Wavelet localisation and local modulation freezing in multifractal random walk unwrapping

Abstract Multifractal random walk (MRW) models describe intermittent multiscale fluctuations through multiplicative modulation, but their conventional unwrapping is formulated primarily through global logarithmic covariance relations. We develop a localised wavelet formulation in which finite-support wavelet coefficients act as local probes of the multiplicative modulation field. The central approximation is that, when the wavelet support is sufficiently small relative to the local variability of the modulation, the modulation can be treated as locally frozen, yielding an approximate factorisation of the wavelet coefficients—an approximate additive relation between the logarithmic wavelet amplitude, the local modulation, and the residual fluctuation. This provides a localised operator formulation of MRW unwrapping and gives a direct interpretation of the scale range over which the approximation is valid. We further analyse the finite-scale deviations produced by residual modulation variability within the wavelet support and show how these lead to covariance mixing and departures from ideal logarithmic scaling. Numerical experiments with orthonormal wavelet decompositions support the predicted transition between fine-scale freezing and coarse-scale mixing and demonstrate the resulting scale-dependent reconstruction behaviour. The framework therefore provides a local perspective on multiplicative modulation extraction in which wavelet localisation is not merely a representation, but the mechanism enabling local multiscale probing of that modulation.