Multi-parameter inversion (vp, vs, rho)

The §6.2–§6.5 sections inverted a single parameter (middle-layer velocity $c_2$) with $c_1$ and $c_3$ frozen. Real seismic FWI inverts MULTIPLE parameters simultaneously: P-wave velocity $v_p$, S-wave velocity $v_s$, density $\rho$. Each combination of ($\Delta v_p$, $\Delta v_s$, $\Delta \rho$) produces a different recorded waveform — but several DIFFERENT combinations produce the SAME recorded waveform. That is the multi-parameter trade-off.

The simplest acoustic example

For 1D acoustic FWI on a layered model with three free velocity layers $(c_1, c_2, c_3)$, the recorded wavefield at a single receiver is

which depends on the LINE INTEGRAL of slowness from source to receiver. Specifically, the first-arrival time at the receiver is

summed over the layers the wave passes through. Two velocity perturbations $(\Delta c_1, \Delta c_2)$ that preserve this sum produce the SAME first-arrival time and very similar (but not identical) waveform amplitudes. The inversion cannot distinguish them.

What the trade-off looks like

The misfit $J(c_1, c_2)$ as a 2D function has a "valley" shape: a long, narrow trough whose AXIS is the direction of indeterminacy and whose SHORT axis is the direction of independent constraint. The valley's tilt encodes the coupling: for our 1D problem the valley tilts roughly along the line

where $\Delta x_1, \Delta x_2$ are the layer thicknesses. The $1/c^2$ factor comes from differentiating the slowness $1/c$ with respect to $c$.

Try it: the 2D misfit landscape

The widget computes $J(c_1, c_2)$ on a 24×24 grid (576 forward solves, ~5 s wall-clock). The truth at $(1.0, 1.5)$ is marked with a green crosshair. Notice the elongated dark valley — that is the cross-talk axis. Any combination of $(c_1, c_2)$ along the dark band fits the data nearly equally well. Multiple local minima may also appear from cycle-skipping (§6.5).

Production remedies

• Multi-shot illumination. Adding a SECOND source at the right edge of the model (transmission from both directions) gives the wave a second line-integral constraint with different $\Delta x_k$ weights. The two ill-posed equations together CAN be solved for both $c_1$ and $c_2$ separately. Production seismic surveys use 100–1000 shots to generate dense ill-posed-but-collectively-well-posed constraints.
• Joint inversion with auxiliary data. Gravity data sees $\rho$ but not $v_p$. MT (magnetotelluric) data sees electrical resistivity. Adding these in a joint-inversion loss provides constraints orthogonal to the seismic-FWI valley. CSEM-FWI joint inversion is standard at major service companies.
• Bayesian priors on parameter ratios. The Vp/Vs ratio is constrained for sedimentary rocks (~1.5–2.0). The Vp–ρ relationship follows Gardner's formula $\rho \approx 0.31 v_p^{0.25}$ for many lithologies. Adding these as soft-constraint terms in the loss damps the inversion away from the valley.
• Impedance reparameterisation. Acoustic impedance $Z = \rho c$ is well-resolved by reflection seismic (it controls reflection coefficients directly). Inverting $Z$ + an independent velocity $c$ instead of $(\rho, c)$ separately removes the canonical Vp–ρ trade-off (Tarantola 1986).

Multi-parameter PINN-FWI

The PINN-FWI version of multi-parameter inversion uses MULTIPLE velocity networks: $v_p^{\mathrm{NN}}(x; \theta_{vp})$, $v_s^{\mathrm{NN}}(x; \theta_{vs})$, $\rho^{\mathrm{NN}}(x; \theta_\rho)$. The PDE residual loss embeds all three parameters; Adam optimises all parameter sets simultaneously. The cross-talk valleys still exist; the same remedies apply. PINN-FWI does have one advantage: it is trivially easy to add ANY auxiliary data term (gravity, MT, sonic logs) as another $L_{\mathrm{aux}}$ in the joint loss — and auto-diff handles the joint gradient. Classical FWI requires custom code per data type.

What §6.7 will do

§6.7 covers a different efficiency idea: SOURCE ENCODING. Instead of running 100 forward solves (one per shot), encode all 100 shots into a single random superposition and run ONE forward solve. The price is noisy intermediate gradients, paid back over many epochs by the $100\times$ compute saving.

References

• Operto, S., Gholami, Y., Prieux, V., Ribodetti, A., Brossier, R., Métivier, L., Virieux, J. (2013). A guided tour of multiparameter full-waveform inversion with multicomponent data: From theory to practice. The Leading Edge 32(9), 1040–1054. The canonical reference for multi-parameter FWI.
• Tarantola, A. (1986). A strategy for nonlinear elastic inversion of seismic reflection data. Geophysics 51(10), 1893–1903. The impedance-reparameterisation paper.
• Gardner, G.H.F., Gardner, L.W., Gregory, A.R. (1974). Formation velocity and density—The diagnostic basics for stratigraphic traps. Geophysics 39(6), 770–780. The Vp–ρ Gardner relation.