Gregoire Mariethoz

A practical guide to performing multiple-point statistical simulations with the Direct Sampling algorithm

The Direct Sampling (DS) algorithm is a recently developed multiple-point statistical simulation technique. It directly scans the training image (TI) for a given data event instead of storing the training probability values in a catalogue prior to simulation. By using distances between the given data events and the TI patterns, DS allows to simulate categorical, continuous and multivariate problems. Benefiting from the wide spectrum of potential applications of DS, requires understanding of the user-defined input parameters. Therefore, we list the most important parameters and assess their impact on the generated simulations. Real case TIs are used, including an image of ice-wedge polygons, a marble slice and snow crystals, all three as continuous and categorical images. We also use a 3D categorical TI representing a block of concrete to demonstrate the capacity of DS to generate 3D simulations. First, a quantitative sensitivity analysis is conducted on the three parameters balancing simulation quality and CPU time: the acceptance threshold t, the fraction of TI to scan f and the number of neighbors n. Next to a visual inspection of the generated simulations, the performance is analyzed in terms of speed of calculation and quality of pattern reproduction. Whereas decreasing the CPU time by influencing t and n is at the expense of simulation quality, reducing the scanned fraction of the TI allows substantial computational gains without degrading the quality as long as the TI contains enough reproducible patterns. We also illustrate the quality improvement resulting from post-processing and the potential of DS to simulate bivariate problems and to honor conditioning data. We report a comprehensive guide to performing multiple-point statistical simulations with the DS algorithm and provide recommendations on how to set the input parameters appropriately.

A composite annual-resolution stalagmite record of North Atlantic climate over the last three millennia

Annually laminated stalagmites can be used to construct a precise chronology, and variations in laminae thickness provide an annual growth-rate record that can be used as a proxy for past climate and environmental change. Here, we present and analyse the first composite speleothem annual growth-rate record based on five stalagmites from the same cave system in northwest Scotland, where precipitation is sensitive to North Atlantic climate variability and the winter North Atlantic Oscillation (NAO). Our 3000-year record confirms persistently low growth-rates, reflective of positive NAO states, during the Medieval Climate Anomaly (MCA). Another persistently low growth period occurring at 290-550 CE coincides with the European Migration Period, and a subsequent period of sustained fast growth-rate (negative NAO) from 600-900 AD provides the climate context for the Viking Age in northern and western Europe.

Accelerating geostatistical simulations using graphics processing units (GPU)

Geostatistical simulations have become a widely used tool for modeling of oil and gas reservoirs and the assessment of uncertainty. One important current issue is the development of high-resolution models in a reasonable computational time. A possible solution is based on taking advantage of parallel computational strategies. In this paper we present a new methodology that exploits the benefits of graphics processing units (GPUs) along with the master-slave architecture for geostatistical simulations that are based on random paths. The methodology is a hybrid method in which different levels of master and slave processors are used to distribute the computational grid points and to maximize the use of multiple processors utilized in GPU. It avoids conflicts between concurrently simulated grid points, an important issue in high-resolution and efficient simulations. For the sake of comparison, two distinct parallelization methods are implemented, one of which is specific to pattern-based simulations. To illustrate the efficiency of the method, the algorithm for the simulation of pattern is adapted with the GPU. Performance tests are carried out with three large grid sizes. The results are compared with those obtained based on simulations with central processing units (CPU). The comparison indicates that the use of GPUs reduces the computation time by a factor of 26-85.

Bridges between multiple-point geostatistics and texture synthesis

Multiple-Point Simulations (MPS) is a family of geostatistical tools that has received a lot of attention in recent years for the characterization of spatial phenomena in geosciences. It relies on the definition of training images to represent a given type of spatial variability, or texture. We show that the algorithmic tools used are similar in many ways to techniques developed in computer graphics, where there is a need to generate large amounts of realistic textures for applications such as video games and animated movies. Similarly to MPS, these texture synthesis methods use training images, or exemplars, to generate realistic-looking graphical textures.Both domains of multiple-point geostatistics and example-based texture synthesis present similarities in their historic development and share similar concepts. These disciplines have however remained separated, and as a result significant algorithmic innovations in each discipline have not been universally adopted. Texture synthesis algorithms present drastically increased computational efficiency, patterns reproduction and user control. At the same time, MPS developed ways to condition models to spatial data and to produce 3D stochastic realizations, which have not been thoroughly investigated in the field of texture synthesis.In this paper we review the possible links between these disciplines and show the potential and limitations of using concepts and approaches from texture synthesis in MPS. We also provide guidelines on how recent developments could benefit both fields of research, and what challenges remain open. We provide a comparative historical overview of MPS and texture synthesis.Some concepts that were thought as original in geostatistics have actually been used for a long time in computer graphics.We analyze the considerable computational gains that were achieved in texture synthesis in the last decades.Recommendations are given regarding future research directions and potential cross-fertilization between disciplines.

Multiple-point geostatistical simulation using the bunch-pasting direct sampling method

Multiple-point geostatistics has opened a new field of methodologies by which complex geological phenomena have been modeled efficiently. In this study, a modified form of direct sampling (DS) method is introduced which not only keeps the strength of DS simulation technique but also speeds it up by one or two orders of magnitude. While previous methods are based on pasting only one point at a time, here the simulation is done by pasting a bunch of nodes at a time, effectively combining the flexibility of DS with the computational advantages of patch-based methods. This bears the potential of significantly speeding up the DS method. The proposed simulation method can be used with unilateral or random simulation paths. No overlap occurs in the simulation procedure because the bunch takes the shape of the empty space around the simulated nodes. Systematic tests are carried on different training images including both categorical and continuous variables, showing that the realizations preserve the patterns existent in the training image. To illustrate the method, a Matlab implementation of the method is attached to the paper.

