Faculty of Science School of Statistics snd Acturial Science 9907894x craig.morgan@goldfields.co.za / This dissertation presents a detailed study of geostatistics. Included in this work are details of the development of geostatistics and its usefulness both in and outside of the mining industry, a comprehensive presentation of the theory of geostatistics, and a discussion of the application of this theory to practical situations. A published debate over the validity of geostatistics is also examined. The ultimate goal of this dissertation is to provide a thorough investigation of geostatistics from both a theoretical and a practical perspective. The theory presented in this dissertation is thus tested on various spatial data sets, and from these tests it is concluded that geostatistics can be effectively used in practice provided that the practitioner fully understands the theory of geostatistics and the spatial data being analyzed. A particularly interesting conclusion to come out of this dissertation is the importance of using additive regionalized variables in all geostatistical analyses.

Thesis advisor: Alfredo Urzua / Thesis advisor: John Ebel / Nine empirical equations that estimate saturated hydraulic conductivity as a func- tion of grain size in well-graded sands with gravels having large uniformity coecients (U > 50) are evaluated by comparing their accuracy when predicting observed conduc- tivities in constant head permeability tests. According to the ndings of this thesis, in decreasing order of accuracy these equations are: USBR (Vukovic and Soro, 1992; USBR, 1978), Hazen (Hazen, 1892), Slichter (Slichter, 1898), Kozeny-Carman (Carrier, 2003), Fair and Hatch (Fair and Hatch, 1933), Terzaghi (Vukovic and Soro, 1992), Beyer (Beyer, 1966), Kruger (Vukovic and Soro, 1992), and Zunker (Zunker, 1932). These re- sults are based on multiple constant head permeability tests on two samples of granular material corresponding to well-graded sands with gravels. Using the USBR equation sat- urated hydraulic conductivities for a statistical population of 874 samples of well-graded sands with gravels forming 29 loads from a heap leaching mine in northern Chile are calculated. Results indicate that, using the USBR equation, on average the hydraulic conductivity of the leaching heaps has a two standard deviation range between 0.18 and 0.15 cm/s. Permeability tests on the actual material used in the heaps provided by the mine shows that the results presented in this thesis are consistent with actual observa- tions and represent saturated conductivities in heaps up to 3 m high under a pressures of up to 62 Kpa. In future work hydraulic conductivities can be combined with water retention curves, discharge rates, irrigation rates, porosities, and consolidation so as to evaluate the relationship between copper yields and the hydraulic conductivities of the heap. / Thesis (MS) — Boston College, 2016. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Geology and Geophysics.

The main objective of our study is to employ copula methodology to develop Bayesian hierarchical models to study the dependencies exhibited by temporal, spatial and spatio-temporal processes. We develop hierarchical models for both discrete and continuous outcomes. In doing so we expect to address the dearth of copula based Bayesian hierarchical models to study hydro-meteorological events and other physical processes yielding discrete responses. First, we present Bayesian methods of analysis for longitudinal binary outcomes using Generalized Linear Mixed models (GLMM). We allow flexible marginal association among the repeated outcomes from different time-points. An unique property of this copula-based GLMM is that if the marginal link function is integrated over the distribution of the random effects, its form remains same as that of the conditional link function. This unique property enables us to retain the physical interpretation of the fixed effects under conditional and marginal model and yield proper posterior distribution. We illustrate the performance of the posited model using real life AIDS data and demonstrate its superiority over the traditional Gaussian random effects model. We develop a semiparametric extension of our GLMM and re-analyze the data from the AIDS study. Next, we propose a general class of models to handle non-Gaussian spatial data. The proposed model can deal with geostatistical data that can accommodate skewness, tail-heaviness, multimodality. We fix the distribution of the marginal processes and induce dependence via copulas. We illustrate the superior predictive performance of our approach in modeling precipitation data as compared to other kriging variants. Thereafter, we employ mixture kernels as the copula function to accommodate non-stationary data. We demonstrate the adequacy of this non-stationary model by analyzing permeability data. In both cases we perform extensive simulation studies to investigate the performances of the posited models under misspecification. Finally, we take up the important problem of modeling multivariate extreme values with copulas. We describe, in detail, how dependences can be induced in the block maxima approach and peak over threshold approach by an extreme value copula. We prove the ability of the posited model to handle both strong and weak extremal dependence and derive the conditions for posterior propriety. We analyze the extreme precipitation events in the continental United States for the past 98 years and come up with a suite of predictive maps.

