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In our previous work, we developed field reconstruction methods in wireless sensor networks based on shift-invariant spaces. In this paper, we use amplify-and-forward for the transmission of the sensor node measurements to the fusion center and we derive the mean square error (MSE) of the reconstructed field as a function of the measurement noise, the channel gains between sensor nodes and fusion center, the receiver noise variance, and the sensor placement. Imposing a sum-power constraint, we formulate the MSE-optimal power allocation as a convex optimization problem that can be solved numerically. For the case of critical sampling we derive a closed-form expression for the optimal power allocation. For Gaussian channels and Rayleigh-fading channels, we compare the performance of the proposed power allocation schemes and uniform power allocation. The power allocation schemes provide new insights for our field reconstruction scheme and feature excellent performance while being easy to implement.