A general parallelization strategy for random path based geostatistical simulation methods

The size of simulation grids used for numerical models has increased by many orders of magnitude in the past years, and this trend is likely to continue. Efficient pixel-based geostatistical simulation algorithms have been developed, but for very large grids and complex spatial models, the computational burden remains heavy. As cluster computers become widely available, using parallel strategies is a natural step for increasing the usable grid size and the complexity of the models. These strategies must profit from of the possibilities offered by machines with a large number of processors. On such machines, the bottleneck is often the communication time between processors. We present a strategy distributing grid nodes among all available processors while minimizing communication and latency times. It consists in centralizing the simulation on a master processor that calls other slave processors as if they were functions simulating one node every time. The key is to decouple the sending and the receiving operations to avoid synchronization. Centralization allows having a conflict management system ensuring that nodes being simulated simultaneously do not interfere in terms of neighborhood. The strategy is computationally efficient and is versatile enough to be applicable to all random path based simulation methods.

Simulation of Earth textures by conditional image quilting

Training image-based approaches for stochastic simulations have recently gained attention in surface and subsurface hydrology. This family of methods allows the creation of multiple realizations of a study domain, with a spatial continuity based on a training image (TI) that contains the variability, connectivity, and structural properties deemed realistic. A major drawback of these methods is their computational and/or memory cost, making certain applications challenging. It was found that similar methods, also based on training images or exemplars, have been proposed in computer graphics. One such method, image quilting (IQ), is introduced in this paper and adapted for hydrogeological applications. The main difficulty is that Image Quilting was originally not designed to produce conditional simulations and was restricted to 2-D images. In this paper, the original method developed in computer graphics has been modified to accommodate conditioning data and 3-D problems. This new conditional image quilting method (CIQ) is patch based, does not require constructing a pattern databases, and can be used with both categorical and continuous training images. The main concept is to optimally cut the patches such that they overlap with minimum discontinuity. The optimal cut is determined using a dynamic programming algorithm. Conditioning is accomplished by prior selection of patches that are compatible with the conditioning data. The performance of CIQ is tested for a variety of hydrogeological test cases. The results, when compared with previous multiple-point statistics (MPS) methods, indicate an improvement in CPU time by a factor of at least 50.

Evaporative cooling of speleothem drip water

This study describes the first use of concurrent high-precision temperature and drip rate monitoring to explore what controls the temperature of speleothem forming drip water. Two contrasting sites, one with fast transient and one with slow constant dripping, in a temperate semi-arid location (Wellington, NSW, Australia), exhibit drip water temperatures which deviate significantly from the cave air temperature. We confirm the hypothesis that evaporative cooling is the dominant, but so far unattributed, control causing significant disequilibrium between drip water and host rock/air temperatures. The amount of cooling is dependent on the drip rate, relative humidity and ventilation. Our results have implications for the interpretation of temperature-sensitive, speleothem climate proxies such as δ18O, cave microecology and the use of heat as a tracer in karst. Understanding the processes controlling the temperature of speleothem-forming cave drip waters is vital for assessing the reliability of such deposits as archives of climate change.

A space and time scale‐dependent nonlinear geostatistical approach for downscaling daily precipitation and temperature

A geostatistical framework is proposed to downscale daily precipitation and temperature. The methodology is based on multiple-point geostatistics (MPS), where a multivariate training image is used to represent the spatial relationship between daily precipitation and daily temperature over several years. Here the training image consists of daily rainfall and temperature outputs from the Weather Research and Forecasting (WRF) model at 50 and 10 km resolution for a 20 year period ranging from 1985 to 2004. The data are used to predict downscaled climate variables for the year 2005. The result, for each downscaled pixel, is daily time series of precipitation and temperature that are spatially dependent. Comparison of predicted precipitation and temperature against a reference data set indicates that both the seasonal average climate response together with the temporal variability are well reproduced. The explicit inclusion of time dependence is explored by considering the climate properties of the previous day as an additional variable. Comparison of simulations with and without inclusion of time dependence shows that the temporal dependence only slightly improves the daily prediction because the temporal variability is already well represented in the conditioning data. Overall, the study shows that the multiple-point geostatistics approach is an efficient tool to be used for statistical downscaling to obtain local-scale estimates of precipitation and temperature from General Circulation Models.

Training Images from Process-Imitating Methods

The lack of a suitable training image is one of the main limitations of the application of multiple-point statistics (MPS) for the characterization of heterogeneity in real case studies. Process-imitating facies modeling techniques can potentially provide training images. However, the parameterization of these process-imitating techniques is not straightforward. Moreover, reproducing the resulting heterogeneous patterns with standard MPS can be challenging. Here the statistical properties of the paleoclimatic data set are used to select the best parameter sets for the process-imitating methods. The data set is composed of 278 lithological logs drilled in the lower Namoi catchment, New South Wales, Australia. A good understanding of the hydrogeological connectivity of this aquifer is needed to tackle groundwater management issues. The spatial variability of the facies within the lithological logs and calculated models is measured using fractal dimension, transition probability, and vertical facies proportion. To accommodate the vertical proportions trend of the data set, four different training images are simulated. The grain size is simulated alongside the lithological codes and used as an auxiliary variable in the direct sampling implementation of MPS. In this way, one can obtain conditional MPS simulations that preserve the quality and the realism of the training images simulated with the process-imitating method. The main outcome of this study is the possibility of obtaining MPS simulations that respect the statistical properties observed in the real data set and honor the observed conditioning data, while preserving the complex heterogeneity generated by the process-imitating method. In addition, it is demonstrated that an equilibrium of good fit among all the statistical properties of the data set should be considered when selecting a suitable set of parameters for the process-imitating simulations.