  The standard geostatistical problem is to predict the values of a spatially
continuous phenomenon, $S(x)$ say, at locations $x$ using data
$(y_i,x_i):i=1,..,n$ where $y_i$ is the realization at location $x_i$ of
$S(x_i)$, or of a random variable $Y_i$ that is stochastically related to
$S(x_i)$. In this paper we address the inverse problem of predicting the
locations of observed measurements $y$. We discuss how knowledge of the
sampling mechanism can and should inform a prior specification, $\pi(x)$ say,
for the joint distribution of the measurement locations $X = \{x_i:
i=1,...,n\}$, and propose an efficient Metropolis-Hastings algorithm for
drawing samples from the resulting predictive distribution of the missing
elements of $X$. An important feature in many applied settings is that this
predictive distribution is multi-modal, which severely limits the usefulness of
simple summary measures such as the mean or median. We present two simulated
examples to demonstrate the importance of the specification for $\pi(x)$, and
analyze rainfall data from Paran\'a State, Brazil to show how, under additional
assumptions, an empirical of estimate of $\pi(x)$ can be used when no prior
information on the sampling design is available.
