Interpretation and nonuniqueness of CTRW transition distributions: Insights from an alternative solute transport formulation

The continuous time random walk (CTRW) has both an elegant mathematical theory and a successful record at modeling solute transport in the subsurface. However, there are some interpretation ambiguities relating to the relationship between the discrete CTRW transition distributions and the underlying continuous movement of solute that have not been addressed in existing literature. These include the exact definition of “transition”, and the extent to which transition probability distributions are unique/quantifiable from data. Here, we present some theoretical results which address these uncertainties in systems with an advective bias. Simultaneously, we present an alternative, reduced parameter CTRW formulation for general advective transport in heterogeneous porous media, which models early- and late-time transport by use of random transition times between sparse, imaginary planes normal to flow. We show that even in the context of this reduced-parameter formulation there is nonuniqueness in the definitions of both transition lengths and waiting time distributions, and that neither may be uniquely determined from experimental data. For practical use of this formulation, we suggest Pareto transition time distributions, leading to a two-degree-of-freedom modeling approach. We then demonstrate the power of this approach in fitting two sets of existing experimental data. While the primary focus is the presentation of new results, the discussion is designed to be pedagogical and to provide a good entry point into practical modeling of solute transport with the CTRW.     

欧洲扇区不同纬度电离层暴特征的统计分析

本文利用Madrigal数据库的TEC数据对2001—2010年间的156次单主相型磁暴事件,统计分析了欧洲扇区从赤道到极光带共5个纬度区域的电离层暴特征,结果表明:(1)电离层暴有明显的纬度分布特征,正负暴出现次数的比例随纬度的降低呈现明显的增加趋势,但夏季赤道地区趋势相反,正负暴比例比更高纬度的反而降低;(2)与主相相比,恢复相期间大部分纬度地区正暴数量减少,负暴数量增加,但赤道地区恢复相期间正暴数量反而增加;(3)中低纬地区电离层暴随磁暴MPO地方时分布特征明显,正暴所对应的MPO主要分布在白天,而MPO发生在夜间容易引起负暴;(4)电离层负暴主要发生在夜间,中、高纬地区负暴的开始时间存在‘时间禁区’,但不同纬度‘时间禁区’的地方时分布有一定差异,正暴分布则相对分散.     

Characterization of the Frequency of Extreme Earthquake Events by the Generalized Pareto Distribution

Recent results in extreme value theory suggest a new technique for statistical estimation of distribution tails (Embrechts et al., 1997), based on a limit theorem known as the Gnedenko-Pickands-Balkema-de Haan theorem. This theorem gives a natural limit law for peak-over-threshold values in the form of the Generalized Pareto Distribution (GPD), which is a family of distributions with two parameters. The GPD has been successfully applied in a number of statistical problems related to finance, insurance, hydrology, and other domains. Here, we apply the GPD approach to the well-known seismological problem of earthquake energy distribution described by the Gutenberg-Richter seismic moment-frequency law. We analyze shallow earthquakes (depth h<70 km) in the Harvard catalog over the period 1977–2000 in 12 seismic zones. The GPD is found to approximate the tails of the seismic moment distributions quite well over the lower threshold approximately M 10²⁴ dyne-cm, or somewhat above (i.e., moment-magnitudes larger than m _W =5.3). We confirm that the b-value is very different (b=2.06 ± 0.30) in mid-ocean ridges compared to other zones (b=1.00 ± 0.04) with a very high statistical confidence and propose a physical mechanism contrasting crack-type rupture with dislocation-type behavior. The GPD can as well be applied in many problems of seismic hazard assessment on a regional scale. However, in certain cases, deviations from the GPD at the very end of the tail may occur, in particular for large samples signaling a novel regime.     

Linear combinations of order statistics to estimate the quantiles of generalized pareto and extreme values distributions

 Ad hoc techniques for estimating the quantiles of the Generalized Pareto (GP) and the Generalized Extreme Values (GEV) distributions are introduced. The estimators proposed are based on new estimators of the position and the scale parameters recently introduced in the Literature. They provide valuable estimates of the quantiles of interest both when the shape parameter is known and when it is unknown (this latter case being of great relevance in practical applications). In addition, weakly-consistent estimators are introduced, whose calculation does not require the knowledge of any parameter. The procedures are tested on simulated data, and comparisons with other techniques are shown. The research was partially supported by Contract n. ENV4-CT97-0529 within the project “FRAMEWORK” of the European Community – D.G. XII. Grants by “Progetto Giovani Ricercatori” are also acknowledged.     

