Inversion basics: from reflectivity to impedance

The forward model: one equation to rule them all

Every post-stack seismic trace obeys the convolutional model (introduced in §1.2):

$s(t) = r(t) * w(t) + n(t)$

where $s(t)$ is the observed seismic trace, $r(t)$ is the reflectivity series, $w(t)$ is the source wavelet, $*$ is convolution, and $n(t)$ is noise.

The reflectivity itself comes from impedance contrasts at each boundary (§1.1):

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In the small-contrast limit, reflectivity is half the fractional change in impedance — the derivative of a log-impedance trace. Each layer boundary is a spike; everything between boundaries is zero.

The forward chain is therefore:

Ip(t) → r(t) = (d ln Ip)/2 → s(t) = r * w + n

If you know the impedance log at a well, you can predict the seismic trace at that well exactly (up to noise). The inverse problem — predicting Ip from s — is what this section is about.

Exercise — walk the four stages

• Open the widget in Forward model view. You see three tracks: the true impedance trace (yellow), the reflectivity (orange spikes at each boundary), and the seismic trace (blue). Notice how every layer boundary produces a reflectivity spike whose polarity encodes whether Ip goes UP (positive R) or DOWN (negative R), and how the wavelet smears each spike into a seismic wiggle. This is nature’s forward model at work.
• Drag the Wavelet slider from 25 Hz down to 10 Hz. Watch the seismic trace become smoother and broader — each reflector now produces a wiggle that extends over a larger time window. Thin beds that were separated at 25 Hz now blend together. This is the RESOLUTION problem (§1.7) — the wavelet BANDWIDTH controls how finely you can see.
• Now drag the slider up to 60 Hz. The seismic becomes sharper; individual reflectors emerge clearly. But REAL seismic data is almost always in the 10–40 Hz band — because Earth attenuates high frequencies with depth. You usually inherit a wavelet; you don’t choose it.
• Switch to Bandlimited seismic view. This is just the third track from the forward panel, isolated as “the inversion input.” Every QI inversion starts here. Note what’s MISSING: the DC trend (average Ip level) and the fine detail — only the middle-frequency “edges” remain.
• Switch to Naive inversion (no LF model). Four tracks now: the seismic input (dimmed), the bandlimited reflectivity estimate (orange), the naive inverted impedance (pink), and the TRUE impedance (yellow dimmed). Compare the pink and yellow: the SHAPE matches — every boundary is in the right place, every step has the right direction — but the ABSOLUTE VALUES are wrong, offset by a variable amount.
• Switch to Full inversion (with LF model). Again four tracks: bandlimited R_est (dimmed), the LF model (purple, smooth trend from well logs), the recovered impedance (green), and the true impedance (yellow dimmed). Now compare green and yellow: both shape and absolute values match — the LF model supplied the DC trend that the seismic could not. This is what a GOOD industrial inversion delivers.

Why the low-frequency model is the linchpin

A wavelet acts as a bandpass filter. Typical marine seismic wavelets pass frequencies from roughly 5 Hz to 80 Hz — anything below 5 Hz or above 80 Hz is not in the seismic data because it was never imprinted there in the first place. Inversion therefore CANNOT recover:

• The DC level (0 Hz): absolute impedance value. Without this, you know the SHAPE of Ip(t) but not where it sits on the y-axis.
• Frequencies below ~5 Hz: the long-wavelength trend. This is the difference between a shale at 6500 Ip and a shale at 7500 Ip — seismic can’t tell them apart from amplitudes alone.
• Frequencies above ~80 Hz: sub-wavelength detail. Thin-bed architecture below the vertical resolution is smeared together.

The low-frequency model (LF model, also called background model or trend model) supplies the missing 0–5 Hz band. It is BUILT, not derived from seismic:

• Start with Ip logs at every well that penetrates the interval. Compute Ip = ρ · Vp at each depth sample.
• Filter each well log to retain only the low frequencies — typically a 0–5 Hz or 0–8 Hz cutoff.
• INTERPOLATE between wells along mapped horizons (from Parts 2–3): the impedance trend follows the geology, so horizons are the natural guides.
• The result is a 3D volume of LOW-FREQUENCY IMPEDANCE that fills in the DC + long-wavelength content of the subsurface.

