Spatial Markov Model Research Articles

Upscaling Mixing in Highly Heterogeneous Porous Media via a Spatial Markov Model

In this work, we develop a novel Lagrangian model able to predict solute mixing in heterogeneous porous media. The Spatial Markov model has previously been used to predict effective mean conservative transport in flows through heterogeneous porous media. In predicting effective measures of mixing on larger scales, knowledge of only the mean transport is insufficient. Mixing is a small scale process driven by diffusion and the deformation of a plume by a non-uniform flow. In order to capture these small scale processes that are associated with mixing, the upscaled Spatial Markov model must be extended in such a way that it can adequately represent fluctuations in concentration. To address this problem, we develop downscaling procedures within the upscaled model to predict measures of mixing and dilution of a solute moving through an idealized heterogeneous porous medium. The upscaled model results are compared to measurements from a fully resolved simulation and found to be in good agreement.

Predicting Downstream Concentration Histories From Upstream Data in Column Experiments

AbstractThe scales of heterogeneity present in geologic media make modeling solute transport extremely challenging, even in idealized laboratory settings. The spatial Markov model (SMM) is an anomalous transport model that has been shown to accurately capture solute transport in a broad range of highly complex and heterogeneous hydrogeologic settings. However, to date, its applications are almost entirely limited to synthetic, numerically simulated systems due to the dense data required to parameterize it, which are typically unobtainable in real experiments. Here we apply a novel SMM inverse model that required only breakthrough curve measurements from laboratory transport experiments in zeolite‐packed columns that are known to display anomalous transport. We introduce an experimental design that allows for simultaneous measurements of breakthrough curves at multiple sampling locations within a one‐dimensional column setup. For the first time, we apply a fully parameterized SMM to successfully predict downgradient breakthrough curves. Results show that breakthrough curve prediction accuracy significantly improves when accounting for correlation effects in these experiments, a feature that the SMM is specifically designed to capture but that most traditional anomalous transport frameworks ignore. We do so for two different Péclet numbers, providing a parsimonious framework that can potentially account for correlation statistics in different field‐scale studies.

Semantic-Enhanced Multi-Dimensional Markov Chains on Semantic Trajectories for Predicting Future Locations.

In this work, we investigate the performance of Markov Chains with respect to modelling semantic trajectories and predicting future locations. In the first part, we examine whether and to what degree the semantic level of semantic trajectories affects the predictive performance of a spatial Markov model. It can be shown that the choice of the semantic level when describing trajectories has a significant impact on the accuracy of the models. High-level descriptions lead to better results than low-level ones. The second part introduces a multi-dimensional Markov Chain construct that considers, besides locations, additional context information, such as time, day and the users’ activity. While the respective approach is able to outperform our baseline, we could also identify some limitations. These are mainly attributed to its sensitivity towards small-sized training datasets. We attempt to overcome this issue, among others, by adding a semantic similarity analysis component to our model that takes the varying role of locations due each time to the respective purpose of visiting the particular location explicitly into consideration. To capture the aforementioned dynamics, we define an entity, which we refer to as Purpose-of-Visit-Dependent Frame (PoVDF). In the third part of this work, we describe in detail the PoVDF-based approach and we evaluate it against the multi-dimensional Markov Chain model as well as with a semantic trajectory mining and prefix tree based model. Our evaluation shows that the PoVDF-based approach outperforms its competition and lays a solid foundation for further investigation.

A Spatial Markov Model for Climate Extremes

ABSTRACTSpatial climate data are often presented as summaries of areal regions such as grid cells, either because they are the output of numerical climate models or to facilitate comparison with numerical climate model output. Extreme value analysis can benefit greatly from spatial methods that borrow information across regions. For Gaussian outcomes, a host of methods that respect the areal nature of the data are available, including conditional and simultaneous autoregressive models. However, to our knowledge, there is no such method in the spatial extreme value analysis literature. In this article, we propose a new method for areal extremes that accounts for spatial dependence using latent clustering of neighboring regions. We show that the proposed model has desirable asymptotic dependence properties and leads to relatively simple computation. Applying the proposed method to North American climate data reveals several local and continental-scale changes in the distribution of precipitation and temperature extremes over time. Supplementary material for this article is available online.

