Estimating the local mean for Bayesian maximum entropy by generalized least squares and maximum likelihood, and an application to the spatial analysis of a censored soil variable

The Bayesian maximum entropy (BME) method is a valuable tool, with rigorous theoretical underpinnings, with which to predict with soft (imprecise) data. The methodology uses a general knowledge base to derive a joint prior distribution of the data and the prediction by the criterion of maximum entropy; the hard (precise) and soft data are then processed using this prior distribution to yield a posterior distribution that provides the BME prediction. The general knowledge base commonly consists of the mean and covariance functions, which may be extracted from the data. The common method for extracting the mean function from the data is a generalized least squares (GLS) approach. However, when the soft data take the form of intervals of plausible values, this method can result in errors in the BME predictions. This paper suggests a maximum likelihood (ML) method for fitting the local mean. The two methods are compared in terms of their predictions, firstly on simulated random fields and then on a case study to predict the depth of soil using some censored data. The results show that the ML method can result in more accurate BME predictions; the degree of improvement over the GLS method depends on the parameters of the spatial covariance model.     

综合BME和BNN法的农田土壤水分与养分分布空间插值

掌握农田土壤水分和养分的空间分布特征是实现农田土壤精确管理及实施精确农业的重要依据。以有限的采样信息为基础,通过多种空间分析理论的融合,形成优势互补的综合方法,对提高土壤变量空间分布模拟和绘图精度具有重要意义。该文将贝叶斯最大熵法(Bayesian maximum entropy,BME)和贝叶斯人工神经网络方法(Bayesian neural networks,BNN)结合形成一种空间插值新方法,即用BNN法表达估值的不确定性,并将其结果融入现代地质统计学BME法中,用融入BNN法结果的BME法(Bayesian maximum entropy method combined with Bayesian neural networks,BMENN)模拟土壤变量的空间分布。以江苏省扬州市区北部某田块的土壤水分、有机质、全氮、碱解氮、速效钾和速效磷6种土壤特性的采样数据为例,运用交叉验证法,将BMENN法对土壤变量的估值精度与BNN法、普通克立格法(ordinary Kriging,OK)进行了比较。结果表明:与OK法和BNN法相比,BMENN法将估计方差(mean squared error,MSE)缩小2.26%~23.54%,具有最小的估计方差和接近于0的平均绝对误差(mean error,ME);BMENN法的估计值与实测值相关系数更大(r=0.62~0.89),具有更高的相关程度;MSE的组成分析表明,BMENN法再现变量波动程度和波动大小的能力更强;从模拟的空间分布图来看,BMENN法绘制的空间分布图更连续,"牛眼"较少,更符合土壤变量的地学规律。BMENN法对于利用有限数据信息提高土壤变量空间分布模拟精度具有重要意义,并可为土壤管理、精准农业的实施以及区域环境规划等提供科学依据。     

贝叶斯最大熵地统计学方法及其在土壤和环境科学上的应用

贝叶斯最大熵(Bayesian Maximum Entropy,BME)地统计学方法是近年来出现的一种时空地统计学新方法。相对于传统的克里金方法,该法具有坚实的认识论框架和方法学基础。它不需要作线性估值、空间匀质和正态分布的假设,能够融入先验知识和软数据,并且不会损失其中蕴含的有用信息,提高了分析精度。本文首先介绍了BME的基本理论及其估值方法,随后简单描述了该方法的理论发展过程及其在土壤和环境科学上的应用情况,最后对该方法的应用做了总结与展望。经过国外研究者多年的开发和实践,BME方法已经被证明是一个理论上较为成熟,能够应用到实际研究中的优秀地统计学方法,在资源环境评估上有着广泛的应用前景。     

Bayesian Maximum Entropy prediction of soil categories using a traditional soil map as soft information

