Calculation of geological uncertainties associated with 3-D geological models

3-D geological models are built with data collected in the field such as boreholes, geophysical measurements, pilot shafts or geological mapping. Unfortunately, these data are always limited in number. It implies that geological information is sparse and subsurface models are thus always built of both subjective interpretation and mathematical interpolation/extrapolation techniques. These models are therefore uncertain and this uncertainty is rarely pointed out in a geological prognosis. Our study proposes to bring a new methodology for the evaluation of geological uncertainties related to 3-D subsurface models and to test its potential use. The methodology we propose is based on the 3-D subsurface model, which is here considered as the most probable prediction (notion of best guess). The various geological interfaces that compose the subsurface model are handled individually as Gaussian random fields. At each location of an interface, the random function Z(u) describing the position of this interface is composed of a deterministic part m(u) which represents the expected position, and a random part σ(u)ε(u) which describes fluctuations around the predicted position. Then, a model of spatial variability (a variogram function γ(h)) is proposed in order to condition the random field according to available observations. Several structural constraints, such as the shape of folds and the thickness of layers can also be accounted for in this model. At this point, we are able to estimate the local variance all over the study area by the application of the kriging technique. Finally, the variability is converted into three-dimensional information by calculating probabilities, this describes the occurrence of the various rock masses that are present in the study area. The probabilities are calculated according to intersection rules that govern the stratigraphic sequence of the subsurface model, and they allow us to build a probabilistic model of subsurface structures in the form of a three-dimensional probability field. All of this has been incorporated in a computer program.

A Probabilistic Model for Chord Progressions

Samy Bengio, Jean-François Paiement

Chord progressions are the building blocks from which tonal music is constructed. Inferring chord progressions is thus an essential step towards modeling long term dependencies in music. In this paper, a distributed representation for chords is designed such that Euclidean distances roughly correspond to psychoacoustic dissimilarities. Estimated probabilities of chord substitutions are derived from this representation and are used to introduce smoothing in graphical models observing chord progressions. Parameters in the graphical models are learnt with the EM algorithm and the classical Junction Tree algorithm is used for inference. Various model architectures are compared in terms of conditional out-of-sample likelihood. Both perceptual and statistical evidence show that binary trees related to meter are well suited to capture chord dependencies.

2005

Chord Representations for Probabilistic Models

Samy Bengio, Jean-François Paiement

Chord progressions are the building blocks from which tonal music is constructed. Inferring chord progressions is thus an essential step towards modeling long term dependencies in music. In this paper, three different representations for chords are designed. In a first representation, Euclidean distances roughly correspond to psychoacoustic dissimilarities between chords. Estimated probabilities of chord substitutions are then derived from these distances and are used to introduce smoothing in graphical models observing another chord representation. Finally, a third representation where we model directly each chord components leads to a probabilistic model considering the interaction between melodies and chord progressions. Parameters in the graphical models are learnt with the EM algorithm and the classical Junction Tree algorithm is used for inference. Various model architectures are compared in terms of conditional out-of-sample likelihood. Both perceptual and statistical evidence show that binary trees related to meter are well suited to capture chord dependencies.

IDIAP2005

The Contact Process

Daniel Rodrigues Valesin

This work is a study of three interacting particle systems that are modified versions of the contact process. The contact process is a spin system defined on a graph and is commonly taken as a model for the spread of an infection in a population; transmission of the infection happens by proximity (contact). The first two models we consider – the grass-bushes-trees model and the multitype contact process – are models for competition between species in Ecology. The third model – annealed approximation to boolean networks – approximately describes the transmission of information among genes in a cell. We consider the grass-bushes-trees model on the set of integers, ℤ. Each point of ℤ is a region of space. In the continuous-time dynamics, at each instant each region can be either empty (state 0) or occupied by an individual of one of two existing species (states 1 and 2). Occupants of both species die at rate 1, leaving their regions empty, and send descendents to neighboring regions at rate λ. An individuals of type 1 may be born on a region previously occupied by an individual of type 2, but the converse is forbidden. We take the "heaviside" initial configuration in which all sites to the left of the origin are occupied by type 1 individuals and all sites to the right of the origin are occupied by type 2 individuals. If the birth of new individuals is allowed to occur at sites that are not adjacent to the parent, and if the rate λ is supercritical for the usual contact process on ℤ, we see the formation of an interface region in which both types coexist. Addressing a conjecture of Cox and Durrett (1995), we prove that the size of this region is stochastically tight. The multitype contact process on ℤ is a process identical to the grass-bushes-trees model in every respect except that no births can occur at previously occupied sites; in particular, the model is symmetric for both species. We again start the process from the heaviside configuration and prove that the size of the interface region is tight. In addition, we prove that the position of the interface, when properly rescaled, converges to Brownian motion. Finally, we give necessary and sufficient conditions on the initial configuration so that one of the two species becomes extinct with probability one and also so that both species are present at all times with positive probability. Lastly, we consider a model proposed by Derrida and Pomeau (1986) and recently studied by Chatterjee and Durrett (2009); it is defined as an approximation to S. Kauffman's boolean networks (1969). The model starts with the choice of a random directed graph on n vertices; each node has r input nodes pointing at it. A discrete time threshold contact process is then considered on this graph: at each instant, each site has probability q of choosing to receive input; if it does, and if at least one of its inputs were occupied by a 1 at the previous instant, then it is labeled with a 1; in all other cases, it is labeled with a 0. r and q are kept fixed and n is taken to infinity. Improving a result of Chatterjee and Durrett, we show that if qr > 1, then the time of persistence of the dynamics is exponential in n.

EPFL2011

The Contact Process

Daniel Rodrigues Valesin

EPFL2011