When Convolution Is Enough

The Boundary of the Model

Part 2 built the convolutional model from impedance to a noisy well tie, and it is a genuinely powerful tool: fast, transparent, and enough for a great deal of real work. But a good modeller knows a tool by its limits, and convolution has a sharp one. It produces zero-offset primaries and nothing else. Four kinds of physics live outside it entirely.

• Lateral wave propagation: diffractions off edges, the bowtie under a syncline. Convolution has no coupling between traces.
• Multiples: energy that reflects more than once. A reflectivity series reflects each interface exactly once.
• Angle dependence (AVO): amplitude that changes with offset. Convolution gives one normal-incidence amplitude.
• Mode conversion: P energy becoming S at a boundary. An acoustic model has no shear.

The Rule, One More Time

Pick a task and the verdict is mechanical: if it needs none of the four, convolution is enough, and reaching higher only spends compute. If it needs even one, a convolutional synthetic would silently omit the very feature you are studying, which is worse than slow, it is wrong in a way that looks fine. Well ties, tuning studies, and fault-detection training sets sit comfortably inside convolution. Diffractions and multiples send you to the wave equation, the next two parts; AVO and mode conversion send you to elastic modelling in Part 6.

This is exactly the fit-for-purpose rule from the modeling ladder in Part 0, now drawn as the boundary of a model you have built yourself. You know what convolution does, what it costs, and precisely what it leaves out. That is the knowledge that lets you choose the cheapest engine that still answers the question, which is the whole craft of synthetic modelling.