Bayesian Maximum Entropy was used to estimate the probabilities of occurrence of soil categories in the Netherlands, and to simulate realizations from the associated multi‐point pdf. Besides the hard observations (H) of the categories at 8369 locations, the soil map of the Netherlands 1:50 000 was used as soft information (S). The category with the maximum estimated probability was used as the predicted category. The quality of the resulting BME(HS)‐map was compared with that of the BME(H)‐map obtained by using only the hard data in BME‐estimation, and with the existing soil map. Validation with a probability sample showed that the use of the soft information in BME‐estimation leads to a considerable and significant increase of map purity by 15%. This increase of map purity was due to the high purity of the existing soil map (71.3%). The purity of the BME(HS) was only slightly larger than that of the existing soil map. This was due to the small correlation length of the soil categories. The theoretical purity of the BME‐maps overestimated the actual map purity, which can be partly explained by the biased estimates of the one‐point bivariate probabilities of hard and soft categories of the same label. Part of the hard data is collected to describe characteristic soil profiles of the map units which explains the bias. Therefore, care must be taken when using the purposively selected data in soil information systems for calibrating the probability model. It is concluded that BME is a valuable method for spatial prediction and simulation of soil categories when the number of categories is rather small (say < 10). For larger numbers of categories, the computational burden becomes prohibitive, and large samples are needed for calibration of the probability model.     

A geostatistical method to account for the number of aliquots in composite samples for normal and lognormal random variables

Geostatistical methods can be used to calculate predictions of soil variables at unsampled locations, but the methodology is typically based on samples collected on identical sample supports. In this paper, we provide and test theory that allows the inclusion of data from mixed sample supports in a single analysis. In particular, we consider composite sample supports that are defined by the number of aliquots used to form a single composite sample, n_i, and the set of locations, x _i, from which the aliquots were collected. We allow both n_i and x _i to vary between samples ( x _i can vary in the extent and geometry of the aliquot locations), and thereby show how point data (a special case of composite data, defined by n_i = 1 and x _i as the known sample point) can be included in the same geostatistical analysis as composite data. A further complication arises when data are not normally distributed, rather their logarithm is. When composite sampling is used for such lognormal data, the sample support affects not only the variance but also the mean. We give the theory for normally distributed variables, and also derive an approximation that can be used when the point‐support variable is lognormal. We focus on this latter case, and test the approach with a series of simulation experiments. Finally, we illustrate the approach on a dataset of soil organic carbon (SOC) values from a grazing property in Queensland, Australia, where soil information from two measurement phases was obtained on different supports.     

Predicting Freshwater Critical Loads of Acidification at the Catchment Scale: An Empirical Model

Current applications of the critical loads concept are geared primarily toward targeting emission control strategies at a national and international level. Maps of critical loads for freshwaters have been produced in grid form based on water samples of representative sites within each grid square. However, the water chemistry data required to calculate freshwater critical loads are not always readily available at a national level and maps are therefore limited to catchments where such data exist. This paper describes the development of an approach that uses nationally available secondary data to predict freshwater critical loads for catchments lacking the appropriate water chemistry information. An empirical statistical model is calibrated using data from 78 catchments throughout Scotland. Water chemistry for each catchment has been determined. Each catchment is characterized according to a number of attributes. Redundancy analysis of these data shows clear relationships between catchment attributes and the critical load derived from the water chemistry. The key variables that explain most of the variation in critical load relate to soil, geology and land use within the catchment. Using these variables as predictors in a regression analysis, the critical load can be predicted across a broad gradient of sensitivity (R² _adj=0.81). The predictive power of the model was maintained when different combinations of explanatory variables were used. This accords the approach a degree of flexibility in that model parameterization can be geared toward availability of secondary data. There are limitations with the model as presently calibrated. However, the approach offers considerable scope for environmental managers to undertake national inventories of catchment sensitivity and specific assessments of individual catchments.     