Investigating hydrologic responses to spatio‐temporal characteristics of storms using a Dynamic Moving Storm generator

A synthetic storm generator—Dynamic Moving Storm (DMS)—is developed in this study to represent spatio‐temporal variabilities of rainfall and storm movement in synthetic storms. Using an urban watershed as the testbed, the authors investigate the hydrologic responses to the DMS parameters and their interactions. In order to reveal the complex nature of rainfall–run‐off processes, previously simplified assumptions are relaxed in this study regarding (a) temporal variability of rainfall intensity and (b) time‐invariant flow velocity in channel routing. The results of this study demonstrate the significant contribution of storm moving velocity to the variation of peak discharge based on a global sensitivity analysis. Furthermore, a pairwise sensitivity analysis is conducted to elucidate not only the patterns in individual contributions from parameters to hydrologic responses but also their interactions with storm moving velocity. The intricacies of peak discharges resulting from sensitivity analyses are then dissected into independent hydrologic metrics, that is, run‐off volume and standard deviation of run‐off timings, for deeper insights. It is confirmed that peak discharge is increased when storms travel downstream along the main channel at the speed that corresponds to a temporal superposition of run‐off. Spatial concentration of catchment rainfall is found to be a critical linkage through which characteristics of moving storms affect peak discharges. In addition, altering peak timing of rainfall intensity in conjunction with storm movement results in varied storm core locations in the channel network, which further changes the flow attenuation effects from channel routing. For future directions, the DMS generator will be embedded in a stochastic modelling framework and applied in rainfall/flow frequency analysis.     

Sensitivity of goodness-of-fit statistics to rainfall data rounding off

An analysis based on the L-moments theory suggests of adopting the generalized Pareto distribution to interpret daily rainfall depths recorded by the rain-gauge network of the Hydrological Survey of the Sardinia Region. Nevertheless, a big problem, not yet completely resolved, arises in the estimation of a left-censoring threshold able to assure a good fitting of rainfall data with the generalized Pareto distribution. In order to detect an optimal threshold, keeping the largest possible number of data, we chose to apply a “failure-to-reject” method based on goodness-of-fit tests, as it was proposed by Choulakian and Stephens [Choulakian, V., Stephens, M.A., 2001. Goodness-of-fit tests for the generalized Pareto distribution. Technometrics 43, 478-484]. Unfortunately, the application of the test, using percentage points provided by Choulakian and Stephens (2001), did not succeed in detecting a useful threshold value in most analyzed time series. A deeper analysis revealed that these failures are mainly due to the presence of large quantities of rounding off values among sample data, affecting the distribution of goodness-of-fit statistics and leading to significant departures from percentage points expected for continuous random variables. A procedure based on Monte Carlo simulations is thus proposed to overcome these problems.     

Revised runoff curve number for runoff prediction in the Loess Plateau of China

The soil conservation service (now Natural Resources Conservation Service) Curve Number (SCS-CN), one of the most commonly used methods for surface runoff prediction. The runoff calculated by this method was very sensitive to CN values. In this study, CN values were calculated by both arithmetic mean (CN_C) and least square fit method (CN_F) using observed rainfall-runoff data from 43 sites in the Loess Plateau region, which are considerably different from the CN₂ values obtained from the USDA-SCS handbook table (CN_T). The results showed that using CN_C instead of CN_T for each watershed produce little improvement, while replacing CN_T with CN_F improves the performance of the original SCS-CN method, but still performs poorly in most study sites. This is mainly due to the SCS-CN method using a constant CN value and discounting of the temporal variation in rainfall-runoff process. Therefore, three factors—soil moisture, rainfall depth and intensity—affecting the surface runoff variability are considered to reflect the variation of CN in each watershed, and a new CN value was developed. The reliability of the proposed method was tested with data from 38 watersheds, and then applied to the remaining five typical watersheds using the optimized parameters. The results indicated that the proposed method, which boosted the model efficiencies to 81.83% and 74.23% during calibration and validation cases, respectively, performed better than the original SCS-CN and the Shi and Wang (2020b) method, a modified SCS-CN method based on tabulated CN value. Thus, the proposed method incorporating the influence of the temporal variability of soil moisture, rainfall depth, and intensity factors suggests an accurate runoff prediction for general applications under different hydrological and climatic conditions on the Loess Plateau region.     