PESA: a predictive energy‐saving approach based on an OSHMM

Reducing energy consumption of the internet becomes more and more important. The authors propose a novel approach, called predictive energy-saving approach (PESA), to put near-idle nodes of a network into sleep mode to save energy. In PESA, they propose an online spatial hidden Markov model (OSHMM) to describe the distribution of the network traffic and predict the lowest-loaded nodes. Those nodes will be put into sleep mode under the constraints that the network's full connectivity is kept and no new congested nodes are generated. To reduce the computation, they leverage on an online training technique and M-algorithm in OSHMM. They evaluate the approach over a randomly generated topology and a real ISP backbone topology with a real traffic dataset. The results show that the proposed approach is energy efficient with low computation complexity.

Parameterizing the Spatial Markov Model From Breakthrough Curve Data Alone

AbstractThe spatial Markov model (SMM) is an upscaled Lagrangian model that effectively captures anomalous transport across a diverse range of hydrologic systems. The distinct feature of the SMM relative to other random walk models is that successive steps are correlated. To date, with some notable exceptions, the model has primarily been applied to data from high‐resolution numerical simulations and correlation effects have been measured from simulated particle trajectories. In real systems such knowledge is practically unattainable and the best one might hope for is breakthrough curves (BTCs) at successive downstream locations. We introduce a novel methodology to quantify velocity correlation from BTC data alone. By discretizing two measured BTCs into a set of arrival times and developing an inverse model, we estimate velocity correlation, thereby enabling parameterization of the SMM in studies where detailed Lagrangian velocity statistics are unavailable. The proposed methodology is applied to two synthetic numerical problems, where we measure all details and thus test the veracity of the approach by comparison of estimated parameters with known simulated values. Our results suggest that our estimated transition probabilities agree with simulated values and using the SMM with this estimated parameterization accurately predicts BTCs downstream. Our methodology naturally allows for estimates of uncertainty by calculating lower and upper bounds of velocity correlation, enabling prediction of a range of BTCs. The measured BTCs fall within the range of predicted BTCs. This novel method to parameterize the SMM from BTC data alone is quite parsimonious, thereby widening the SMM's practical applicability.

A Lagrangian Transport Eulerian Reaction Spatial (LATERS) Markov Model for Prediction of Effective Bimolecular Reactive Transport

AbstractPrediction of effective transport for mixing‐driven reactive systems at larger scales, requires accurate representation of mixing at small scales, which poses a significant upscaling challenge. Depending on the problem at hand, there can be benefits to using a Lagrangian framework, while in others an Eulerian might have advantages. Here we propose and test a novel hybrid model which attempts to leverage benefits of each. Specifically, our framework provides a Lagrangian closure required for a volume‐averaging procedure of the advection diffusion reaction equation. This hybrid model is a LAgrangian Transport Eulerian Reaction Spatial Markov model (LATERS Markov model), which extends previous implementations of the Lagrangian Spatial Markov model and maps concentrations to an Eulerian grid to quantify closure terms required to calculate the volume‐averaged reaction terms. The advantage of this approach is that the Spatial Markov model is known to provide accurate predictions of transport, particularly at preasymptotic early times, when assumptions required by traditional volume‐averaging closures are least likely to hold; likewise, the Eulerian reaction method is efficient, because it does not require calculation of distances between particles. This manuscript introduces the LATERS Markov model and demonstrates by example its ability to accurately predict bimolecular reactive transport in a simple benchmark 2‐D porous medium.

Spatial hidden Markov models and species distributions

ABSTRACTA spatial hidden Markov model (SHMM) is introduced to analyse the distribution of a species on an atlas, taking into account that false observations and false non-detections of the species can occur during the survey, blurring the true map of presence and absence of the species. The reconstruction of the true map is tackled as the restoration of a degraded pixel image, where the true map is an autologistic model, hidden behind the observed map, whose normalizing constant is efficiently computed by simulating an auxiliary map. The distribution of the species is explained under the Bayesian paradigm and Markov chain Monte Carlo (MCMC) algorithms are developed. We are interested in the spatial distribution of the bird species Greywing Francolin in the south of Africa. Many climatic and land-use explanatory variables are also available: they are included in the SHMM and a subset of them is selected by the mutation operators within the MCMC algorithm.

