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This paper considers power allocation problem in wireless sensor networks where distributed sensors amplify and forward their observations of a Gaussian random source to a remote fusion center (FC) which reconstructs the underlying source. The sensor networks are characterized by the availability of limited energy. Motivated by this fact, we design a power allocation scheme where our objective is to minimize the network power consumption such that the reconstruction distortion does not exceed a target value. The reconstruction distortion is quantified based on linear minimum mean-squared error (LMMSE) estimation rule. For power allocation, we propose a novel design based on successive approximation of the distortion function. The resulting algorithm turns out to be simple, computationally efficient and exhibits good convergence properties. The design is based on perfect knowledge of fading channel gains. We also address the case where only estimates of the channel gains are available. The simulation examples illustrate that the proposed design holds considerable performance gain compared to a uniform power allocation scheme.