  We consider the problem of describing excursion sets of a real-valued
function $f$, i.e. the set of inputs where $f$ is above a fixed threshold. Such
regions are hard to visualize if the input space dimension, $d$, is higher than
2. For a given projection matrix from the input space to a lower dimensional
(usually $1,2$) subspace, we introduce profile sup (inf) functions that
associate to each point in the projection's image the sup (inf) of the function
constrained over the pre-image of this point by the considered projection.
Plots of profile extrema functions convey a simple, although intrinsically
partial, visualization of the set. We consider expensive to evaluate functions
where only a very limited number of evaluations, $n$, is available, e.g.
$n<100d$, and we surrogate $f$ with a posterior quantity of a Gaussian process
(GP) model. We first compute profile extrema functions for the posterior mean
given $n$ evaluations of $f$. We quantify the uncertainty on such estimates by
studying the distribution of GP profile extrema with posterior
quasi-realizations obtained from an approximating process. We control such
approximation with a bound inherited from the Borell-TIS inequality. The
technique is applied to analytical functions ($d=2,3$) and to a $5$-dimensional
coastal flooding test case for a site located on the Atlantic French coast.
Here $f$ is a numerical model returning the area of flooded surface in the
coastal region given some offshore conditions. Profile extrema functions
allowed us to better understand which offshore conditions impact large flooding
events.