基于贝叶斯最大熵的黄河三角洲土壤含盐量空间分布预测

黄河三角洲土壤盐渍化问题是制约当地农业生产和生态稳定的关键因素。为了准确掌握盐渍土的空间分布,提高土壤含盐量的空间预测精度,本研究根据2022年5月黄河三角洲的193个采样点两个深度土壤含盐量分析数据,结合高程和Landsat9遥感影像等数据,采用地理加权回归（Geographically Weighted Regression, GWR）模型构造区间型软数据,进而建立贝叶斯最大熵（Bayesian Maximum Entropy, BME）模型对研究区土壤含盐量的分布进行了预测,并同传统的地统计模型普通克里金（Ordinary Kriging, OK）及GWR模型的预测结果进行了比较。结果表明：BME模型对土壤含盐量的预测精度要高于另外两种模型。与OK相比, BME的预测误差在土壤表层和底层分别降低25%和21%,R2分别提高了0.5432和0.3527,BME作为本研究最佳土壤含盐量空间预测模型,展现了多源数据整合及非线性估计的优势。黄河三角洲表层土壤盐渍化率（88%）高于底层（68%）,大体呈现由西南到东北方向上土壤含盐量递增的趋势,沿海地区大于内陆地区,黄河三角洲北部是整个区域盐渍化最为严重的地区。     

Estimating the water retention curve from soil properties: comparison of linear, nonlinear and concomitant variable methods

The unsaturated soil hydraulic functions involving the soil–water retention curve (SWRC) and the hydraulic conductivity provide useful integrated indices of soil quality. Existing and newly devised methods were used to formulate pedotransfer functions (PTFs) that predict the SWRC from readily available soil data. The PTFs were calibrated using a large soils database from Hungary. The database contains measured soil–water retention data, the dry bulk density, sand, silt and clay percentages, and the organic matter content of 305 soil layers from some 80 soil profiles. A three-parameter van Genuchten type function was fitted to the measured retention data to obtain SWRC parameters for each soil sample in the database. Using a quasi-random procedure, the database was divided into “evaluation” (EVAL) and “test” (TEST) parts containing 225 and 80 soil samples, respectively. Linear PTFs for the SWRC parameters were calculated for the EVAL database. The PTFs used for this purpose particle-size percentages, dry bulk density, organic matter content, and the sand/silt ratio, as well as simple transforms (such as logarithms and products) of these independent variables. Of the various independent variables, the eight most significant were used to calculate the different PTFs. A nonlinear (NL) predictive method was obtained by substituting the linear PTFs directly into the SWRC equation, and subsequently adjusting the PTF parameters to all retention data of the EVAL database. The estimation error (SSQ) and efficiency (EE) were used to compare the effectiveness of the linear and nonlinearly adjusted PTFs. We found that EE of the EVAL and the TEST databases increased by 4 and 7%, respectively, using the second nonlinear optimization approach. To further increase EE, one measured retention data point was used as an additional (concomitant) variable in the PTFs. Using the 20 kPa water retention data point in the linear PTFs improved the EE by about 25% for the TEST data set. Nonlinear adjustment of the concomitant variable PTF using the 20 kPa retention data point as concomitant variable produced the best PTF. This PTF produced EE values of 93 and 88% for the EVAL and TEST soil data sets, respectively.     

Effect of Conditioning Agents on the Structure of Tempered and Steam-Flaked Sorghum

Tempering agents affect the pericarp of steamed grain and the endosperm of steam-flaked sorghum. Examination by environmental scanning electron microscopy (ESEM) clearly showed that β-mercaptoethanol (BME), sulfurous acid (SA), phosphoric acid, and cellulase significantly altered the structure of the pericarp and endosperm during tempering. Lime and protease had a lesser effect on the pericarp structure of the kernels, and had little effect on the endosperm. Steamed flakes from the SA treatment were more translucent and durable than all other treatments. Flakes with BME were translucent and high in quality, but were more fragile than the SA flakes. Nontempered, water only, and commercial conditioner treatments produced flakes with the lowest quality. Peak, final, and breakdown Rapid Visco Analyzer (RVA) viscosities were lowest for the SA flakes due to disruption of the starch chains by the weak acid. BME viscosity values were higher than SA, but lower than the water and commercial tempering treatments. The commercial conditioner (15×normal concentration) did not alter kernel structure and did not cause any changes in the RVA profiles over kernels tempered with water alone. Starch gelatinization, measured by enzyme susceptible starch (ESS), was highest in the SA and BME flakes and lowest in the nontempered flakes. Feedlot operators may be able to save money by avoiding the use of chemical additives that do nothing to the grain. By using chemicals proven to have positive effects on flake quality, operators could save money by reducing the processing time and energy needed to produce good quality flakes.     