A probabilistic framework to evaluate the uncertainty of design hydrograph: case study of Swannanoa River watershed

ABSTRACT

This study presents a probabilistic framework to evaluate the impact of uncertainty of design rainfall depth and temporal pattern as well as antecedent moisture condition (AMC) on design hydrograph attributes – peak, time to peak, duration and volume, as well as falling and rising limb slopes – using an event-based hydrological model in the Swannanoa River watershed in North Carolina, USA. Of the six hydrograph attributes, falling limb slope is the most sensitive to the aforementioned uncertainties, while duration is the least sensitive. In general, the uncertainty of hydrograph attributes decreases in higher recurrence intervals. Our multivariate analysis revealed that in most of the return periods, AMC is the most important driver for peak, duration and volume, while time to peak and falling limb slope are most influenced by rainfall pattern. In higher return periods, the importance of rainfall depth and pattern increases, while the importance of AMC decreases.     

基于POT模型的昆仑山地区地震统计特征分析

极值统计是研究较少发生但一旦发生即产生极大影响的随机事件的有效方法。本文以地震活动频繁的昆仑山地区作为研究区域,建立了基于广义帕累托分布的超阈值（POT）模型,并讨论了该地区若干地震活动性参数,包括强震震级分布、潜在震级上限、强震平均复发间隔、一定周期内的强震发震概率、一定时期内的重现水平和超定值重现震级。经统计分析得到：该地区震级阈值选定为MS5.5,超阈值期望震级为MS6.81,潜在震级上限高达MS9.08,MS8.0的平均复发间隔仅为66.8年,未来3年该地区发生MS5.5～MS6.5的概率在80%以上,百年重现水平即可达到历史最大震级MS8.1。     

基于广义帕累托分布的马尼拉海沟俯冲带地震危险性估计

选取马尼拉海沟俯冲带作为潜源区, 基于广义帕累托分布, 通过对一定时段内超过某一阈值的震级数据进行拟合, 建立该潜源区地震危险性估计模型, 估计强震重现水平和震级上限, 并对估计结果的不确定性进行了分析, 得到马尼拉海沟俯冲带震级上限为9.0级, 10 a、 50 a、 100 a、 200 a马尼拉海沟俯冲带的震级重现水平期望值分别为7.1级、 7.6级、 7.7级、 7.9级。     

Statistical characterization of impact of system variables on temporal dynamics of groundwater in highly weathered regoliths

In situations where the water table fluctuates during the rainy season the characterization of the impact of system variables on the temporal dynamics of the groundwater (GW) is essential to improve the understanding at catchment or regional scale behaviour of GW. In this study the appropriateness of the statistical parameters; mean, median, the 80^th percentile (PC80), coefficient of variation (CV), correlation coefficient (r), and multiple regression models were assessed to characterize the impact of system variables on the temporal dynamics of hydraulic head relative to ground surface (HH) during rainy seasons. The study was conducted from 1999 to 2003 in the wet tropical Johnstone River catchment (JRC) in north‐east Queensland, Australia. Piezometer wells were installed at 32 sites under cropping to 5–90 m depth on different soil types, landscape positions, and varying proximity to surface water bodies (i.e. four system variables). The HH was measured, at least at 10–15 day intervals during 1–5 consecutive rainy seasons. The HH in the 32 wells fluctuated throughout each of the five rainy seasons. The mean HH averaged over the seasons ranged from 1·1 to 17·2 m across the wells, the median from 0·9 to 17·3 m, and the PC80 from 0·3 to 16·1 m. The temporal behaviour of HH characterization by mean of means of HH, the mean of medians of HH, and the mean of PC80 of HH, indicated the HH can be classified to belong to three different groups for each one of these parameters. The impact of the system variables on temporal dynamics, explored using multiple regression procedure, indicated that the model for median was marginally better than mean. The CV was found to be most appropriate parameter to characterize the impact of GW system variable (aquifer type), a component of the system variables, on temporal dynamics. The interactions of GW (i) belonging to different GW system and (ii) at shoulder with footslope in a landscape were best characterized by simple linear correlations. Copyright © 2007 John Wiley & Sons, Ltd.     

