  Many records in environmental sciences exhibit asymmetric trajectories and
there is a need for simple and tractable models which can reproduce such
features. In this paper we explore an approach based on applying both a time
change and a marginal transformation on Gaussian processes. The main
originality of the proposed model is that the time change depends on the
observed trajectory. We first show that the proposed model is stationary and
ergodic and provide an explicit characterization of the stationary
distribution. This result is then used to build both parametric and
non-parametric estimates of the time change function whereas the estimation of
the marginal transformation is based on up-crossings. Simulation results are
provided to assess the quality of the estimates. The model is applied to
shallow water wave data and it is shown that the fitted model is able to
reproduce important statistics of the data such as its spectrum and marginal
distribution which are important quantities for practical applications. An
important benefit of the proposed model is its ability to reproduce the
observed asymmetries between the crest and the troughs and between the front
and the back of the waves by accelerating the chronometer in the crests and in
the front of the waves.