Anomalous transport in disordered fracture networks: Spatial Markov model for dispersion with variable injection modes

We investigate tracer transport on random discrete fracture networks that are characterized by the statistics of the fracture geometry and hydraulic conductivity. While it is well known that tracer transport through fractured media can be anomalous and particle injection modes can have major impact on dispersion, the incorporation of injection modes into effective transport modeling has remained an open issue. The fundamental reason behind this challenge is that—even if the Eulerian fluid velocity is steady—the Lagrangian velocity distribution experienced by tracer particles evolves with time from its initial distribution, which is dictated by the injection mode, to a stationary velocity distribution. We quantify this evolution by a Markov model for particle velocities that are equidistantly sampled along trajectories. This stochastic approach allows for the systematic incorporation of the initial velocity distribution and quantifies the interplay between velocity distribution and spatial and temporal correlation. The proposed spatial Markov model is characterized by the initial velocity distribution, which is determined by the particle injection mode, the stationary Lagrangian velocity distribution, which is derived from the Eulerian velocity distribution, and the spatial velocity correlation length, which is related to the characteristic fracture length. This effective model leads to a time-domain random walk for the evolution of particle positions and velocities, whose joint distribution follows a Boltzmann equation. Finally, we demonstrate that the proposed model can successfully predict anomalous transport through discrete fracture networks with different levels of heterogeneity and arbitrary tracer injection modes.

Upscaling of dilution and mixing using a trajectory based Spatial Markov random walk model in a periodic flow domain

The Spatial Markov Model (SMM) is an upscaled model that has been used successfully to predict effective mean transport across a broad range of hydrologic settings. Here we propose a novel variant of the SMM, applicable to spatially periodic systems. This SMM is built using particle trajectories, rather than travel times. By applying the proposed SMM to a simple benchmark problem we demonstrate that it can predict mean effective transport, when compared to data from fully resolved direct numerical simulations. Next we propose a methodology for using this SMM framework to predict measures of mixing and dilution, that do not just depend on mean concentrations, but are strongly impacted by pore-scale concentration fluctuations. We use information from trajectories of particles to downscale and reconstruct pore-scale approximate concentration fields from which mixing and dilution measures are then calculated. The comparison between measurements from fully resolved simulations and predictions with the SMM agree very favorably.

Testing the limits of the spatial Markov model for upscaling transport: The role of nonmonotonic effective velocity autocorrelations

The spatial Markov model is a Lagrangian random walk model, widely and successfully used for upscaling transport in heterogeneous flows across a broad range of problems. It is particularly useful at early or pre-asymptotic times when many other conventional upscaling approaches may not be valid. However, as with all upscaled models, it must have its limits. In particular, the question of what the smallest scale at which it can be legitimately applied, without violating implicit assumptions, remains. Here we address this issue by considering one of the most classical transport upscaling problems: Taylor dispersion in a bounded shear flow. We demonstrate that the smallest scale for the spatial Markov model depends on the transverse width of the domain, the variability of the flow field as quantified by a coefficient of variation, and the competition of longitudinal and transverse diffusion coefficients. We show that this scale is a factor of the Peclet number smaller than the classical Taylor dispersion scale, meaning that for advection-dominated systems where Peclet numbers are large, this model can be applied at much smaller scales than classical Taylor-Aris dispersion theories.

Emergence of anomalous transport in stressed rough fractures

We report the emergence of anomalous (non-Fickian) transport through a rough-walled fracture as a result of increasing normal stress on the fracture. We show that the origin of this anomalous transport behavior can be traced to the emergence of a heterogeneous flow field dominated by preferential channels and stagnation zones, as a result of the larger number of contacts in a highly stressed fracture. We show that the velocity distribution determines the late-time scaling of particle spreading, and velocity correlation determines the magnitude of spreading and the transition time from the initial ballistic regime to the asymptotic anomalous behavior. We also propose a spatial Markov model that reproduces the transport behavior at the scale of the entire fracture with only three physical parameters. Our results point to a heretofore unrecognized link between geomechanics and particle transport in fractured media.

