Stochastic overland flows

A theoretical solution framework to the nonlinear stochastic partial differential equations (SPDE) of the kinematic wave and diffusion wave models of overland flows under stochastic inflows/outflows, stochastic surface roughness field and stochastic state of flows was obtained. This development was realized by means of an eigenfunction representation of the time-space overland flow depths, and by transforming the problem into the phase space. By using Van Kampen's lemma and the cumulant expansion theory of Kubo-Van Kampen-Fox, the deterministic partial differential equation (PDE) for the evolutionary probability density function (pdf) of overland flow depths was finally obtained. Once this deterministic PDE is solved for the time-varying pdf of overland flow depths, then the time-space varying pdf of overland flow depths can be obtained by a transformation given in the text. In this solution framework it is possible to incorporate the stochastic dynamic behavior of the parameters and of the forcing functions of the overland flow process. For example, not only the individual rainfall duration and fluctuating rain intensity characteristics but also the sequential behavior of rainfall patterns is incorporated into the evolutionary probability density function of overland flow depths.     

Stochastic overland flows Part 1: Physics-based evolutionary probability distributions

A theoretical solution framework to the nonlinear stochastic partial differential equations (SPDE) of the kinematic wave and diffusion wave models of overland flows under stochastic inflows/outflows, stochastic surface roughness field and stochastic state of flows was obtained. This development was realized by means of an eigenfunction representation of the time-space overland flow depths, and by transforming the problem into the phase space. By using Van Kampen's lemma and the cumulant expansion theory of Kubo-Van Kampen-Fox, the deterministic partial differential equation (PDE) for the evolutionary probability density function (pdf) of overland flow depths was finally obtained. Once this deterministic PDE is solved for the time-varying pdf of overland flow depths, then the time-space varying pdf of overland flow depths can be obtained by a transformation given in the text. In this solution framework it is possible to incorporate the stochastic dynamic behavior of the parameters and of the forcing functions of the overland flow process. For example, not only the individual rainfall duration and fluctuating rain intensity characteristics but also the sequential behavior of rainfall patterns is incorporated into the evolutionary probability density function of overland flow depths.     

Relevance of Deterministic Structures for Modeling of Transport: The Lauswiesen Case Study

Knowledge of site‐specific contaminant transport processes is an essential requirement for performing various tasks concerning the protection and management of groundwater resources. However, prediction of their behavior is often difficult, especially in heterogeneous aquifers because of the lack of information about flow‐ and transport‐governing subsurface structures and parameters. Hence, stochastic approaches have been developed and frequently used. However, extensive modeling studies on sedimentary structures have shown that consideration of hydrogeological subunits and their distribution can be essential for transport modeling. A case study from the intensely investigated Lauswiesen site is used to demonstrate that more accurate predictions are possible with improved knowledge of deterministic structures. Results of this case study using direct‐push injection logging (DPIL) provide a more reliable characterization of hydraulic conductivity than sieve and flow meter data.     

Separation of deterministic signals using independent component analysis (ICA)

Independent Component Analysis (ICA) represents a higher-order statistical technique that is often used to separate mixtures of stochastic random signals into statistically independent sources. Its benefit is that it only relies on the information contained in the observations, i.e. no parametric a-priori models are prescribed to extract the source signals. The mathematical foundation of ICA, however, is rooted in the theory of random signals. This has led to questions whether the application of ICA to deterministic signals can be justified at all? In this context, the possibility of using ICA to separate deterministic signals such as complex sinusoidal cycles has been subjected to previous studies. In many geophysical and geodetic applications, however, understanding long-term trend in the presence of periodical components of an observed phenomenon is desirable. In this study, therefore, we extend the previous studies with mathematically proving that the ICA algorithm with diagonalizing the 4th order cumulant tensor, through the rotation of experimental orthogonal functions, will indeed perfectly separate an unknown mixture of trend and sinusoidal signals in the data, provided that the length of the data set is infinite. In other words, we justify the application of ICA to those deterministic signals that are most relevant in geodetic and geophysical applications.     

