Deformed SPDE models with an application to spatial modeling of significant wave height

A non-stationary Gaussian random field model is developed based on a combination of the stochastic partial differential equation (SPDE) approach and the classical deformation method. With the deformation method, a stationary field is defined on a domain which is deformed so that the field becomes non-stationary. We show that if the stationary field is a Mat'ern field defined as a solution to a fractional SPDE, the resulting non-stationary model can be represented as the solution to another fractional SPDE on the deformed domain. By defining the model in this way, the computational advantages of the SPDE approach can be combined with the deformation method's more intuitive parameterisation of non-stationarity. In particular it allows for independent control over the non-stationary practical correlation range and the variance, which has not been possible with previously proposed non-stationary SPDE models. The model is tested on spatial data of significant wave height, a characteristic of ocean surface conditions which is important when estimating the wear and risks associated with a planned journey of a ship. The model parameters are estimated to data from the north Atlantic using a maximum likelihood approach. The fitted model is used to compute wave height exceedance probabilities and the distribution of accumulated fatigue damage for ships traveling a popular shipping route. The model results agree well with the data, indicating that the model could be used for route optimization in naval logistics.Comment: 22 pages, 12 figure

Crossings of Second-order Response Processes Subjected to LMA Loadings

The focus of this paper is on the estimation of the crossing intensities of responses for second order dynamical systems, subjected to stationary, non-Gaussian external loadings. A new model for random loadings – the Laplace driven moving average (LMA) – is used. The model is non-Gaussian, strictly stationary, can model any spectrum and has additional flexibility to model the skewness and kurtosis of the marginal distribution. The system response can be expressed as a second-order combination of the LMA processes. A numerical technique for estimating the level crossing intensities for such processes is developed. The proposed method is a hybrid method which combines the saddle-point approximation with limited Monte Carlo simulations. The performance and the accuracy of the proposed method is illustrated through a set of numerical examples

Joint spatial modeling of significant wave height and wave period using the SPDE approach

The ocean wave distribution in a specific region of space and time is described by its sea state. Knowledge about the sea states a ship encounters on a journey can be used to assess various parameters of risk and wear associated with this journey. Two important characteristics of the sea state are significant wave height and mean wave period. We propose a joint spatial model of these two quantities on the north Atlantic ocean. The model describes the distribution of the logarithm of the two quantities as a bivariate Gaussian random field, modeled as a solution to a system of coupled fractional stochastic partial differential equations. The bivariate random field is non-stationary and allows for arbitrary, and different, smoothness for the two marginal fields. The parameters of the model are estimated from data using a stepwise maximum likelihood method. The fitted model is used to derive the distribution of accumulated fatigue damage for a ship sailing a transatlantic route. Also, a method for estimating the risk of capsizing due to broaching-to based on the joint distribution of the two sea state characteristics is investigated. The risks are calculated for a transatlantic route between America and Europe using both data and the fitted model. The results show that the model compares well with observed data. It further shows that the bivariate model is needed and cannot simply be approximated by a model of significant wave height alone

Load Description and Damage Evaluation Using Vehicle Independent Driving Events

AbstractWe consider the loads that are related to steering events, and focus on the events that cause high forces on steering components. The load is simplified by keeping the extreme force value for each driving event. We define a simplified stochastic model for the load by modeling the extreme value for each driving event by a random variable. We give formulas to compute the theoretical load spectrum and the expected fatigue damage caused by the driving events. Further, in a sensitivity study we investigate how much the expected damage depends on the variability of parameters of the proposed model

Palm distributions of wave characteristics in encountering seas

Distributions of wave characteristics of ocean waves, such as wave slope, waveheight or wavelength, are an important tool in a variety of oceanographic applications such as safety of ocean structures or in the study of ship stability, as will be the focus in this paper. We derive Palm distributions of several wave characteristics that can be related to steepness of waves for two different cases, namely for waves observed along a line at a fixed time point and for waves encountering a ship sailing on the ocean. The relation between the distributions obtained in the two cases is also given physical interpretation in terms of a ``Doppler shift'' that is related to the velocity of the ship and the velocities of the individual waves.Comment: Published in at http://dx.doi.org/10.1214/07-AAP480 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spatio-temporal modelling of wind speed variations and extremes in the Caribbean and the Gulf of Mexico

The wind speed variability in the North Atlantic has been successfully modelled using a spatio-temporal transformedGaussian field. However, this type of model does not correctly describe the extreme wind speeds attributed to tropicalstorms and hurricanes. In this study, the transformed Gaussian model is further developed to include the occurrence ofsevere storms. In this new model, random components are added to the transformed Gaussian field to model rare events withextreme wind speeds. The resulting random field is locally stationary and homogeneous. The localized dependence structureis described by time- and space-dependent parameters. The parameters have a natural physical interpretation. To exemplifyits application, the model is fitted to the ECMWF ERA-Interim reanalysis data set. The model is applied to compute longtermwind speed distributions and return values, e.g., 100- or 1000-year extreme wind speeds, and to simulate random windspeed time series at a fixed location or spatio-temporal wind fields around that location

Safety of intravenous ferric carboxymaltose versus oral iron in patients with nondialysis-dependent CKD: an analysis of the 1-year FIND-CKD trial.

Background: The evidence base regarding the safety of intravenous (IV) iron therapy in patients with chronic kidney disease (CKD) is incomplete and largely based on small studies of relatively short duration. Methods: FIND-CKD (ClinicalTrials.gov number NCT00994318) was a 1-year, open-label, multicenter, prospective study of patients with nondialysis-dependent CKD, anemia and iron deficiency randomized (1:1:2) to IV ferric carboxymaltose (FCM), targeting higher (400-600 µg/L) or lower (100-200 µg/L) ferritin, or oral iron. A post hoc analysis of adverse event rates per 100 patient-years was performed to assess the safety of FCM versus oral iron over an extended period. Results: The safety population included 616 patients. The incidence of one or more adverse events was 91.0, 100.0 and 105.0 per 100 patient-years in the high ferritin FCM, low ferritin FCM and oral iron groups, respectively. The incidence of adverse events with a suspected relation to study drug was 15.9, 17.8 and 36.7 per 100 patient-years in the three groups; for serious adverse events, the incidence was 28.2, 27.9 and 24.3 per 100 patient-years. The incidence of cardiac disorders and infections was similar between groups. At least one ferritin level ≥800 µg/L occurred in 26.6% of high ferritin FCM patients, with no associated increase in adverse events. No patient with ferritin ≥800 µg/L discontinued the study drug due to adverse events. Estimated glomerular filtration rate remained the stable in all groups. Conclusions: These results further support the conclusion that correction of iron deficiency anemia with IV FCM is safe in patients with nondialysis-dependent CKD