shift or a phase error, which then contaminates the whole volume.
• A short extraction window. With too few reflections the wavelet is underdetermined; windows of several hundred milliseconds are usual.
• Tuning. Interfering reflections within the window bias the estimate; a geologically simple interval helps.
• Non-stationarity. Attenuation lowers the frequency content with depth, so a single wavelet may not fit a long interval; estimate in several windows or use a time-variant wavelet.

Judging an inversion: blind wells and the residual

An inversion is validated against wells that were not used to build its low-frequency model or estimate its wavelet:

• Extract the inverted impedance at the blind well.
• Take the well’s impedance log at the same location and sampling.
• Compute their correlation and rms difference, as the table of Figure 7.3 does against the true impedance.

The two measures are complementary. Correlation is blind to a constant offset or scale, so a relative inversion can correlate moderately well while every value is wrong; the rms difference catches the level. In Figure 7.3 with a trend for the low frequencies, the correlation is about 0.56 while the gas sand reads about 21 % high. Reserving several wells for blind tests, and leaving one out of the model at a time, is how a workflow shows it generalises beyond its calibration wells.

The second check is the residual: the recorded trace minus the trace the inverted impedance predicts. It should look like the noise, with no coherent events left at the reservoir, and be about as large as the noise. A residual far below the noise means the inversion is fitting noise and needs more regularisation; far above, it is leaving signal unexplained. Choosing the regularisation so the residual matches the noise is the discrepancy principle (Morozov 1966). A residual at the noise level is necessary, not sufficient: in Figure 7.3 it can be mostly unfitted signal, and a wrong wavelet or a wrong low-frequency model can fit the trace and still be wrong, which only the blind wells reveal.

What inversion cannot do

• Restore what was never recorded. If the seismic holds little below 8 Hz, the band from 0 to 8 Hz must come from the low-frequency model or from an assumption about the earth. When the model is wrong the inversion is wrong, and no engine fixes it.
• See below the noise. Where a change in impedance produces less amplitude than the noise, inversion cannot tell it from noise, however it is regularised; in Figure 7.3 the noise shares the signal’s band, so regularisation only trades fitting noise for smoothing signal.
• Resolve beds thinner than the band allows. Beds thinner than about a quarter of the dominant wavelength, $\lambda/4$ (Section 1.7), interfere; inversion recovers their average impedance, not each bed, unless an assumption such as sparse reflectivity is right for them.
• Fix wrong physics. If the convolutional model breaks down (strong attenuation, mode conversion, anisotropy left out of the forward model), the inversion returns an answer that fits the data and is physically wrong.

A typical industrial inversion project

A production pre-stack inversion on a developed field (five to ten wells, a 3D survey) runs roughly as follows:

• Weeks 1 and 2: data conditioning: gather flattening, multiple removal, offset balancing, noise attenuation. The inversion is only as good as its input, and conditioning is often the bottleneck.
• Weeks 3 and 4: wavelet estimation at each well and for each angle stack, with stationarity and consistency checks, usually iterated with the rock physicist over the well ties.
• Weeks 5 and 6: the low-frequency model: logs in time, horizon-guided interpolation, edge handling, and blind-well tests of the model alone.
• Weeks 7 and 8: the inversion itself, model-based or simultaneous, usually in several passes with different regularisation; the pass that best matches the blind wells is kept.
• Week 9: quality control: blind-well correlations, residual analysis (the residual should look like noise), and the ranges of the outputs.
• Week 10: deliverables: acoustic and shear impedance (or $V_{\mathrm{P}}/V_{\mathrm{S}}$) and density volumes, uncertainty volumes if the inversion is stochastic, and a QC report.

The team is typically an inversion specialist, a rock physicist, a seismic processor, an interpreter for the horizon framework, and a data manager. Ten weeks is the fast case; sixteen to twenty is common when conditioning or wavelet estimation turns up problems.