Recently developed models of the acid fracturing process have shown that the differential etching necessary to create lasting fracture conductivity is caused by the heterogeneous distributions of permeability and mineralogy along the fracture faces. To predict the conductivity that can be created by acid in a particular formation, the models require information about these formation properties. This research aims to quantify correlation lengths using a geostatistical description of small scale heterogeneity to ascertain the distribution of permeability and mineralogy in a carbonate formation. The correlation length parameters are a first step in being able to couple acid transport and rock dissolution models at reservoir scale with a model of fracture conductivity based on channels and roughness features caused by small scale heterogeneity. Geostatistical parameters of small scale heterogeneity affecting wells in the Hugoton Field are developed. Data leading to their derivation are obtained from a combination of well logs and cores. The permeability of slabbed core is measured to yield vertical correlation length. Well logs are used to estimate permeability via an empirical relationship between core plug permeability and well log data for calculation of horizontal correlation length. A fracture simulator computes the acid etched fracture width for known treatment conditions. The resulting geostatistical parameters and acid etched width are used to predict acid fracture performance for a well in the Hugoton Field. Application of new model conductivity correlations results in a unique prediction for the acid fracture case study that differs from the industry standard. Improvements in low cost stimulation treatments such as acid fracturing are the key to revitalizing production in mature carbonate reservoirs like the Hugoton Field. Planning and development of new wells in any carbonate formation necessarily must consider acid fracturing as a production stimulation technique. Reliable models that accurately predict acid fracture conductivity can be used to make an informed investment decision.

The variogram plays a central role in the analysis of geostatistical data. A valid variogram model is selected and the parameters of that model are estimated before kriging (spatial prediction) is performed. These inference procedures are generally based upon examination of the empirical variogram, which consists of average squared differences of data taken at sites lagged the same distance apart in the same direction. The ability of the analyst to estimate variogram parameters efficiently is affected significantly by the sampling design, i.e., the spatial configuration of sites where measurements are taken. In this paper, we propose design criteria that, in contrast to some previously proposed criteria oriented towards kriging with a known variogram, emphasize the accurate estimation of the variogram. These criteria are modifications of design criteria that are popular in the context of (nonlinear) regression models. The two main distinguishing features of the present context are that the addition of a single site to the design produces as many new lags as there are existing sites and hence also produces that many new squared differences from which the variograrn is estimated. Secondly, those squared differences are generally correlated, which inhibits the use of many standard design methods that rest upon the assumption of uncorrelated errors. Several approaches to design construction which account for these features are described and illustrated with two examples. We compare their efficiency to simple random sampling and regular and space-filling designs and find considerable improvements. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

We use geostatistical and pedotrasnfer functions to estimate the three-dimensional distributions of soil types and hydraulic properties in a relatively large volume of vadose zone underlying the Maricopa Agriculture Center near Phoenix, Arizona. Soil texture and bulk density data from the site are analyzed geostatistically to reveal the underlying stratigraphy as well as finer features of their three-dimensional variability in space. Such fine features are revealed by cokriging soil texture and water content measured prior to large-scale long-term infiltration experiments. Resultant estimates of soil texture and bulk density data across the site are then used as input into a pedotransfer function to produce estimates of soil hydraulic parameter (saturated and residual water content θs and θr, saturated hydraulic conductivity Ks, van Genuchten parameters αand n) distributions across the site in three dimensions. We compare these estimates with laboratory-measured values of these same hydraulic parameters and find the estimated parameters match the measured well for θs, n and Ks but not well for θr nor α, while some measured extreme values are not captured. Finally the estimated soil hydraulic parameters are put into a numerical simulator to test the reliability of the models. Resultant simulated water contents do not agree well with those observed, indicating inverse calibration is required to improve the modeling performance. The results of this research conform to a previous work by Wang et al. at 2003. Also this research covers the gaps of Wang’s work in sense of generating 3-D heterogeneous fields of soil texture and bulk density by cokriging and providing comparisons between estimated and measured soil hydraulic parameters with new field and laboratory measurements of water retentions datasets.

