Closing the loop: synthetic seismograms and inversion

The convolutional model (recap from §1.2)

A seismic trace at a single location is approximately the convolution of the earth’s reflectivity series with the wavelet:

$s(t) = r(t) ,\circledast, w(t)$

where:

• r(t) is the reflectivity series: a sequence of spikes, one at each subsurface boundary, with amplitude equal to that boundary’s reflection coefficient R₀.
• w(t) is the wavelet: the impulse response of the seismic source (or, more practically, the source plus the propagation effects, often modeled as a Ricker wavelet).
• s(t) is the synthetic trace: what the seismic would look like at this location if the convolutional model were exactly correct.

The convolution operation $\circledast$ in time means: at each output time, sum up wavelet copies centered at every reflector, scaled by that reflector’s R₀. Bright reflections produce bright wavelets; weak reflections produce small wavelets. The whole trace is the superposition.

Building a synthetic from a well log

The standard well-tie procedure:

• Read Vp, Vs, and ρ from the well logs (sonic, density). At each depth, compute the acoustic impedance Z = ρ·Vp.
• Compute the reflectivity series at each log-sample boundary using R₀ = (Z₂ − Z₁) / (Z₂ + Z₁). The result is a long, dense reflectivity sequence in DEPTH.
• Convert depth to two-way time using the well’s velocity profile (the same Vp data, integrated). The reflectivity becomes a TIME series.
• Estimate or extract a wavelet from the seismic. Typical approaches: Ricker wavelet at the survey’s peak frequency, or a more sophisticated Wiener-extracted wavelet from a window of seismic near the well.
• Convolve reflectivity × wavelet → synthetic trace.
• Display the synthetic next to the recorded seismic at the well location. They should look similar; the differences highlight where the convolutional model breaks down (typically thin beds, strong noise, or bad wavelet estimates).

The result is the seismic-to-well tie — a single side-by-side display of synthetic vs recorded that lets the interpreter calibrate everything that follows: which seismic event corresponds to which formation, what polarity convention applies, what the wavelet looks like at this part of the survey, how much amplitude scaling to apply for inversion.

Exercise — read the synthetic

• The widget defaults to a brine-sand / gas-sand stack. Look at the four columns: layered earth on the left, then reflectivity spikes (red for positive, blue for negative), then the wavelet, then the synthetic trace. The synthetic is just the convolution of the spikes with the wavelet.
• Notice the gas-sand boundaries produce much bigger reflectivity spikes than the brine-sand boundaries (compare the two pairs of red+blue spikes at different depths). After convolution, the gas-sand reflectors produce visibly brighter peaks and troughs in the synthetic. This is the bright-spot DHI signature — quantified by rock physics, made visible by the synthetic.
• Slide the wavelet frequency from 30 Hz down to 12 Hz. The wavelet broadens; reflectors that were resolved as separate events at 30 Hz now merge into composite peaks. This is tuning (§1.7) made visible. At 12 Hz, the brine sand and gas sand are barely distinguishable on the trace; at 30 Hz they’re obvious.
• Slide the frequency up to 60 Hz. The wavelet sharpens; you can now see the top and base of each thin sand as separate events, where at lower frequencies they were tuned together. This is why high-frequency surveys cost more but resolve thinner reservoirs.
• Switch to the Carbonate atop shale preset. A single strong positive peak appears at the carbonate top; the magnitude is much larger than any of the sand-sequence reflections. Carbonates are easy to identify on stack precisely because their impedance contrasts are large.
• Try the Thin-bed tuning demo: two thin gas sands separated by a thin shale. At low frequencies they merge into one composite reflection; at high frequencies they resolve as three distinct reflections. The transition between resolution regimes happens around the wavelength = bed thickness / 4 limit (§1.7).
• Try the Salt over reservoir preset. The salt boundaries dominate; the underlying gas-sand reflection is much smaller in comparison. This illustrates the practical challenge of imaging beneath salt: the strong salt energy obscures the weaker reservoir reflections beneath, and special processing techniques (sub-salt imaging) are needed to recover them.

