Inverse transform sampling

Abstract: At scale lengths less than 100 km or so, statistical descriptions of seafloor morphology can be usefully employed to characterize ridge crest processes, off-ridge tectonics and vulcanism, sedimentation, and postdepositional transport. We seek to develop methods for the estimation of seafloor statistics that take into account the finite precision, resolution, and sampling obtained by actual echo sounding systems. In this initial paper we restrict our attention to the problem of recovering second-order statistics from data sets collected by multibeam devices such as Sea Beam. The seafloor is modeled as a stationary, zero-mean, Gaussian random field completely specified by its two-point covariance function. We introduce an anisotropic two-point covariance function that has five free parameters describing the amplitude, orientation, characteristic wave numbers, and Hausdorff (fractal) dimension of seafloor topography. We formulate the general forward problem relating this model to the statistics of an ideal multibeam echo sounder, in particular the along-track autocovariance functions of individual beams and the cross-covariance functions between beams of arbitrary separation. Using these second moments as data functionals, we then pose the inverse problem of estimating the seafloor parameters from realistic, noisy data sets with finite sampling and beamwidth, and we solve this inverse problem by an iterative, linearized, least squares method. The inversion method is applied to Sea Beam transit data from both the Pacific and Atlantic oceans. Sea Beam system noise stands out as a sharp spike on the along-track autocovariance function and can be modeled as a white noise process whose amplitude generally increases with beam angle. The five parameters in our second-order model can be estimated from the inversion of data sets comprising ∼100–200 km of track length. In general, the cross-track wave number is the most poorly determined, although uncertainties in the assumed Sea Beam response may bias the values of the fractal dimension. Using the assumed beamwidth, the measured noise values, and the seafloor parameters recovered from the inversion, we generate Sea Beam “synthetics” whose statistical character can be directly compared with raw Sea Beam data. For most of the track segments we have processed thus far the synthetics are similar to the data. In the case of one Atlantic profile, however, the comparison clearly indicates the necessity of incorporating higher-order statistics. The space domain procedures described in this paper can be extended for this purpose.

Abstract: SUMMARY We developed a new inversion method to reconstruct static images of seismic sources from geodetic data, using Akaike’s Bayesian Information Criterion (ABIC). Coseismic surface displacements are generally related with a slip distribution on a fault surface by linear integral equations. Parametric expansion of the fault slip distribution by a finite number of known basis functions yields a set of observation equations expressed in a simple vector form. Incorporating prior constraints on the smoothness of slip distribution with the observation equations, we construct a Bayesian model with unknown hyperparameters. The optimal values of the hyperparameters, which control the structure of the Bayesian model, are objectively determined from observed data by using ABIC. Once the values of hyperparameters are determined, we can use the maximum likelihood method to find the optimal distribution of fault slip. We examined the validity of this method through a numerical experiment using theoretical data with random noise. We analysed geodetic data associated with the 1946 Nankaido earthquake (Ms = 8.2) by using this method. The result shows that the fault slip distribution of this earthquake has two main peaks of 4 and 6 m, located off Kii Peninsula and Muroto Promontory. These two high-slip areas are clearly separated by a low-slip zone extending along Kii Strait. Such a slip distribution corresponds with the fact that the rupture process of this earthquake in the western part is notably different from that in the eastern part.

Abstract: The short-term wind power scenarios have a significant impact on the operation cost and power system reliability due to the stochastic generation scheduling of wind-integrated power systems. In order to obtain the scenarios containing the information of forecast error distribution and fluctuation distribution for short-term wind power, a scenario generation method is proposed. This paper characterizes forecast error via empirical distributions of a set of forecast bins and assumes that wind power fluctuations over unit interval follow t location-scale distribution. An inverse transform sampling from a multivariate normal distribution is adopted to generate a large number of wind power scenarios. The covariance matrix of the multivariate normal distribution is estimated to fit the distribution of historical wind power fluctuations. The proposed scenario generation method is applied to the actual aggregate wind power data in the whole regions of Ireland's Power System. The results indicate that the variability of wind power scenarios can be adjusted by estimating the key range parameter in the exponential covariance structure of a multivariate normal distribution.

Abstract: This paper deals with the problem of making inferences about the mean of an exponential distribution when the sample is “time-censored”. The exact sampling distribution of the maximum likelihood estimate is obtained and used to show that the asymptotic sampling theory is inadequate unless the sample size is very large. An approximation to the distribution is proposed for use in small samples and compared with a method suggested by Bartlett (1953a). An alternative estimate is suggested which is both simple and highly efficient in certain circumstances. The methods are illustrated by examples.