Pedotransfer functions for estimating plant available water and bulk density in Swedish agricultural soils

Abstract

Pedotransfer functions (PTFs) to estimate plant available water were developed from a database of arable soils in Sweden. The PTFs were developed to fulfil the minimum requirements of any agro-hydrological application, i.e., soil water content at wilting point (θ _wp ) and field capacity (θ _fc ), from information that frequently is available from soil surveys such as texture and soil organic carbon content (SOC). From the same variables we also estimated bulk density (ρ) and porosity (ε), which seldom are included in surveys, but are needed for calculating element mass balances. The seven particle-size classes given in the data set were aggregated in different ways to match information commonly gained from surveys. Analysis of covariance and stepwise multiple linear regression were used for quantifying the influence of depth, particle size class, textural class and soil organic carbon on the characteristic variables. PTFs developed from other data sets were also tested and their goodness-of-fit and bias was evaluated. These functions and those developed for the Swedish database were also tested on an independent data set and finally ranked according to their goodness of fit. Among single independent variables, clay was the best predictor for θ _wp , sand (or the sum of clay and silt) for θ _fc and SOC for ρ and ε. A large fraction of the variation in θ _wp and θ _fc is explained by soil texture and SOC (up to 90%) and root mean square errors (RMSEs) were as small as 0.03 m³ water m^?3 soil in the best models. For the prediction of ρ and ε in the test data set, the best PTF could only explain 40–43% of the total variance with corresponding RMSEs of 0.14 g cm^?3 and 5.3% by volume, respectively. Recently presented PTFs derived from a North American database performed very well for estimating θ _wp (low error and bias) and could be recommended for Swedish soils if measurements of clay, sand and SOC were available. Although somewhat less accurately, also θ _fc could be estimated satisfactorily. This indicates that the determination of plant available water by texture and SOC is rather independent of soil genesis and that certain PTFs are transferable between continents.     

Evaluation of pedotransfer functions predicting hydraulic properties of soils and deeper sediments

Eight pedotransfer functions (PTF) originally calibrated to soil data are used for evaluation of hydraulic properties of soils and deeper sediments. Only PTFs are considered which had shown good results in previous investigations. Two data sets were used for this purpose: a data set of measured pressure heads vs. water contents of 347 soil horizons (802 measured pairs) from Bavaria (Southern Germany) and a data set of 39 undisturbed samples of tertiary sediments from deeper ground (down to 100 m depth) in the molasse basin north of the Alps, containing 840 measured water contents vs. pressure head and unsaturated hydraulic conductivity. A statistical analysis of the PTFs shows that their performance is quite similar with respect to predicting soil water contents. Less satisfactory results were obtained when the PTFs were applied to prediction of water content of sediments from deeper ground. The predicted unsaturated hydraulic conductivities show about the same uncertainty as for soils in a previous study. Systematic deviations of predicted values indicate that an adaptation of the PTFs to the specific conditions of deeper ground should be possible in order to improve predictions.     

Rooting patterns of four crop legumes in response to seed-placement depths in the dry season

Abstract

On sandy paddy fields, key factors for successful crops in the dry season without irrigation are a shallow water table and practices such as deep seed-placement but only some legume species are adapted to such conditions. To understand the adaptation of legume species to deep seed-placement over shallow water tables, we studied their rooting patterns on two sandy soils. Cowpea (Vigna unguiculata), mungbean (Vigna radiata), peanut (Arachis hypogaea) and soybean (Glycine max) seeds were sown shallow (～5 cm) or deep (～15 cm) in deep sandy soils after harvesting rice in two shallow water table locations in north-east Thailand. The legumes depended mainly on capillary water rising from the water table and none experienced water deficit throughout the growing season. Generally, deeper seed-placement decreased overall root dry weight, but it increased the root surface area to weight ratio. Deep seed-placement promoted a greater fraction of root growth into the subsoil for cowpea (86–99% of total root length), mungbean (61–93% of total root length) and peanut (78–98% of total root length) where the soil contained more water throughout the growing season. Moreover, deep seed-placement at the site with the lower water table promoted deeper penetration of roots of cowpea (～20 cm deeper), mungbean (～20–40 cm deeper) and peanut (～20–40 cm deeper) which improved water access, especially late during the growing season when topsoils dried to close to wilting point. Unlike other species, the soybean rooting pattern did not respond much to seed-placement depths, or soil moisture.     