Rainfall data simulation by hidden Markov model and discrete wavelet transformation

In many regions, monthly (or bimonthly) rainfall data can be considered as deterministic while daily rainfall data may be treated as random. As a result, deterministic models may not sufficiently fit the daily data because of the strong stochastic nature, while stochastic models may also not reliably fit into daily rainfall time series because of the deterministic nature at the large scale (i.e. coarse scale). Although there are different approaches for simulating daily rainfall, mixing of deterministic and stochastic models (towards possible representation of both deterministic and stochastic properties) has not hitherto been proposed. An attempt is made in this study to simulate daily rainfall data by utilizing discrete wavelet transformation and hidden Markov model. We use a deterministic model to obtain large-scale data, and a stochastic model to simulate the wavelet tree coefficients. The simulated daily rainfall is obtained by inverse transformation. We then compare the accumulated simulated and accumulated observed data from the Chao Phraya Basin in Thailand. Because of the stochastic nature at the small scale, the simulated daily rainfall on a point to point comparison show deviations with the observed data. However the accumulated simulated data do show some level of agreement with the observed data.     

A stochastic model reduction method for nonlinear unconfined flow with multiple random input fields

In this paper we present a stochastic model reduction method for efficiently solving nonlinear unconfined flow problems in heterogeneous random porous media. The input random fields of flow model are parameterized in a stochastic space for simulation. This often results in high stochastic dimensionality due to small correlation length of the covariance functions of the input fields. To efficiently treat the high-dimensional stochastic problem, we extend a recently proposed hybrid high-dimensional model representation (HDMR) technique to high-dimensional problems with multiple random input fields and integrate it with a sparse grid stochastic collocation method (SGSCM). Hybrid HDMR can decompose the high-dimensional model into a moderate M-dimensional model and a few one-dimensional models. The moderate dimensional model only depends on the most M important random dimensions, which are identified from the full stochastic space by sensitivity analysis. To extend the hybrid HDMR, we consider two different criteria for sensitivity test. Each of the derived low-dimensional stochastic models is solved by the SGSCM. This leads to a set of uncoupled deterministic problems at the collocation points, which can be solved by a deterministic solver. To demonstrate the efficiency and accuracy of the proposed method, a few numerical experiments are carried out for the unconfined flow problems in heterogeneous porous media with different correlation lengths. The results show that a good trade-off between computational complexity and approximation accuracy can be achieved for stochastic unconfined flow problems by selecting a suitable number of the most important dimensions in the M-dimensional model of hybrid HDMR.     

A stochastic based approach for a new site classification method: application to the Algerian seismic code

Building codes have widely considered the shear wave velocity to make a reliable subsoil seismic classification, based on the knowledge of the mechanical properties of material deposits down to bedrock. This approach has limitations because geophysical data are often very expensive to obtain. Recently, other alternatives have been proposed based on measurements of background noise and estimation of the H/V amplification curve. However, the use of this technique needs a regulatory framework before it can become a realistic site classification procedure. This paper proposes a new formulation for characterizing design sites in accordance with the Algerian seismic building code(RPA99/ver.2003), through transfer functions, by following a stochastic approach combined to a statistical study. For each soil type, the deterministic calculation of the average transfer function is performed over a wide sample of 1-D soil profiles, where the average shear wave(S-W) velocity, V s, in soil layers is simulated using random field theory. Average transfer functions are also used to calculate average site factors and normalized acceleration response spectra to highlight the amplification potential of each site type, since frequency content of the transfer function is significantly similar to that of the H/V amplification curve. Comparison is done with the RPA99/ver.2003 and Eurocode8(EC8) design response spectra, respectively. In the absence of geophysical data, the proposed classification approach together with micro-tremor measures can be used toward a better soil classification.     

