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Simulating Only What Changes: How to Recompute Elastic Wavefields Without Recomputing the Whole World

Wave simulations show up everywhere: earthquake science, seismic imaging, non-destructive testing, and medical ultrasound. The goal is simple: you specify a source (an earthquake, a pulse, a transducer) and a model of the material, and you compute how waves travel, scatter, and arrive at sensors.

The hard part is that realistic models are big, and realistic wave physics is expensive. Even worse, many workflows need not one simulation, but hundreds or thousands of very similar simulations. In imaging and inversion, for example, you keep updating the model—often changing only a small target zone each time. Running a full-domain simulation every time is like re-rendering an entire movie scene because you moved one prop.

This blog explains the idea behind immersive wavefield modelling: a way to recompute wavefields after a local model change by simulating only a local region, while still reproducing the same answer you would get from a full-domain simulation (including long-range interactions).

Two key references:

The promise is:

Schematic of immersive boundary conditions showing a local simulation box, a recording surface, and the surrounding full model.
Figure 1 — The “small box inside a big world” idea
We only simulate waves inside the local box (red dashed line). A second surface (blue dashed line) is where we record what the outside world would do. Using precomputed responses (think: “how waves travel from blue to red”), the boundary can inject the correct incoming waves and absorb outgoing ones—so the small simulation behaves like it is still embedded in the full model.

1) Why repeated wave simulations become painful

Many wave-modelling tasks are iterative by nature. Full waveform inversion (FWI), survey design, uncertainty studies, and time-lapse analysis all demand many wave simulations. Even if each update is small, rerunning a global simulation recomputes propagation everywhere—again and again.

If most of the model stays unchanged, it is natural to ask: can we simulate only the part that changes—without losing the influence of the unchanged surroundings?

2) Conventional local replay: what it gets right—and what it can miss

A standard local strategy is:

  1. Run a global simulation once.
  2. Record wavefields on a closed surface around a target zone.
  3. Change the model inside that surface.
  4. Re-inject the recorded wavefield to reproduce the “incoming” energy inside the target zone.

This is useful, but it can miss an important effect: waves scattered by the modified target can travel into the exterior, reflect/scatter from distant structures, and then come back—sometimes multiple times. One-shot replay does not fully reproduce these higher-order, long-range interactions.

3) The core idea: Immersive Boundary Conditions (IBCs)

Immersive Boundary Conditions make a small local simulation behave as if it were still sitting inside the full model. The boundary is active: it absorbs waves leaving the local box (to avoid artificial reflections) and it emits the correct waves coming in from the unchanged exterior. Because this happens continuously in time, the local region remains two-way coupled to the outside world.

In practice, the boundary updates rely on a library of precomputed wave responses in the unchanged background model (often described using Green’s functions). You compute that library once, then reuse it across many local reruns.

4) Two implementation routes: FD injection (exact) vs MPS (efficient)

Once you accept the IBC concept, the practical question becomes: how do we record and inject wavefields on a finite-difference grid without adding numerical artefacts?

4.1 Finite-difference injection (FD injection): “exact by construction”

FD injection makes the boundary consistent with the discrete finite-difference solver itself. That is why, for acoustic waves, it can match a full-domain finite-difference run down to machine precision.

Strength: extremely accurate (often machine-precision agreement).
Trade-off: more bookkeeping and overhead, especially for higher-order finite-difference stencils.

Reference: van Manen et al. (2020), GJI, DOI: 10.1093/gji/ggaa317.

Acoustic finite-difference models used in boundary and immersion examples.
Figure 2 — Example acoustic models (what the computer ‘thinks’ the world looks like)
These are simple test worlds used to check the boundary method. The patterns (dots/objects/lines) act like strong scatterers, so waves bounce and scatter in many directions. This is a tough test: a good immersive boundary must still handle all that scattering without creating fake reflections at the edge of the local box.
Acoustic wavefield snapshots comparing immersive modelling to a full-domain reference, including difference panels.
Figure 3 — Immersive vs full simulation (and the tiny difference)
The snapshots compare an immersive simulation (local box + IBCs) against a full-domain reference run. The “difference” panels are scaled up heavily; without that exaggeration, the mismatch is essentially invisible. This is the key point: the local simulation can reproduce what the full simulation would have done, while only computing a small region.

4.2 Method of Multiple Point Sources (MPS): “physics-like and cheaper” (elastic case)

For elastic waves (P-waves and S-waves), simulations track several coupled wave components. Many elastic solvers use a staggered grid (different components live at slightly different grid locations), which makes clean recording/injection harder—especially near corners.

MPS represents the boundary effect using many point sources distributed around the boundary. In practice it can be much lighter than FD injection, especially as finite-difference stencils become higher order.

Reference: Li et al. (2022), JCP, DOI: 10.1016/j.jcp.2021.110826.

Elastic full model and local subdomain for immersive boundary condition tests, showing recording and emitting surfaces.
Figure 4 — Elastic test setup: full model vs local model
The full model contains scatterers (green) that reflect and redirect waves. The immersive simulation only runs inside the local box, but it is connected to the outside world via two surfaces: a recording surface (where we measure what the exterior would do) and an emitting boundary (where we inject boundary sources so the local run stays fully coupled to the exterior).
Elastic wavefield snapshots comparing reference simulation with immersive simulations using FD injection and MPS, including differences.
Figure 5 — Elastic results: two boundary implementations, one target answer
The first column is the full-domain reference wavefield. The next columns show immersive simulations using FD injection and MPS. The difference panels are scaled up strongly to make any mismatch visible. The main takeaway is that both approaches can reproduce the full-domain behaviour inside the local box—while MPS can be cheaper to run and store.

5) What the workflow looks like in practice

Immersive modelling is most valuable when you need many reruns where only a small region changes. A typical workflow is:

  1. Pick a target zone: draw a box around the part you expect to modify.
  2. Precompute exterior responses (once): in the unchanged background model, compute the information needed to update the boundary through time.
  3. Run many local simulations: for each update, simulate only the local box while the boundary supplies the “outside world.”
  4. Extract outputs: inside the local region (and at receivers), you obtain the same result as a full-domain run.

6) When is immersive modelling worth it?

Immersive modelling tends to pay off when:

It is less attractive when:

7) Why this matters (beyond one method)

The broader point is not just a new boundary condition. It’s a way of thinking about wave simulation that matches how many problems are structured: a large environment with a small region of interest that changes.

References