Field scale variability of soil texture can influence crop yield and movement of soil water in the field. The objective of this study was to investigate the spatial structure of soil texture in relation to the variability of nitrate-N leaching using geostatistics. Soil textural fractions showed strong spatial autocorrelations from surface to 60 cm depth. Random variability of soil texture increased with depth. Soil water content, as well as total carbon, total nitrogen and soil organic carbon of top 15 cm, also showed spatial autocorrelations similar to soil texture. Elevation, relative slope position and vertical distance to channel network showed significant influence on the distribution of soil texture. Soil texture at 90 cm depth correlated best with cumulative percolated water and cumulative nitrate leached in field lysimeters. Our results showed that soil layers with low hydraulic conductivity control the water and nitrate movement through the soil profile.

Made available in DSpace on 2014-06-11T19:23:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-12-22Bitstream added on 2014-06-13T19:49:59Z : No. of bitstreams: 1 teixeira_db_me_jabo.pdf: 508270 bytes, checksum: 93d2af6f4bf67e9aab6a84ffe3f4ac1a (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / A emissão de CO2 do solo (FCO2) apresenta alta variabilidade espacial, sendo devida a grande dependência espacial existente nas propriedades do solo que a influenciam. Neste estudo objetivou-se (i) caracterizar e relacionar a variabilidade e a distribuição espacial da FCO2, temperatura do solo, porosidade livre de água (PLA), teor de matéria orgânica do solo (MO) e densidade do solo (Ds), (ii) avaliar a acurácia dos resultados fornecidos pelo método da krigagem ordinária (KO) e simulação sequencial Gaussiana (SSG), e (iii) avaliar a incerteza na predição da variabilidade espacial das FCO2 e demais propriedades utilizando a SSG. O estudo foi conduzido em uma malha amostral regular de 60 x 60 m2 com 141 pontos, com espaçamento mínimo variando de 0,50 a 10 m, instalada em área de cana-de-açúcar. Nestes pontos foram avaliados a FCO2, temperatura do solo, PLA, determinadas com base na média de 07 dias de avaliação, MO e Ds. Todas as variáveis apresentaram estrutura de dependência espacial, sendo ajustados modelos Gaussianos, esféricos e exponenciais. A configuração da malha amostral e possivelmente a presença de espessa camada de resíduos da cultura sobre o solo influenciaram a estrutura de variabilidade espacial da FCO2, temperatura e MO. FCO2 apresentou correlações positivas com a MO (r = 0,25, p < 0,05) e PLA (r = 0,27, p < 0,01) e negativa com a Ds (r = - 0,41, p < 0,01). No entanto, quando os valores digitais estimados espacialmente (N=8.833) são considerados, a PLA passa a ser a principal variável responsável pelas características espaciais da FCO2, apresentando correlação de 0,26 (p < 0,01). As simulações individuais propiciaram, para todas as variáveis analisadas, melhor reprodução das funções de distribuição acumuladas (fdac), e dos variogramas em comparação... / The soil CO2 emission (FCO2) has high spatial variability, which caused due to the strong spatial dependence in soil properties that influence it. This study aimed to (i) to characterize the variability and spatial distribution of FCO2, soil temperature, air-filled pore space (AFPS), soil organic matter (OM) and soil bulk density (BD) and related properties, (ii) evaluate the accuracy of the results provided by the method of ordinary kriging (OK) and sequential Gaussian simulation (SGS), and (iii) evaluate the uncertainty in predicting the spatial variability of FCO2 and other properties using the SSG. The study was conducted on an regular sampling grid with 141 points, with spacing ranging from 0.50 to 10 m, installed in a sugarcane area. In this place were evaluated FCO2, soil temperature, AFPS, were based on the average of 07 days of evaluation, OM and BD. All variables showed spatial dependence structure, and models adjusted Gaussian, spherical and exponential. The configuration of the sampling grid and the presence of intense layer of crop residues in the soil influenced the structure of spatial variability of FCO2, temperature, and OM. The FCO2 showed positive correlations with OM (r = 0.25, p <0.05) and AFPS (r = 0.27, p <0.01) and negatively with Ds (r = - 0.41, p <0.01). However, when the estimated spatially values are considered, the AFPS becomes the main variable responsible for the spatial characteristics of FCO2, showing correlation of 0.26 (p <0.01). The individual simulations led to all variables, better reproduction of the cumulative distribution functions (cdf), and variograms compared to OK and E-type estimate. The analysis results show strong similarities between the E-type estimates to those generated by the procedure of OK. The major uncertainties in predicting FCO2 were associated with areas with the highest... (Complete abstract click electronic access below)