Fractal travel time estimates for dispersive contaminants

Alternative fractional models of contaminant transport lead to a new travel time formula for arbitrary concentration levels. For an evolving contaminant plume in a highly heterogeneous aquifer, the new formula predicts much earlier arrival at low concentrations. Travel times of contaminant fronts and plumes are often obtained from Darcy's law calculations using estimates of average pore velocities. These estimates only provide information about the travel time of the average concentration (or peak, for contaminant pulses). Recently, it has been shown that finding the travel times of arbitrary concentration levels is a straightforward process, and equations were developed for other portions of the breakthrough curve for a nonreactive contaminant. In this paper, we generalize those equations to include alternative fractional models of contaminant transport.     

Non-stationary response of structures excited by random seismic processes with time variable frequency content

Seismic random processes are characterized by high non-stationarity and, in most cases, by a marked variability of frequency content. The hypothesis modeling seismic signal as a simple product of a stationary signal and a deterministic modulation function, consequently, is hardly ever applicable. Several mathematical models aimed at expressing the recorded process by means of a system of stationary random processes and deterministic amplitude and frequency modulations are proposed. Models oriented into the frequency domain with subsequent response analysis based on integral spectral resolution and models oriented into the time domain based on the multicomponent resolution are investigated. The resolution into individual components (non-stationary signals) is carried out by three methods. The resolution into intrinsic mode functions seems to possess the best characteristics and yields results almost not differing from the results obtained by stochastic simulation. An example of the seismic response of an existing bridge obtained by two older models and three variants of multicomponent resolution is given.     

Disturbed periodic model for river meanders

Although river meanders are not perfectly regular their serial statistics show periodic tendencies which cannot be explained by previous stochastic models. The regular and random approaches to meander geometry can be reconciled in a disturbed periodic model with separate scale, sinuosity, and irregularity parameters. Meandering is viewed as a deterministic oscillation but irregularity is introduced by quasi-random variability in valley-floor topography and materials. For stability such a model needs either a Bagnold type limit on bend curvature or frictional damping of the oscillatory response to individual disturbances. Realistic statistical properties are derived for the second case. The differential equation for direction can be approximated by a second-order autoregression, which generates realistic simulated patterns and gives a good fit to natural direction series.     

Evidence of Spatio-Temporal Variations in Contaminants Discharging to a Peri-Urban Stream

Xenobiotic organic compounds can be discharged from contaminated groundwater inflow and may seep into streams from multiple pathways with very different dynamics, some not fully understood. In this study, we investigated the spatio-temporal variation of chlorinated ethenes discharging from a former industrial site (with two main contaminant sources, A and B) into a stream system in a heterogeneous clay till setting in eastern Denmark. The investigated reach and near-stream surroundings are representative of peri-urban settings, with a mix of high channel alteration and more natural stream environment. We therefore propose an approach for risk assessing impacts arising from such complex contamination patterns, accounting for potential spatio-temporal fluctuations and presence of multiple pathways. Our study revealed substantial variations in pathway contributions and overall contaminant mass discharge to the stream. Variable contaminant contributions arising from both groundwater seepage and urban drains were identified in the channelized part of the north stream, primarily from source A. Furthermore, variations in the hyporheic and shallow groundwater flows were found to enhance contaminant transport from source B. These processes result in an increase of the overall mass of contaminant discharged, correlating with the channels' flow. Thus, an in-stream control plane approach was found to be an effective method for integrating multiple and variable discharge contributions quantitatively, although information on specific contaminant sources is lost. This study highlights the complexity and variability of contaminant fluxes occurring at the interface between groundwater and peri-urban streams, and calls for the consideration of these variations when designing monitoring programs and remedial actions for contaminated sites with the potential to impact streams.     

