We consider a wireless sensor network equipped with energy harvesting technology. It contains M sensors that observe a random process and transmit an amplified uncoded analog version of the observed signal through fading wireless channels to a remote station. The remote station, often called the fusion center, estimates the realization of the random process by using a best linear unbiased estimator. In this paper, we consider the optimal energy allocation policy that minimizes total distortion over a finite time horizon subject to energy harvesting constraints at the sensors. We focus on two types of available side information at the sensor, i.e. (1) causal side information involving the present and previous channel states and the previous values of the harvested energy and (2) full (non-causal) side information, under both finite and infinite energy storage capacity at each sensor's battery. The derivations and some structural properties of the optimal energy allocation schemes are discussed, and numerical results presented.