Spatial interpolation methods were analyzed to determine the best fit for subsidence rates and to create a predictive surface for the lower Mississippi River Industrial corridor (LMRIC). Empirical Bayesian kriging, ordinary kriging, universal kriging, and Inverse Distance Weighted interpolation methods were applied to the 2004 National Oceanic and Atmospheric Administration (NOAA) published Technical Report #50 dataset and cross validation methods were utilized to determine the accuracy of each method. The mean error and root mean square error were calculated for each interpolation method, then used to detect bias and compare the predicted value with the actual observation value. Cross-validation estimates are comparable for each method statistically and visually; however, the results indicate the empirical Bayesian kriging interpolation method is the most accurate of the methods using the lowest root mean square scores. Digital elevation models for the years 2025, 2050, and 2075 were developed based on the predictive surface of subsidence rates using the results from the empirical Bayesian kriging interpolation method. Results indicate that by 2025, 30.9% of landmass in the LMRIC will be below sea level, with 41.9% below sea level by 2050, and 53.5% by 2075. Subsidence rates in the LMRIC range from approximately 28 mm to 2 mm per year. Eighteen of the 153 industrial complexes located in the LMRIC are estimated to be below sea level by the year 2075.

There has been an increased interest in characterizing and interpreting ecological heterogeneity over space and time in the past two decades. This is mainly due to the renewed recognition of the significance of heterogeneity in ecological theories. However, studies that have combined both spatial and temporal aspects of heterogeneity have been rare. A unified approach to define and quantify heterogeneity has also been lacking. Designed to overcome these problems, this study was conducted in a 40-year-old Pinus elliottii Engelm. forest at the Savannah River Site near Aiken, SC, USA with the following specific objectives: 1) to characterize the spatial patterns of soil and forest floor variables (moisture, pH, soil available nitrogen and phosphate, forest floor and soil carbon and nitrogen), 2) to examine the dynamics of these spatial patterns in response to two types of disturbance: whole-tree harvesting and girdling, and 3) to evaluate some of the current methods for quantifying ecological heterogeneity. In response to both disturbance treatments, spatial heterogeneity measured by sample variance showed a marked "increase and then decline" temporal pattern in soil moisture, soil available nitrogen and phosphorus. Similar patterns were not found in total soil C and N, and total litter C and N. Harvesting resulted in greater and more drastic changes in the variations of soil nutrients and water than did girdling. Despite the popularity of semivariogram analysis in recent ecological studies, the technique did not provide consistent results on patterns of heterogeneity in our system. A simulation experiment demonstrated that semivariogram analysis may suffer from many problems when it is used to characterize patchiness, one form of heterogeneity. The results from this study have a number of implications. First, spatial patterns of soil resources are high dynamic. The dynamics of patterns in soil resources may partly account for the weak correlation between vegetation and soil observed in ecological literature. Second, heterogeneity may be most effectively quantified by first identifying quantifiable components and then quantifying these components individually. A common pattern can be sought by comparing patterns of different components of heterogeneity for a given ecological property and by comparing patterns of different ecological variables for a given component of heterogeneity. Third, compared to surveys, field manipulative experiments can provide information that link patterns with ecological processes. As such, this study adds to ecological literature valuable information on temporal changes of soil heterogeneity following disturbance and conceptual advances in the quantification of ecological heterogeneity. / Ph. D.