Stochastic methods in hydraulics and hydrology of streamflow

Stochastic methods in hydraulics and hydrology of streamflow are presented. The hydraulics part consists of mechanics of streamflow and sediment transport. A technique presented herein enables one to analyze a limited amount of field data to determine the stochastic structure of irregular stream geometry so that cross-sections and slopes of a stream may be simulated wherever, or as many as, needed. It provides the rational basis of efficient use, interpolation, and extrapolation of field data of irregular stream geometry for any studies to understand and control transport processes in streams. Stochastic modelings of motion of a single sediment particle, either in suspension or on the stream bed, help in understanding the complex mechanism governing sediment transport and, hence, improving techniques for calculating the spatial distribution and transport rate of sediment. For practical applications, however, the technique combining the stochastic and deterministic methods should be most effective.In the hydrology part, Markov and non-Markov models are presented which may be used to simulate streamflow data. Markov models, which dominated stochastic hydrology in the past, have short memories and, therefore, cannot preserve or simulate long-term persistence characterizing physiscal streamflows. Non-Markov models which are currently being developed, and may or may not belong to the Brownian domain, have very long or infinite memories.This paper is dedicated to the idea of coupling the stochastic and deterministic methods in hydraulics and hydrology, so that the two methods may contribute their strengths while complementing each other for their weaknesses.     

Deterministic and probabilistic representation of near-field pulse-like ground motion

The response and damage assessment of engineering structures under near-field ground motions is currently of great interest. Near-field ground motion with directivity focusing or fling effects produces pulse-like ground motion that has characteristics different from those of ordinary records. This paper develops simple deterministic and probabilistic models for near-field pulse-like ground motions. These models belong to the class of engineering models that aim to replicate some of the gross features observed in near-field records. The ground velocity is expressed as a steady-state function or a stationary random process modulated by an envelope function. Both models account for the non-stationarity and the multiple pulses in the ground velocity. While the deterministic model is similar to some of the models developed earlier, the probabilistic model facilitates handling uncertainties in the ground motion and variability in the structure's properties. For instance, this model combined with structural reliability methods can be used for reliability assessment of structures under near-field random ground motion. The reduction of the structural response by adding supplemental dampers is also investigated.     

Solution of random structural system subject to nonstationary excitation: transforming the equation with random coefficients to one with deterministic coefficients and random initial conditions

This paper deals with the lower order (first four) nonstationary statistical moments of the response of linear systems with random stiffness and random damping properties subject to random nonstationary excitation modeled as white noise multiplied by an envelope function. The method of analysis is based on a Markov approach using stochastic differential equations (SDE). The linear SDE with random coefficients subject to random excitation with deterministic initial conditions are transformed to an equivalent nonlinear SDE with deterministic coefficients and random initial conditions subject to random excitation. In this procedure, new SDE with random initial conditions, deterministic coefficients and zero forcing functions are introduced to represent the random variables. The joint statistical moments of the response are determined by considering an augmented dynamic system with state variables made up of the displacement and velocity vectors and the random variables of the structural system. The zero time-lag joint statistical moment equations for the augmented state vector are derived from the Itô differential formula. The statistical moment equations are ordinary nonlinear differential equations where hierarchy of moments appear. The hierarchy is closed by the cumulant neglect closure method applied at the fourth order statistical moment level. General formulation is given for multi-degree-of-freedom (MDOF) systems and the performance of the method in problems with nonstationary excitations and large variabilities is illustrated for a single-degree-of-freedom (SDOF) oscillator.     

Fluid flow dynamics under location uncertainty

We present a derivation of a stochastic model of Navier Stokes equations that relies on a decomposition of the velocity fields into a differentiable drift component and a time uncorrelated uncertainty random term. This type of decomposition is reminiscent in spirit to the classical Reynolds decomposition. However, the random velocity fluctuations considered here are not differentiable with respect to time, and they must be handled through stochastic calculus. The dynamics associated with the differentiable drift component is derived from a stochastic version of the Reynolds transport theorem. It includes in its general form an uncertainty dependent subgrid bulk formula that cannot be immediately related to the usual Boussinesq eddy viscosity assumption constructed from thermal molecular agitation analogy. This formulation, emerging from uncertainties on the fluid parcels location, explains with another viewpoint some subgrid eddy diffusion models currently used in computational fluid dynamics or in geophysical sciences and paves the way for new large-scales flow modeling. We finally describe an applications of our formalism to the derivation of stochastic versions of the Shallow water equations or to the definition of reduced order dynamical systems.     

