We develop a localised wavelet formulation of multifractal random walk unwrapping based on the local multiplicative modulation freezing. The framework is motivated by the observation that finite-support wavelet localisation may induce approximate local factorisation of multiplicatively modulated stochastic fields, allowing the modulation component to become effectively frozen within sufficiently localised probing domains. Within this regime, logarithmic wavelet amplitudes admit an approximate additive decomposition linking local wavelet statistics directly to the underlying modulation field. This viewpoint reformulates covariance-based MRW unwrapping as a localised multiscale operator problem in which wavelet coefficients act as finite-support probes of multiplicative organisation. The validity of the approximation depends explicitly on support geometry, scale-dependent overlap, and residual multiscale mixing generated by internal modulation variability. We show that these effects naturally produce finite-scale deviations from ideal logarithmic covariance scaling and lead to structured covariance distortions whose form depends on the interaction between the modulation field and the geometry of the wavelet representation. In the resulting framework, localisation itself becomes the operational mechanism enabling multiscale probing of local stochastic organisation. Numerical investigations using orthonormal wavelet decompositions support the proposed interpretation and demonstrate the emergence of scale-dependent freezing regimes, residual covariance mixing, and finite-support breakdown effects consistent with the theory. The proposed framework suggests a broader connection between wavelet localisation, local regularity organisation, and finite-support multiscale stochastic operators. Wavelet localisation becomes an operational mechanism for probing localised multiscale structure.

翻译：我们提出了一种基于局部乘性调制冻结的多重分形随机游走展开的局部化小波方法。该框架的动机在于：有限支撑的小波局部化可导致乘性调制随机场近似局部分解，使得调制分量在充分局部化的探测域内有效冻结。在此条件下，对数小波振幅允许近似加性分解，从而将局部小波统计量与底层调制场直接关联。这一观点将基于协方差的MRW展开重新表述为局部化多尺度算子问题，其中小波系数作为乘性组织结构的有限支撑探针。近似的有效性明确依赖于支撑几何结构、尺度相关重叠以及由内部调制变异性产生的残余多尺度混叠。我们证明，这些效应自然产生有限尺度下理想对数协方差标度的偏差，并导致结构化的协方差畸变，其形式取决于调制场与小波表示几何结构之间的相互作用。在所提出框架中，局部化本身成为探测局部随机组织结构的操作性机制。使用正交小波分解的数值实验支持该解释，并展示了与理论一致的尺度相关冻结区域、残余协方差混叠及有限支撑破坏效应的出现。所提框架揭示了小波局部化、局部正则性组织与有限支撑多尺度随机算子之间的深层联系——小波局部化成为探测局部化多尺度结构的操作机制。