Paper IV extends the work of Paper III. Here, not only the significant wave height, Hs, but also the wave period, T , is modeled spatially. It turns out that
also the data of log T is explained well by a Gaussian random field. Hence, the univariate spatial models of Hs and T independently are equivalent to that
of Paper III. That is, both log Hs and log T independently can be modeled
as solutions to the same class of SPDEs, but with different parameter values. Here, the FEM implementation is extended to allow for arbitrary smoothness as well as considering the spatial domain to be on the sphere instead of on the plane of longitude-latitude projections. That is, the mesh is created on a subset of the sphere.
7.4. Paper IV: Joint spatial modeling of significant wave height and wave
period using SPDEs 65
Paper IV also introduces a bivariate random field model of log Hs and
log T jointly. The model is constructed such that the marginal distribution of the two, univariate, random fields independently is equivalent to the model of Paper III. The dependency structure between the two random fields are introduced by a system of coupled SPDEs (Bolin and Wallin, 2018; Hu and Steinsland, 2016), p 1 + ρ2Lα/2 X X − ρL β/2 Y Y = W Lβ/2Y Y = V.
Here, X(s) = log Hs(s) and Y (s) = log T (s). The differential operators,
LX and LY, are both of the class defined in Equation (7.1). The two spa-
tial Wiener noise fields, W and V, are identically distributed and independent. The parameter ρ explains the cross-correlation structure between the two ran- dom fields, X and Y . Since ρ is allowed to be spatially varying, it allows for a flexible bivariate random field model. It should be noted that the parameter ρ is not identical to the pointwise cross-correlation between the two fields. How- ever, it is related and a negative ρ corresponds to a negative cross-correlation and vice versa.
Just as for the model of Paper III, the bivariate model can be approxi- mated by the finite element method. This gives the same important beneficial properties as in the standard SPDE approach (Lindgren et al., 2011). Fig- ure 7.6 shows a realization of such a bivariate random field with a negative cross-correlation, and with different anisotropy structures and smoothness in the two fields. As can be seen, even though the two realizations have quite different structure, for instance elongated in directions perpendicular to each other, the regions with high values in one of them tend to be regions with low values in the other one. This is an effect of ρ < 0.
The univariate models for Hs and T independently were shown to agree
well with data of the north Atlantic during April month in the years 1979- 2018. For the joint model, some degree of model-misspecification is present in the cross-correlation structure. This seems to be largely due to the dynamic nature of the sea states in space and time. The spatial points at which the maximum cross-correlation is reached between Hs and T are not aligned in
space. Since the model of cross-correlation structure assumes an alignment between points of maximum cross-correlation, the ML estimates of the cross- correlation structure do not yield agreement with data. However, by fitting the cross-correlation structure as to explain the pointwise cross-correlation, the fitted model could explain the joint distribution relatively well.
Figure 7.6: Realization of a bivariate, anisotropic and stationary Gaussian random field. The left field has a correlation range of 25 in the direction of the principal axis at 45◦ and a correlation range of 14 in the perpendicular direction. The right field has the principal direction at an angle of −45◦with the correlation range 30, the perpendicular direction has a range of 15. The correlation between the fields are controlled by ρ = −5.
The fitted model was used to reevaluate the fatigue damage analysis of Paper III. In Paper III, data for T was not available, instead, the fatigue damage was computed using only Hs. By using a formula dependent on
both Hs and T , a better approximation of true fatigue damage is acquired.
In Paper IV, it is shown that the joint model explains the fatigue damage distribution better than just using Hs. However, similar results could be
achieved by considering a univariate model of Hs and pointwise conditional
means, T |Hs.
Also, a method for analyzing the risk of capsizing due to broaching-to was investigated. In the method, the intensity of encountering a “dangerous” wave is derived as a function of the sea state and the ship path. Furthermore, the risk of capsizing given a “dangerous” wave is evaluated as a log-linear function of Hs, T , and constants associated with the ship design. Putting these two
components together, the risk of a capsizing event due to broaching-to for a ship traversing a route can be evaluated.
The risk of capsizing due to broaching-to is evaluated for a fictitious ship and the route of Figure 7.5. The risk is computed, both from the available data and from the spatial model. The distribution of capsize-risk using the bivariate spatial model compare well with the data.