Adding this LF volume to the bandlimited inversion output gives the full-band impedance estimate. The critical property: the LF model is AS GOOD AS YOUR WELL COVERAGE. Sparse wells (one per 10 km) give you a smoother, less-reliable LF model; dense wells (one per 1 km) give you a sharper, better-constrained one. In a frontier basin with a single well, LF model uncertainty is one of the largest error sources in the inversion deliverable.

Inversion engine families

Industry inversion engines fall into four broad families. All try to solve the same forward equation, but they make different mathematical choices:

• Recursive inversion (trace-by-trace). The simplest flavor: $I_{p,i+1} = I_{p,i} \cdot (1+r_i)/(1-r_i)$, marching down the trace. Given the reflectivity estimate (from deconvolution) and a starting Ip, it integrates the reflectivity back into an impedance trace. FAST and intuitive, but UNSTABLE — errors accumulate downward. Used as a first-pass reconnaissance tool or as a subcomponent of more sophisticated engines.
• Model-based inversion. Perturb an initial impedance model iteratively; at each step, forward-model the predicted seismic, compare to observed, and update to reduce misfit. Regularized by a smoothness prior or a well-log-derived prior. STABLE and widely used in industry; outputs full-band Ip directly if the LF model is supplied as the prior. Commercial tools: Jason, Hampson-Russell STRATA, CGG Jason.
• Sparse-spike inversion. Assumes the true reflectivity is a sparse set of large spikes (layered Earth); finds the sparsest r(t) that, when convolved with the wavelet, matches the seismic. Produces sharp, “blocky” impedance traces with crisp boundaries. Good for clear layered sedimentary sections; over-simplifies in heterogeneous zones.
• Bayesian / stochastic inversion. Treat Ip as a random field with a rock-physics-defined prior; compute the posterior distribution consistent with the seismic + prior + well constraints. Outputs NOT a single impedance model but a SUITE of plausible models (realizations), each consistent with the data. Enables proper uncertainty quantification — the only way to answer “how confident am I in this Ip value?” correctly. Slower and more complex but increasingly the standard for high-value fields.

In a real project you often run several engines on the same data and compare. Consensus across engines = trust; disagreement = flag for QC.

Pre-stack vs post-stack inversion

The forward model above is for POST-STACK data — a single seismic trace at each location, already stacked over all offsets. Post-stack inversion recovers only ONE elastic quantity: Ip. That’s enough to discriminate LITHOLOGY (sand vs shale) but not FLUID (gas vs brine) reliably — because fluid sensitivity lives mostly in Vs and density, not in Vp alone.

Pre-stack inversion uses data BEFORE stacking — the seismic at multiple angles of incidence (near, mid, far). Each angle stack responds to a different weighted combination of ΔIp/Ip, Δρ/ρ, and ΔVs/Vs (via the Aki-Richards approximation, §5.4). Simultaneously inverting all the angle stacks gives you THREE outputs: Ip, Is (or Vp/Vs), and density. With three elastic quantities you can distinguish gas sand from brine sand (gas drops Ip and Vp/Vs; brine doesn’t).

The practical split:

• Post-stack acoustic inversion: cheap, fast, lithology-sensitive. Used for broad-scale screening, well-constrained lithologic mapping, and pre-QI feasibility studies.
• Pre-stack elastic inversion: more expensive (requires angle gathers + more careful wavelet estimation per angle) but fluid-diagnostic. The industry standard for rigorous QI; outputs the inputs to §7.4 (reading inversion products) and §7.5 (facies classification).

For the rest of Part 7 we’ll mostly be talking about pre-stack elastic inversion outputs, because those are what enable the downstream property prediction.

Wavelet estimation

The wavelet is INFERRED from the data, not measured directly. At each well the workflow is:

• Take the well-log Ip trace. Compute its reflectivity.
• Take the seismic trace AT the well location.
• Find the wavelet $w(t)$ such that $r_{well} * w \approx s_{well}$ — i.e., convolution of well-derived reflectivity with the wavelet reproduces the observed seismic.
• Verify: does the estimated wavelet have a plausible shape (compact, zero-phase or constant-phase, band-limited in the expected range)?

