Michel Dekking

A Markov Chain Model for Subsurface Characterization: Theory and Applications

This paper proposes an extension of a single coupled Markov chain model to characterize heterogeneity of geological formations, and to make conditioning on any number of well data possible. The methodology is based on the concept of conditioning a Markov chain on the future states. Because the conditioning is performed in an explicit way, the methodology is efficient in terms of computer time and storage. Applications to synthetic and field data show good results.

Multi-Scale and Multi-Resolution Stochastic Modeling of Subsurface Heterogeneity by Tree-Indexed Markov Chains

A new methodology is proposed to handle multi-scale heterogeneous structures. It can be of importance in the field of hydrogeology and for petroleum engineers who are interested in characterizing subsurface heterogeneity at various scales. The framework of this methodology is based on a coarse to fine scale representation of the heterogeneous structures on trees. Different depths in the tree correspond to different spatial scales in representing the heterogeneous structures on trees. On these trees a Markov chain is used to describe scale to scale transitions and to account for the uncertainty in the stochastically generated images.We focus in this work on the description and application of the methodology to synthetic data that are geologically realistic. The methodology is flexible. Conditioning on field data and measurements is straightforward. Non-stationary and stationary fields, compound and nested structures can be addressed.

Random Cantor Sets and Their Projections

We discuss two types of random Cantor sets, M-adic random Cantor sets, and Larsson’s random Cantor sets. We will discuss the properties of their ninety and fortyfive degree projections, and for both we give answers to the question whether the algebraic difference of two independent copies of such sets will contain an interval or not.

Which ABO-matching rule should be the decisive factor in the choice between a highly urgent and an elective patient?

Abstract ABO blood group matching policy between donor and recipient is a key element of organ allocation. Unequal distribution of the ABO blood groups in the population can lead to inequities in the distribution of organs to potential recipients. Furthermore, High Urgency liver transplant candidates might compromise the chances of transplantation for the elective patients. To compare the influence of the various ABO blood group matching policies on the transplantation rate of HU patients and on the subsequent donor liver availability for elective patients, a simulation study was undertaken. The study shows that in the Eurotransplant liver allocation program, a restricted ABO‐compatible matching policy for HU liver patients offers the highest probability of acquiring a liver transplant, for both high Urgency‐ and elective patients, irrespective of their ABO blood group. A simulation study once again proved to be an elegant tool for objectively analysing various options in a complex organ allocation algorithm.

Paperfolding morphisms, planefilling curves, and fractal tiles

An interesting class of automatic sequences emerges from iterated paperfolding. The sequences generate curves in the plane with an almost periodic structure. We generalize the results obtained by Davis and Knuth on the self-avoiding and planefilling properties of these curves, giving simple geometric criteria for a complete classification. Finally, we show how the automatic structure of the sequences leads to self-similarity of the curves, which turns the planefilling curves in a scaling limit into fractal tiles. For some of these tiles we give a particularly simple formula for the Hausdorff dimension of their boundary.

Understanding the Non-Gaussian Nature of Linear Reactive Solute Transport in 1D and 2D: From Particle Dynamics to the Partial Differential Equations

In the present study, we examine non-Gaussian spreading of solutes subject to advection, dispersion and kinetic sorption (adsorption/desorption). We start considering the behavior of a single particle and apply a random walk to describe advection/dispersion plus a Markov chain to describe kinetic sorption. We show in a rigorous way that this model leads to a set of differential equations. For this combination of stochastic processes, such a derivation is new. Then, to illustrate the mechanism that leads to non-Gaussian spreading, we analyze this set of equations at first leaving out the Gaussian dispersion term (microdispersion). The set of equations now transforms to the telegrapher’s equation. Characteristic for this system is a longitudinal spreading that becomes Gaussian only in the longtime limit. We refer to this as kinetics-induced spreading. When the microdispersion process is included back again, the characteristics of the telegraph equations are still present. Now, two spreading phenomena are active, the Gaussian microdispersive spreading plus the kinetics-induced non-Gaussian spreading. In the long run, the latter becomes Gaussian as well. Another non-Gaussian feature shows itself in the 2D situation. Here, the lateral spread and the longitudinal displacement are no longer independent, as should be the case for a 2D Gaussian spreading process. In a displacing plume, this interdependence is displayed as a ‘tailing’ effect. We also analyze marginal and conditional moments, which confirm this result. With respect to effective properties (velocity and dispersion), we conclude that effective parameters can be defined properly only for large times (asymptotic times). In the two-dimensional case, it appears that the transverse spreading depends on the longitudinal coordinate. This results in ‘cigar-shaped’ contours.

Multimodality of the Markov binomial distribution

We study the shape of the probability mass function of the Markov bi- nomial distribution, and give necessary and sufficient conditions for the probability mass function to be unimodal, bimodal or trimodal. These are useful to analyze the double-peaking results from a PDE reactive transport model from the engineering lit- erature. Moreover, we give a closed form expression for the variance of the Markov binomial distribution, and expressions for the mean and the variance conditioned on the state at time n.

Correlated Fractal Percolation and the Palis Conjecture

Let F1 and F2 be independent copies of one-dimensional correlated fractal percolation, with almost sure Hausdorff dimensions dim H(F1) and dim H(F2). Consider the following question: does dim H(F1)+dim H(F2)>1 imply that their algebraic difference F1−F2 will contain an interval? The well known Palis conjecture states that ‘generically’ this should be true. Recent work by Kuijvenhoven and the first author (Dekking and Kuijvenhoven in J. Eur. Math. Soc., to appear) on random Cantor sets cannot answer this question as their condition on the joint survival distributions of the generating process is not satisfied by correlated fractal percolation. We develop a new condition which permits us to solve the problem, and we prove that the condition of Dekking and Kuijvenhoven (J. Eur. Math. Soc., to appear) implies our condition. Independently of this we give a solution to the critical case, yielding that a strong version of the Palis conjecture holds for fractal percolation and correlated fractal percolation: the algebraic difference contains an interval almost surely if and only if the sum of the Hausdorff dimensions of the random Cantor sets exceeds one.

A simple stochastic kinetic transport model

We introduce a discrete-time microscopic single-particle model for kinetic transport. The kinetics are modeled by a two-state Markov chain, and the transport is modeled by deterministic advection plus a random space step. The position of the particle after n time steps is given by a random sum of space steps, where the size of the sum is given by a Markov binomial distribution (MBD). We prove that by letting the length of the time steps and the intensity of the switching between states tend to 0 linearly, we obtain a random variable S(t), which is closely connected to a well-known (deterministic) partial differential equation (PDE), reactive transport model from the civil engineering literature. Our model explains (via bimodality of the MBD) the double peaking behavior of the concentration of the free part of solutes in the PDE model. Moreover, we show for instantaneous injection of the solute that the partial densities of the free and adsorbed parts of the solute at time t do exist, and satisfy the PDEs.

The algebraic difference of two random Cantor sets: The Larsson family

In this paper, we consider a family of random Cantor sets on the line and consider the question of whether the condition that the sum of the Hausdorff dimensions is larger than one implies the existence of interior points in the difference set of two independent copies. We give a new and complete proof that this is the case for the random Cantor sets introduced by Per Larsson.