A spatial HMM approach for green networking

Green networking becomes more and more important, especially with the rapid development of data centers in recent years. In this paper, in order to minimize the energy consumption of a network, we present a novel energy saving approach, called Predictive Green Networking Approach (PGNA), based on a spatial Hidden Markov Model (sHMM). The sHMM is proposed to describe both the topology and the traffic distribution of the network. Loads of links in the network can be predicted based on the sHMM, and the links that most likely become near idle can be put into sleep mode to save energy. A deep-sleep method is proposed to maximize the energy saving while the network satisfies the connectivity and the maximum utilization constraints. We test the performance of PGNA with two real ISP backbone topologies and real traffic demands. The results show that our approach is effective and works better than related approaches.

Spatial dynamics of urban neighborhood quality of life

This paper investigates the spatial process of neighborhood improvement and decline with respect to its quality of life (QoL) profile. In particular, it seeks to determine to what extent a neighborhood’s geographic situation impacts its probability of improving or declining and is spatial proximity equally advantageous and detrimental. To answer these spatial dependence questions, we employ and expand upon a spatial Markov model. Aside from the standard weight matrix typically utilized in spatial Markov studies comprised of the mean value of a neighborhood’s surrounding area, this paper furthers this conceptualization by exploring alternate specifications including the mode and upper and lower quartiles. Results suggest that spatial dependence plays a strong role in impacting biennial transitions, and when examining the upper and lower quartiles, this dependence is shown to be nonlinear and amplified at the extreme ends of the QoL spectrum. A case study of neighborhoods in the city of Charlotte, North Carolina, during the 2000–2010 decade is used to illustrate these concepts to the study of neighborhood dynamics.

Upscaling transport of a reacting solute through a peridocially converging–diverging channel at pre-asymptotic times

In this study we extend the Spatial Markov model, which has been successfully used to upscale conservative transport across a diverse range of porous media flows, to test if it can accurately upscale reactive transport, defined by a spatially heterogeneous first order degradation rate. We test the model in a well known highly simplified geometry, commonly considered as an idealized pore or fracture structure, a periodic channel with wavy boundaries. The edges of the flow domain have a layer through which there is no flow, but in which diffusion of a solute still occurs. Reactions are confined to this region. We demonstrate that the Spatial Markov model, an upscaled random walk model that enforces correlation between successive jumps, can reproduce breakthrough curves measured from microscale simulations that explicitly resolve all pertinent processes. We also demonstrate that a similar random walk model that does not enforce successive correlations is unable to reproduce all features of the measured breakthrough curves.

Pre-asymptotic Transport Upscaling in Inertial and Unsteady Flows Through Porous Media

In most classical formulations of flow and transport through porous media Reynolds numbers are assumed to be small ( $$\mathrm{Re}<1$$ ), meaning that the role of inertia is considered negligible. However, many examples of practical relevance exist where this is not the case and inertial effects can be important leading to changes in flow structure and even giving rise to unsteady and turbulent flows as Reynolds numbers become larger. This change in flow structure can have a profound impact on how solutes are transported through the porous medium, influencing how effective large-scale transport should be modeled. Here we simulate, using high-resolution numerical models, flow and transport through an idealized porous medium for flow conditions over a range of Reynolds numbers, including steady and unsteady flows. For all these conditions we propose and test three upscaled models for transport—an advection dispersion equation, an uncorrelated spatial model (USM) and a spatial Markov model (SMM). The USM and SMM fall into the wider and more general family of continuous time random walk models. We test these models by their ability to reproduce pre-asymptotic and asymptotic plume second centered moments and breakthrough curves. We demonstrate that for steady flows where inertial effects are strong, the spatial Markov model outperforms the other two, faithfully capturing many of the non-Fickian features of transport, while for unsteady flows the uncorrelated spatial model performs best, due to the fact that unsteadiness in the flow field dampens the role of correlation on large scale transport. We conclude that correlation must be accounted for to properly upscale transport in steady flows, while it can be neglected in unsteady flows.