Root development of winter wheat in erosion‐affected soils depending on the position in a hummocky ground moraine soil landscape

Agricultural soil landscapes of hummocky ground moraines are characterized by 3D spatial patterns of soil types that result from profile modifications due to the combined effect of water and tillage erosion. We hypothesize that crops reflect such soil landscape patterns by increased or reduced plant and root growth. Root development may depend on the thickness and vertical sequence of soil horizons as well as on the structural development state of these horizons at different landscape positions. The hypotheses were tested using field data of the root density (RD) and the root lengths (RL) of winter wheat using the minirhizotron technique. We compared data from plots at the CarboZALF‐D site (NE Germany) that are representing a non‐eroded reference soil profile (Albic Luvisol) at a plateau position, a strongly eroded profile at steep slope (Calcaric Regosol), and a depositional profile at the footslope (Anocolluvic Regosol). At each of these plots, three Plexiglas access tubes were installed down to approx. 1.5 m soil depth. Root measurements were carried out during the growing season of winter wheat (September 2014–August 2015) on six dates. The root length density (RLD) and the root biomass density were derived from RD values assuming a mean specific root length of 100 m g^?1. Values of RD and RLD were highest for the Anocolluvic Regosol and lowest for the Calcaric Regosol. The maximum root penetration depth was lower in the Anocolluvic Regosol because of a relatively high and fluctuating water table at this landscape position. Results revealed positive relations between below‐ground (root) and above‐ground crop parameters (i.e., leaf area index, plant height, biomass, and yield) for the three soil types. Observed root densities and root lengths in soils at the three landscape positions corroborated the hypothesis that the root system was reflecting erosion‐induced soil profile modifications. Soil landscape position dependent root growth should be considered when attempting to quantify landscape scale water and element balances as well as agricultural productivity.     

Transferability of continuous‐ and class‐pedotransfer functions to predict water retention properties of semiarid Syrian soils

Hydraulic properties of soils, particularly water retention, are key for appropriate management of semiarid soils. Very few pedotransfer functions (PTF s) have been developed to predict these properties for soils of Mediterranean regions, where data are particularly scarce. We investigated the transferability of PTF s to semiarid soils. The quality of the prediction was compared to that for soils originating from temperate regions for which most PTF s were developed. We used two soil data sets: one from the Paris basin (French data set, n  = 30) and a Syrian data set (n  = 30). Soil samples were collected in winter when the water content was near field capacity. Composition and water content of the samples were determined at seven water potentials. Continuous‐ and class‐PTF s developed using different predictors were tested using the two data sets and their performance compared to those developed using artificial neural networks (ANN ). The best performance and transferability of the PTF s for both data sets used soil water content at field capacity as predictor after stratification by texture. The quality of prediction was similar to that for ANN ‐PTF s. Continuous‐ and class‐PTF s may be transferable to other countries with performances that vary according to their ability to account for variation in soil composition and structure. Taking into account predictors of composition (particle size distribution, texture, organic carbon content) and structure (bulk density, porosity, field capacity) did not lead to a better performance or the best transferability potential.     

