Mina Ossiander

Time scale and intensity dependency in multiplicative cascades for temporal rainfall disaggregation

[1] Multiplicative random cascades (MRCs) can parsimoniously generate highly intermittent patterns similar to those in rainfall. The elemental MRC model parameter is the cascade weight, which determines how rainfall at one scale is partitioned at the next smallest scale in the cascade. While it is known that the probability density of these weights may vary with both time scale and rainfall intensity, nearly all previous studies have considered either time scale or intensity separately. We examined the simultaneous dependency of the weights on both factors and assessed the impacts of explicitly including these dependencies in the MRC model. On the basis of the observed relationships between cascade weights and time scale and intensity, four progressively more “dependent” models were constructed to disaggregate a long time series of daily rainfall to hourly intervals. We found that inclusion of the intensity dependency on the model parameters that generate dry intervals greatly improved performance. For the relatively small range of time scales over which the rainfall was disaggregated, varying model parameters with time scale resulted in minor improvement.

Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations

A general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d Navier-Stokes equations. The basic idea generalizes a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain weak solutions whose Fourier transform may be represented by an expected value of a stochastic cascade. A functional analytic framework is also developed which partially connects stochastic iterations and certain Picard iterates. Some local existence and uniqueness results are also obtained by contractive mapping conditions on the Picard iteration.

Certain positive-definite kernels

In one way or another, the extension of the standard Brownian motion process {B T ;T∈[0,∞)} to a (Gaussian) random field {B t :t∈R + d } involves a proof of the positive semi-definiteness of the kernel used to generalize ρ(s,t)=cov(B s ,B t )=s∧t to multidimensional time. Simple direct analytical proofs are provided here for the cases of (i) the Levy multiparameter Brownian motion, (ii) the Chentsov Brownian sheet, and (iii) the multiparameter fractional Brownian field

Semi-Markov Cascade Representations of Local Solutions to 3-D Incompressible Navier-Stokes

A probabilistic approach int roduced by LeJan and Sznitrnan (1997) perm its derivation of weak solutions to 3-d incompressible Navier-Stokes equations whose Fourier transform may be rep resented by an expected value of a stochastic cascade. This approach was extended in Bhattacharya et al. (2003) by met hods which would yield unique global solutions by a stochastic representation under “small initial data conditions”. A connection to iterative contraction maps on appropriate function space was also provided which would also yield local existence and uniqueness under “short time” constraint s, but with out stochastic cascade representations. In t he present paper the authors (i.) Provide a stochastic cascade representation for local solutions, and (ii.) Provide time-asymptotics for global solutions from t he stochastic representation.

Conditional Stochastic Simulations of Flow and Transport with Karhunen-Loève Expansions, Stochastic Collocation, and Sequential Gaussian Simulation

We derive a new method of conditional Karhunen-Loeve (KL) expansions for stochastic coefficients in models of flow and transport in the subsurface, and in particular for the heterogeneous random permeability field. Exact values of this field are never known, and thus one must evaluate uncertainty of solutions to the flow and transport models. This is typically done by constructing independent realizations of the permeability field followed by numerical simulations of flow and transport for each realization and assembling statistical estimates of moments of desired quantities of interest. We follow the well-known framework of KL expansions and derive a new method that incorporates known values of the permeability at given locations so that the realizations of the permeability field honor this data exactly. Our method relies on projections to an appropriate subspace of random weights applied to the eigenfunctions of the covariance operator. We use the permeability realizations constructed with our stochastic simulation method in simulations of flow and transport and compare the results to those obtained when realizations are constructed with sequential Gaussian simulation (SGS). We also compare efficiency and stochastic convergence with that of stochastic collocation.

Estimation and simulation for geospatial porosity and permeability data

Reservoir simulation of

Support Fragmentation for Multiplicative Cascade Measures

\hbox {CO}_2

ON ESTIMATION THEORY FOR MULTIPLICATIVE CASCADES

CO2 sequestration, energy recovery, and environmental contamination scenarios must be accompanied by uncertainty quantification. Typically this is done by stochastically modeling porosity and permeability fields, simulating realizations based on the model, and then numerically simulating flow and transport. The challenge is to generate simulated porosity and permeability fields with characteristics as similar as possible to those known of the reservoir under study. In this paper we focus on the first two steps above in analyzing a large 3-dimensional array of geospatial porosity data and using the results to produce simulated data with characteristics mimicking those of the original porosity observations. The spatial covariance is empirically approximated from horizontal cross sections of the data via a kernel principle component analysis yielding dimension reduction. Simulations in three dimensions are produced by linking consecutive parallel cross sections via conditioning on a small subarray of the data. The conditional simulations effectively reproduce observed channeling, an important large scale feature of interest in the sub-surface relevant to transport of contaminates. The original porosity data is non-Gaussian and requires additional analysis and transformation to generate both porosity and permeability fields.