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We give a factorization procedure for a strictly hyperbolic partialdifferential operator of second order with logarithmic slow scale coefficients.From this we can microlocally diagonalize the full wave operator which resultsin a coupled system of two first-order pseudodifferential equations in amicrolocal sense. Under the assumption that the full wave equation ismicrolocal regular in a fixed domain of the phase space, we can approximate theproblem by two one-way wave equations where a dissipative term is added tosuppress singularities outside the given domain. We obtain well-posedness ofthe corresponding Cauchy problem for the approximated one-way wave equationwith a dissipative term.

Factorization of second-order strictly hyperbolic operators with logarithmic slow scale coefficients and generalized microlocal approximations