Inversion: the reverse direction

The synthetic seismogram is forward modeling: rock properties → reflectivity → convolve with wavelet → trace. Seismic inversion is the reverse: trace → deconvolve with wavelet → reflectivity → invert for rock properties (specifically, acoustic impedance).

The basic recipe:

• Estimate the wavelet (from a well tie, or by spectral analysis).
• Deconvolve the seismic trace by the wavelet → reflectivity series $\hat r(t)$.
• Integrate the reflectivity to recover acoustic impedance: $Z(t) = Z(0) \cdot \exp!\left(2 \int_0^t \hat r(\tau),d\tau\right)$. This needs an initial Z(0) value, typically anchored to a well.
• The result is an acoustic-impedance volume — a 3D map of the rock’s Z at every voxel.

Acoustic-impedance volumes are widely used for reservoir characterization because Z is a property of the rock, not of the seismic wavelet — it is the same for all surveys at the same location, and it correlates more directly with porosity and lithology than any single amplitude attribute.

Pre-stack inversion extends this to angle-stack data, recovering Vp, Vs, and ρ separately (rather than just Z). The output is a full set of elastic-property volumes, which then feed Gassmann substitution, AVO classification, and the LMR crossplots of §5.5. Pre-stack inversion is the most complete deliverable of modern QI workflows; it is also the most expensive in computation, processing care, and well-control requirements.

Common synthetic-seismogram pitfalls

• Wavelet estimation matters. The synthetic is only as good as the wavelet estimate. A wrong wavelet produces a synthetic that looks vaguely right but doesn’t match the seismic in detail — leading to systematic errors in subsequent inversion. The seismic-to-well tie itself helps refine the wavelet estimate iteratively.
• Time-depth conversion uncertainty. Well logs are in depth; seismic is in two-way time. The conversion uses the well’s sonic velocity, which has its own measurement noise. Small errors propagate into the seismic-to-well tie.
• Multiples and other non-primary energy. The convolutional model assumes only primary reflections. Real seismic has multiples (waves bouncing off the same boundary multiple times), surface waves, and other coherent noise. The synthetic doesn’t include these; the recorded seismic does. The mismatch is real and unavoidable.
• Anisotropy. The convolutional model assumes vertical incidence. Real seismic has incidence from many angles, and anisotropic shales (Vp varies with angle) distort the simple R₀ picture.
• The wavelet is rarely zero-phase. Recorded seismic wavelets often have residual phase from acquisition and processing. The synthetic uses an idealized zero-phase Ricker; the real wavelet may be minimum-phase or mixed-phase. Phase mismatches need correction before the tie can be quantitative.

Closing Part 5: the QI workflow recap

Putting all six sections together, here is the canonical quantitative-interpretation workflow:

• §5.1 — Acoustic impedance and reflection coefficient. Z·Vp·ρ and R₀ = (Z₂−Z₁)/(Z₂+Z₁). The minimum elastic description of a boundary.
• §5.2 — Moduli and Vp/Vs. K and μ determine velocities; Vp/Vs is the lithology/fluid diagnostic; Castagna’s mudrock line provides the brine-saturated reference.
• §5.3 — Gassmann fluid substitution. Predict how velocities change when pore fluid changes. Distinguish brine, oil, and gas at the same lithology.
• §5.4 — AVO basics. Aki-Richards approximation extracts (R₀, G) from angle-dependent reflection coefficients; G is the fluid-sensitive gradient.
• §5.5 — (R₀, G) crossplot and AVO classification. Background trend + class regions; anomaly polygons identify candidate prospects.
• §5.6 — Synthetic seismograms close the loop. Forward model from rock properties through convolution; calibrate against well control; use inversion in the reverse direction to recover rock properties from seismic.

Together with the structural toolkit (Part 2) and the attribute toolkit (Part 6), Part 5 completes the foundational quantitative interpretation framework. Most professional QI deliverables — reservoir-property volumes, fluid maps, prospect-ranking reports — use exactly this chain at some step. The math is now in your hands; the rest is calibration to specific basins and judgment about specific prospects.