We consider random extended surface perturbations in the transverse field Ising model decaying as a power of the distance from the surface towards a pure bulk system. The decay may be linked either to the evolution of the couplings or to their probabilities. Using scaling arguments, we develop a relevance-irrelevance criterion for such perturbations. We study the probability distribution of the surface magnetization, its average and typical critical behaviour for marginal and relevant perturbations. According to analytical results, the surface magnetization follows a log-normal distribution and both the average and typical critical behaviours are characterized by power-law singularities with continuously varying exponents in the marginal case and essential singularities in the relevant case. For enhanced average local couplings, the transition becomes first order with a nonvanishing critical surface magnetization. This occurs above a positive threshold value of the perturbation amplitude in the marginal case.
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