Colin Fox

On the Oblique Reflexion and Transmission of Ocean Waves at Shore Fast Sea Ice

A mathematical model is reported describing the oblique reflexion and penetration of ocean waves into shore fast sea ice. The arbitrary depth model allows all velocity potentials occurring in the open water region to be matched precisely to their counterparts in the ice-covered region. Matching is done using a preconditioned conjugate gradient technique which allows the complete solution to be found to a predefined precision. The model enables the reflexion and transmission coefficients at the ice edge to be found, and examples are reported for ice plates of different thicknesses. A critical angle is predicted beyond which no travelling wave penetrates the ice sheet; in this case the deflexion of the ice is due only to evanescent modes. Critical angle curves are provided for various ice thicknesses on deep, intermediate and shallow water. The strain field which is set up within the ice sheet due to the incoming waves is also discussed; principal strains are provided as are the strains normal to the ice edge. Finally the spreading function within the ice cover, and some consequences of this function to unimodal seas with realistic open water spreading functions, are reported with the aim of generalizing the work to model the effect of shore fast ice on an incoming directional wave spectrum of specified structure.

Dynamic, in situ measurement of sea-ice characteristic length

Abstract We present a method for measuring the characteristic length of sea ice based on fitting to a recently found solution for the flexural response of a floating ice sheet subject to localized periodic loading. Unlike previous techniques, the method enables localized measurements at single frequencies of geophysical interest, and since the measurements may be synchronously demodulated, gives excellent rejection of unwanted measurement signal (e.g. from ocean swell). The loading mechanism is described and we discuss how the effective characteristic length may be determined using a range of localized measurements. The method is used to determine the characteristic length of the sea ice in McMurdo Sound, Antarctica.

Markov Chain Monte Carlo Using an Approximation

This article presents a method for generating samples from an unnormalized posterior distribution f(·) using Markov chain Monte Carlo (MCMC) in which the evaluation of f(·) is very difficult or computationally demanding. Commonly, a less computationally demanding, perhaps local, approximation to f(·) is available, say f**x(·). An algorithm is proposed to generate an MCMC that uses such an approximation to calculate acceptance probabilities at each step of a modified Metropolis–Hastings algorithm. Once a proposal is accepted using the approximation, f(·) is calculated with full precision ensuring convergence to the desired distribution. We give sufficient conditions for the algorithm to converge to f(·) and give both theoretical and practical justifications for its usage. Typical applications are in inverse problems using physical data models where computing time is dominated by complex model simulation. We outline Bayesian inference and computing for inverse problems. A stylized example is given of recovering resistor values in a network from electrical measurements made at the boundary. Although this inverse problem has appeared in studies of underground reservoirs, it has primarily been chosen for pedagogical value because model simulation has precisely the same computational structure as a finite element method solution of the complete electrode model used in conductivity imaging, or “electrical impedance tomography.” This example shows a dramatic decrease in CPU time, compared to a standard Metropolis–Hastings algorithm.

The Bayesian Framework for Inverse Problems in Heat Transfer

The aim of this paper is to provide researchers dealing with inverse heat transfer problems a review of the Bayesian approach to inverse problems, the related modeling issues, and the methods that are used to carry out inference. In Bayesian inversion, the aim is not only to obtain a single point estimate for the unknown, but rather to characterize uncertainties in estimates, or predictions. Before any measurements are available, we have some uncertainty in the unknown. After carrying out measurements, the uncertainty has been reduced, and the task is to quantify this uncertainty, and in addition to give plausible suggestions for answers to questions of interest. The focus of this review is on the modeling-related topics in inverse problems in general, and the methods that are used to compute answers to questions. In particular, we build a scene of how to handle and model the unavoidable uncertainties that arise with real physical measurements. In addition to giving a brief review of existing Bayesian treatments of inverse heat transfer problems, we also describe approaches that might be successful with inverse heat transfer problems.

Strain in shore fast ice due to incoming ocean waves and swell

Using a development from the theoretical model presented by Fox and Squire (1990), this paper investigates the strain field generated in shore fast ice by normally incident ocean waves and swell. After a brief description of the model and its convergence, normalized absolute strain (relative to a 1-m incident wave) is found as a function of distance from the ice edge for various wave periods, ice thicknesses, and water depths. The squared transfer function, giving the relative ability of incident waves of different periods to generate strain in the ice, is calculated, and its consequences are discussed. The ice is then forced with a Pierson-Moskowitz spectrum, and the consequent strain spectra are plotted as a function of penetration into the ice sheet. Finally, rms strain, computed as the incoherent sum of the strains resulting from energy in the open water spectrum, is found. The results have implications to the breakup of shore fast ice and hence to the floe size distribution of the marginal ice zone.