Simulating site-specific nitrogen mineralization dynamics in a Swedish arable field

Abstract

The objective of this work was to test whether a dynamic soil C and N model using site-specific information improved estimates of apparent net N mineralization compared with regressions only based on static soil properties. This comparison was made using data from a 34-point sampling grid within a Swedish arable field during two growing seasons, using a simple carbon balance and nitrogen mineralization model (ICBM/N) for the dynamic approach. Three free model parameters were simultaneously optimized using non-linear regression to obtain the best model fit to the data from all grid points and both years. Calculated annual mean net mineralization (Nm_sim) matched the measured Nm mean exactly, and was 44 and 71 kg N ha^?1 for the two growing seasons 1999 and 2000, respectively. However, the variability in calculated Nm_sim values among the 34 grid points was smaller than that measured, and only a small proportion of the variation within a single year was explained by the model. Despite this, the model explained 56% of the total variation in Nm during the two growing seasons, mainly due to the good fit to the seasonal overall difference. Significant factors influencing net mineralization included the soil environment controlling mineralization, total N in soil organic matter and N in crop residues. Uncertainties in estimation of θ _fc and θ _wp (soil water content at saturation and wilting point) and the possible influence of unknown horizontal and vertical water flows made high-precision calculations of soil water content difficult. The precision and general applicability of the actual measurements thus set limits for estimating critical parameters, and the limitations of both the experimental design and the model are discussed. It is concluded that improvements in precision in sampling and analysis of data from the grid points are needed for more critical hypothesis testing.     

The conversion of permittivity as measured by a PR2 capacitance probe into soil moisture values for Des Moines lobe soils in Iowa

Measurement of soil moisture is essential for irrigation scheduling and capacitance sensors have been widely used to monitor soil moisture at different depths. Two approaches for converting permittivity measures using the capacitance probe (PR2, Delta‐T Devices) to soil water content are to (a) use the default equation and parameters provided by the manufacturer, and (b) use site specific calibration equations. The objective of this study was to evaluate the performance of the manufacturer’s default equation and in‐situ calibrated equations for estimating soil water content. Permittivity measurement using the PR2 probe coincided with soil sampling during the growing seasons in 2006, 2007 and 2008 for Des Moines lobe soils in north‐central Iowa. The default equation provided by Delta‐T Devices for the PR2 probe estimated the soil water content for 3 years with an average root mean square error (RMSE) and index of agreement (IoA) values of 0.097 cm³/cm³ and 0.587, respectively. The default equation was calibrated by a 1‐year (2006) and a 2‐year (2006 + 2007) data set. The resultant statistics indicate that site specific calibration gives more accurate estimates of soil water content compared to the uncalibrated default equation. Three‐year averaged RMSE and IoA values were 0.049 cm³/cm³ and 0.742 for equations calibrated by the 1‐year data set, and 0.034 cm³/cm³ and 0.807 for equations calibrated by the 2‐year data set. The results from this study indicate that a site specific calibration is necessary for the PR2 probe, and equations calibrated by data from a longer period performed better than data from a shorter period. Where a site‐specific field calibration cannot be applied, coefficients are provided for various cropping systems in Des Moines Lobe soils of Iowa based on the results from this study.     

Assessment of the usefulness of particle size distribution measured by laser diffraction for soil water retention modelling

Particle size distribution (PSD) is a major soil characteristic, which is essential and commonly used for the development of pedotransfer functions (PTFs) to estimate the water retention of soils. The laser diffraction method (LDM) became a popular alternative to the standard sieve‐hydrometer method (SHM) of PSD measurement. Unfortunately, PSDs determined with LDM and SHM methods differ sometimes substantially. Moreover, it is claimed that the laser diffraction method underestimates finer fractions in favor of coarser fractions. Several authors have tried to elaborate on methods to recalculate LDM PSD into its SHM counterparts, but no universal methodology has been developed to this date. In this paper, we use PSD determined by LDM directly for PTF development and compare it with the classical PTF approach based on PSD measured by SHM. Four different PTF models based on LDM particle size distribution data were developed, with different PSD characteristics taken as the models' input variables. The possibility of using alternative PSD characteristics, such as deciles, area moment mean and volume moment mean, for PTF development was examined. The accuracy of PTF models constructed on the basis of LDM‐measured PSD was comparable with that of the developed models using texture data obtained from SHM, giving approximately the same RMSE and R² values. Our study shows that LDM‐measured particle size distribution may be directly used for PTF developments without any recalculations to their sieve‐hydrometer counterparts.