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A {\it scenery} is a coloring $\xi$ of the integers. Let $\{S_t\}_{t\geq 0}$be a recurrent random walk on the integers. Observing the scenery $\xi$ alongthe path of this random walk, one sees the color $\chi_t:=\xi(S_t)$ at time$t$. The {\it scenery reconstruction problem} is concerned with recovering thescenery $\xi$, given only the sequence of observations $\chi:=(\chi_t)_{t\geq0}$. The scenery reconstruction methods presented to date require the randomwalk to have bounded increments. Here, we present a new approach for randomwalks with unbounded increments which works when the tail of the incrementdistribution decays exponentially fast enough and the scenery has five colors.

Information recovery from observations by a random walk having jump distribution with exponential tails