## FWI and the "noise" quandary

### Laurence R. Lines

Full waveform inversion (FWI) has been termed the most general inversion method in which we attempt to model every digital sample in the seismic trace by adjusting model parameters so that the discrepancies between data and model response are minimized -often in a least squares sense. FWI attempts to model every data sample, even though the data values be signal contaminated with "noise". This begs the question. What is noise? This paper attempts to define "noise" and how it might affect FWI.

If we consider noise as that part of the seismic data that is not related to physical properties of the Earth's interior, then noise could be due to wind, waves, traffic, instrument effects and power lines. For cases where this type of noise is random, the least-squares implementation of FWI is robust.

Here the synthetic data has signal-to-noise (S/N) ratios of infinity (no noise), S/N=2.5, S/N=5.0. While the error of fit in each case worsens with increasing noise, the inversion estimate for seismic-Q converges to the correct solution in three iterations for each case.

If we considered multiples to be signal that can be inverted, then least-squares inversion is shown to provide reliable estimates of reflection coefficients through fitting the multiple energy. That is, through FWI, multiple energy can be used to estimate the Earth's reflectivity. Examples show FWI of multiples can be used to estimate reflectivity.