Deterministic uncertainty in landscapes

Deterministic complexity (chaos) may be common in geomorphic systems, but traditional definitions may have limited practical utility for empirical geomorphology. These definitions are based on sensitivity to initial conditions, which in geomorphology are both unknown and unknowable. Further, chaos analysis depends on distinguishing deterministic complexity from stochastic complexity. This is problematic in geomorphology because some stochastic complexity is virtually always present in addition to any chaos that may be present. While it is important to recognize that some complex, apparently random patterns may derive from inherent non-linear system dynamics, this is of limited use in explaining process–response relationships or mechanics of landscape evolution. A more general term, which subsumes chaos, is deterministic uncertainty, i.e. uncertainty associated with an identifiable but unknown or uncertain source. An analysis of landscape entropy shows that such underlying constraints produce spatial patterns which are apparently chaotic. For the case of geologic controls, the apparent contribution of deterministic chaos to the landscape entropy is a direct non-linear function of the extent of geologic constraints. However, the underlying constraints and their contribution to observed spatial patterns can also be interpreted in non-chaotic terms. Examples are given, involving geologic constraints on stream channel networks and parent material control of surface soil textures. Because both randomness and chaos may be more apparent than real, the concept of deterministic uncertainty is more useful in process geomorphology than that of chaos.     

A real-time, interactive steering environment for integrated ground water modeling

We present in this note an innovative software environment, called Interactive Ground Water (IGW), for unified deterministic and stochastic ground water modeling. Based on efficient computational algorithms, IGW allows simulating three-dimensional (3D) unsteady flow and transport in saturated media subject to systematic and "random" stresses and geological and chemical heterogeneity. Adopting a new computing paradigm, IGW eliminates the fragmentation in the traditional modeling schemes and allows fully utilizing today's dramatically increased computing power. For many problems, IGW enables real-time modeling, visualization, mapping, and analysis. The software environment functions as a "numerical laboratory" in which an investigator may freely explore the following: creating visually an aquifer system of desired configurations, interactively applying stresses and boundary conditions, and then investigating and visualizing on the fly the geology and flow and transport dynamics. At any time, a researcher can pause to interact dynamically with virtually any aspects of the modeling process and then resume the integrated visual exploration; he or she can initiate, pause, or resume particle tracking, plume modeling, subscale modeling, stochastic modeling, monitoring, and budget analyses. IGW continually provides results that are dynamically processed, overlaid, and displayed. It dynamically merges modeling inputs and outputs into composite two-dimensional/3D images-integrating related data to provide a more complete view of the complex interplay among the geology, hydrology, flow system, and transport. These unique capabilities of real-time modeling, steering, analysis, and mapping expand the utility of models as tools for research, education, and professional investigations.     

Ground water contaminant transport by nondivergence-free, unsteady and nonstationary velocity fields

Pore flow velocity is assumed to be a nondivergence-free, unsteady, and nonstationary random function of space and time for ground water contaminant transport in a heterogeneous medium. The laboratory-scale stochastic contaminant transport equation is up scaled to field scale by taking the ensemble average of the equation by using the cumulant expansion method. A new velocity correction, which is a function of mean pore flow velocity divergence, is obtained due to strict second order cumulant expansion (without omitting any term after the expansion). The field scale transport equations under the divergence-free pore flow velocity field assumption are also derived by simplifying the nondivergence-free field scale equation. The significance of the new velocity correction term is investigated on a two dimensional transport problem driven by a density dependent flow.