Wavelet estimation is done at EVERY well and the results averaged (with outlier rejection). The wavelet will vary laterally — typically slow, smooth changes that can be interpolated. In pre-stack inversion you estimate a DIFFERENT wavelet for each angle stack (near wavelets differ from far wavelets, because the effective frequency content is angle-dependent).

Wavelet quality is the single most common cause of bad inversion results. Common failure modes:

• Time misalignment between log and seismic at the well (log depths not properly time-converted). The wavelet estimate picks up a phase error that contaminates the whole volume.
• Short extraction window. Too few reflectors in the estimation window — the wavelet is under-determined. Use a longer window (typically 500–1000 ms of interval).
• Tuning effects. Interfering reflections within the wavelet’s extent can bias the estimate. Pick a geologically simple interval.
• Non-stationarity. The wavelet changes with depth (typically lower frequency deeper). Either estimate at multiple windows or use a time-variant wavelet model.

Well ties — the inversion’s scoreboard

Every inversion output should be validated against wells NOT used in the building of the LF model or the wavelet. The diagnostic:

• Extract the inverted Ip trace AT the blind well location.
• Extract the well-log Ip trace at the same location (and same time sampling).
• Compute the CORRELATION and the RMS error between them.

A well-run inversion achieves correlations of 0.85–0.95 and RMS errors of 5–10% at blind wells. Correlations below 0.7 are a red flag — the inversion is not capturing the trace character. RMS errors above 15% suggest either wavelet, LF model, or calibration problems.

The best QI projects reserve MULTIPLE wells for blind testing — ideally 20–30% of the wells. Leaving out a well from calibration, running the inversion, then comparing the blind-well result is how you know the workflow is generalizable beyond the calibration sample.

What inversion CANNOT do

• Restore what wasn’t there. If the seismic is severely muted below 8 Hz, no inversion can meaningfully recover the 0–8 Hz band from the data alone — it must come from the LF model. When the LF model is wrong, the inversion is wrong, and no amount of engine sophistication fixes it.
• See past the noise floor. Above some local noise level, changes in the true impedance produce seismic amplitudes smaller than the noise. Inversion cannot distinguish a small real signal from a small noise fluctuation.
• Resolve below the wavelet. Thin beds below λ/4 (§1.7, the tuning thickness) are amplitude-tuned, not individually resolved. No inversion recovers individual thin-bed impedance from a tuned amplitude.
• Fix wrong physics. If the convolutional model breaks down (strong attenuation, major mode conversion, anisotropy neglected in the forward modeling), the inversion will produce a mathematically consistent but PHYSICALLY WRONG answer.

A typical industrial inversion project

Timeline for a production-quality pre-stack inversion on a developed field (5–10 wells, 3D survey):

• Week 1–2: data conditioning. Angle-dependent gather flattening, multiple removal, offset balancing, noise attenuation. Inversion is only as good as its input; gather conditioning is frequently the bottleneck.
• Week 3–4: wavelet estimation at each well, per angle stack. Stationarity + consistency checks. Typically several iterations with the rock physicist to verify log-seismic ties.
• Week 5–6: LF model build. Log-to-time, horizon-guided interpolation, edge-condition handling. Blind-well cross-validation of the LF model alone.
• Week 7–8: run the inversion (model-based or simultaneous). Typically multiple passes with different regularization strengths; pick the pass that best matches blind wells.
• Week 9: QC. Well-tie correlations, residual analysis (seismic minus forward-modeled from inverted Ip — should be featureless noise), output sanity (Ip ranges, Vp/Vs sanity).
• Week 10: deliverables. Full-band Ip, Is (or Vp/Vs), density volumes; per-sample uncertainty maps if Bayesian; QC report.

Team typically: inversion specialist (lead), rock physicist, seismic processor (for conditioning), interpreter (for horizon framework), data manager. 10 weeks is the FAST case; 16–20 weeks is more typical when gather conditioning or wavelet estimation surface problems.