  We describe a new method, full waveform inversion by model extension (FWIME)
that recovers accurate acoustic subsurface velocity models from seismic data,
when conventional methods fail. We leverage the advantageous convergence
properties of wave-equation migration velocity analysis (WEMVA) with the
accuracy and high-resolution nature of acoustic full waveform inversion (FWI)
by combining them into a robust mathematically-consistent workflow with minimal
need for user inputs. The novelty of FWIME resides in the design of a new cost
function using the variable projection method, and a novel optimization
strategy to combine the two techniques, making our approach more efficient and
powerful than applying them sequentially. We observe that FWIME mitigates the
need for accurate initial models and low-frequency long-offset data, which can
be challenging to acquire. We generate three cycle-skipped 2D synthetic
datasets, each containing only one type of wave (transmitted, reflected,
refracted), and we analyze how FWIME successfully recovers accurate solutions
with the same procedure for all three cases. In a second paper, we apply FWIME
to challenging realistic examples where we simultaneously invert all wave
modes.