Toward realism in modeling ocean wave behavior in marginal ice zones

The model of Meylan and Squire [1996], which treats solitary ice floes as floating, flexible circular disks, is incorporated into the equation of transport for the propagation of waves through a scattering medium, assumed to represent open ice pack in a marginal ice zone. The time-independent form of the equation is then solved for homogeneous ice conditions allowing for dissipation due to scattering, together with extra absorption from interactions between floes, losses in the water column, and losses arising from the inelastic character of the sea ice including local brash. The spatial evolution of wave spectra as they progress through the pack is investigated with the aim of explaining the field data of Wadhams et al. [1986]. Specifically, the change toward directional isotropy experienced by waves as they travel into the ice interior is of interest. In accord with observations, directional spread is found to widen with penetration until eventually becoming isotropic, the process being sensitive to wave period. The effect of absorption on the solution is investigated.

MCMC Estimation of Markov Models for Ion Channels

Ion channels are characterized by inherently stochastic behavior which can be represented by continuous-time Markov models (CTMM). Although methods for collecting data from single ion channels are available, translating a time series of open and closed channels to a CTMM remains a challenge. Bayesian statistics combined with Markov chain Monte Carlo (MCMC) sampling provide means for estimating the rate constants of a CTMM directly from single channel data. In this article, different approaches for the MCMC sampling of Markov models are combined. This method, new to our knowledge, detects overparameterizations and gives more accurate results than existing MCMC methods. It shows similar performance as QuB-MIL, which indicates that it also compares well with maximum likelihood estimators. Data collected from an inositol trisphosphate receptor is used to demonstrate how the best model for a given data set can be found in practice.

Lifetime estimation for a land-fast ice sheet subjected to ocean swell

Abstract It is well known that an incoming ocean swell produces a strain field in a land-fast ice sheet. The attenuation and spectral content of this strain field can be calculated and has been measured. The response of the sea ice to this type of cyclic forcing has also been measured, and in particular we are able to estimate the number of cycles to failure for sea ice loaded at constant amplitude. In this paper we consider the response of the land-fast ice sheet or vast floe to a measured ice-coupled wave field of variable amplitude. We use the Palmgren-Miner cumulative damage law and stress-lifetime curves taken from field experiments to predict the lifetime of the sea-ice sheet as a function of significant wave height and sea-ice brine fraction. Calculations are performed to account for the swell entering a land-fast sea-ice sheet at arbitrary angle, and the influence of c-axis alignment and the presence of pre-existing cracks are discussed.

Exact MAP states and expectations from perfect sampling: Greig, porteous and seheult revisited

In 1989 Greig, Porteous and Seheult showed that the maximum a posteriori (MAP) state can be exactly calculated for degraded binary images. Their interest was in assessing the performance of algorithms used to find the MAP state, such as simulated annealing. A secondary conclusion was that the MAP state, at least in the restricted setting of two-color images, does not provide a robust reconstruction of the true image. That result has been interpreted by some as indicating that the Ising MRF used by GPS is not a good prior model for such images. We show that such a judgement is premature as the MAP state does not well summarize the information in the posterior distribution in this case. In particular, the deviation of the MAP state from the mean, particularly at larger smoothing parameters, shows that the MAP state is not representative of the bulk of feasible reconstructions. We calculate other summary statistics that interpret and display the information in the posterior by implementing full Bayesian infe...

Sampling Gaussian Distributions in Krylov Spaces with Conjugate Gradients

This paper introduces a conjugate gradient sampler that is a simple extension of the method of conjugate gradients (CG) for solving linear systems. The CG sampler iteratively generates samples from a Gaussian probability density, using either a symmetric positive definite covariance or precision matrix, whichever is more convenient to model. Similar to how the Lanczos method solves an eigenvalue problem, the CG sampler approximates the covariance or precision matrix in a small dimensional Krylov space. As with any iterative method, the CG sampler is efficient for high dimensional problems where forming the covariance or precision matrix is impractical, but operating by the matrix is feasible. In exact arithmetic, the sampler generates Gaussian samples with a realized covariance that converges to the covariance of interest. In finite precision, the sampler produces a Gaussian sample with a realized covariance that is the best approximation to the desired covariance in the smaller dimensional Krylov space. ...