Effects of Time-Duration on the Performance of the Spatial-Markov Model for Land use Change Forecasting

Markov chain is one of the most widely used methods for land use change forecasting, however, it's a non-spatial model and few papers have discussed the effects of time-duration on its performance. In this paper, we first present the primary methodologies of the Spatial-Markov model, which endows the ordinary Markov chain with spatial dimension using spatial analysis techniques, and then explore the effects of forecasting time-duration on the model's performance. By taking Shandong province, China as a case study area, land use maps in 1990, 1995, 2000, 2005 and 2010 were created using on Landsat images and then the Spatial-Markov model was developed at 1 km spatial scale. In detail, we repeatedly run the model by choosing different initial time points and the same time step (five year interval) to simulate the spatial-temporal dynamics of land use change from 1990 to 2010. The forecasting results of a single run included a stack of ratio scale images and a derived nominal scale image, chi(2) test and Kappa coefficient were adopted to evaluate their accuracy respectively. It turned out that the Spatial-Markov model could achieve very good performance for short period forecasting. For the case study, it was quite qualified for the prediction of three time steps (up to 15 years) or more within which the results had much high reliability, however, time-duration of forecasting had much significant impact on the model's performance, the longer the forecasting duration, the lower the model's accuracy.

Modeling preasymptotic transport in flows with significant inertial and trapping effects – The importance of velocity correlations and a spatial Markov model

We study solute transport in a periodic channel with a sinusoidal wavy boundary when inertial flow effects are sufficiently large to be important, but do not give rise to turbulence. This configuration and setup are known to result in large recirculation zones that can act as traps for solutes; these traps can significantly affect dispersion of the solute as it moves through the domain. Previous studies have considered the effect of inertia on asymptotic dispersion in such geometries. Here we develop an effective spatial Markov model that aims to describe transport all the way from preasymptotic to asymptotic times. In particular we demonstrate that correlation effects must be included in such an effective model when Péclet numbers are larger than O(100) in order to reliably predict observed breakthrough curves and the temporal evolution of second centered moments. For smaller Péclet numbers correlation effects, while present, are weak and do not appear to play a significant role. For many systems of practical interest, if Reynolds numbers are large, it may be typical that Péclet numbers are large also given that Schmidt numbers for typical fluids and solutes can vary between 1 and 500. This suggests that when Reynolds numbers are large, any effective theories of transport should incorporate correlation as part of the upscaling procedure, which many conventional approaches currently do not do. We define a novel parameter to quantify the importance of this correlation. Next, using the theory of CTRWs we explain a to date unexplained phenomenon of why dispersion coefficients for a fixed Péclet number increase with increasing Reynolds number, but saturate above a certain value. Finally we also demonstrate that effective preasymptotic models that do not adequately account for velocity correlations will also not predict asymptotic dispersion coefficients correctly.

Land use and landscape pattern changes in coastal areas of Shandong province, China

In this paper, the Spatial-Markov model, which was based on the theory of Markov process and spatial analysis techniques, was proposed to simulate land use change and landscape dynamics. By the Spatial-Markov model, the study area could be divided into numerous lattices and land use change in each lattices was simulated separately by the Markov process model. The outputs of the model include a set of ratio scale images and a nominal scale image. The whole process of the model was fulfilled by compiling programs with AML in ArcGIS 9.3. The coastal area of Shandong province was selected as the case study area. Land use maps were extracted based on Landsat TM/ETM+ images captured in 2000, 2005, and 2010 respectively. Firstly, characteristics of land use change and landscape dynamics were analyzed. It showed that, from 2000 to 2010, urban area and rural settlement expanded dramatically by massively occupying farmland, which, in turn, drove grassland reclaimed to farmland. At the landscape level, the landscape fragmentation increased, and both the diversity and evenness of the landscape increased. Secondly, using land use maps in 2000 and 2005, the Spatial-Markov model was developed to simulate the land use map in 2010 at a spatial scale of 500 m. At the same time, the CA-Markov model was selected for model comparison, in specific, eleven driving factors were selected and the Logistic regression method was used to create the transitional maps for CA. Both Kappa coefficient and landscape indices were introduced to evaluate and compare the two models. It showed that the Spatial-Markov model not only achieved much higher Kappa coefficient, but also much better landscape indices than the CA-Markov model. Therefore, the Spatial-Markov model was applied to predict land use change and landscape dynamics in the next decade. Moreover, the prediction result shows that, from 2010 to 2020, areas of urban area and rural settlement will go on increasing, while areas of farmland will continue to decline. At the landscape level, all the landscape indices will follow their historical trend except for fractal dimension. As to the landscape indices at the class level, all landscape types will follow the same trend as before except for water and unused land.

Blockwise empirical likelihood for spatial Markov model assessment

Blockwise empirical likelihood for spatial Markov model assessment