A stochastic model of low flows

A model is developed for annual low flow hydrographs. Its two primary components reflect the fact that hydrologic processes during streamflow rise (function of water input) and recession (function of basin storage) are different. Durations of periods of rise (wet intervals) and recession (dry intervals) are modelled by discrete probability distributions — negative binomial for dry intervals and negative binomial or modified logarithmic series for wet intervals depending on goodness of fit. During wet intervals, the total inflow is modelled by the lognormal distribution and daily amounts are allocated according to a pattern-averaged model. During dry intervals, the flow recedes according to a deterministic-stochastic recession model. The model was applied to three Canadian basins with drainage area ranging from 2210 to 22000 km² to generate 50 realizations of low flow hydrographs. The resulting two standard-error confidence band for the simulated probability distribution of annual minimum 7-day flows enclosed the probability distribution estimated from the observed record. A sensitivity analysis for the three basins revealed that in addition to the recession submodel, the most important submodel is that describing seasonality. The state of the basin at the beginning of the low flow period is of marginal importance and the daily distribution of input is unimportant.     

A stochastic model of low flows

A model is developed for annual low flow hydrographs. Its two primary components reflect the fact that hydrologic processes during streamflow rise (function of water input) and recession (function of basin storage) are different. Durations of periods of rise (wet intervals) and recession (dry intervals) are modelled by discrete probability distributions — negative binomial for dry intervals and negative binomial or modified logarithmic series for wet intervals depending on goodness of fit. During wet intervals, the total inflow is modelled by the lognormal distribution and daily amounts are allocated according to a pattern-averaged model. During dry intervals, the flow recedes according to a deterministic-stochastic recession model. The model was applied to three Canadian basins with drainage area ranging from 2210 to 22000 km² to generate 50 realizations of low flow hydrographs. The resulting two standard-error confidence band for the simulated probability distribution of annual minimum 7-day flows enclosed the probability distribution estimated from the observed record. A sensitivity analysis for the three basins revealed that in addition to the recession submodel, the most important submodel is that describing seasonality. The state of the basin at the beginning of the low flow period is of marginal importance and the daily distribution of input is unimportant.     

Estimation of the temporal autocorrelation structure by the collocation technique with an emphasis on soil moisture studies

Abstract

The collocation technique has become a popular tool in oceanography and hydrology for estimating the error variances of different data sources such as in situ sensors, models and remote sensing products. It is also possible to determine calibration constants, for example to account for an off-set between the data sources. So far, the temporal autocorrelation structure of the errors has not been studied, although it is known that it has detrimental effects on the results of the collocation technique, in particular when calibration constants are also determined. This paper shows how the (triple) collocation estimators can be adapted to retrieve the autocovariance functions; the statistical properties as well as the structural deficencies are described. The coupling between the autocorrelation of the error and the estimation of calibration constants is studied in detail, due to its importance for analysing temporal changes. In soil moisture applications, such time variations can be induced, for example, by seasonal changes in the vegetation cover, which affect both models and remote sensing products. The limitations of the proposed technique associated with these considerations are analysed using remote sensing and in situ soil moisture data. The variability of the inter-sensor calibration and the autocovariance are shown to be closely related to temporal patterns of the data.

Editor D. Koutsoyiannis

Citation Zwieback, S., Dorigo, W., and Wagner, W., 2013. Estimation of the temporal autocorrelation structure by the collocation technique with an emphasis on soil moisture studies. Hydrological Sciences Journal, 58 (8), 1729–1747.     

Impacts of forest structure on precipitation interception and run‐off generation in a semiarid region in northern China

Water resource scarcity and uneven distribution are 2 major environmental issues in China today. Forest structure is a dominant factor that influences hydrological processes, but the specific interactions remain uncertain due to the predominant use of individual or 1‐dimensional forest structure metrics in previous studies. In this study, forest structures in 8 run‐off plots on Mount Miaofeng in north China were parameterized by metrics of different dimensionalities. The relation between canopy interception and forest structure, shrub/litter interception, and forest structure as well as run‐off and forest structure were analysed by regression method and validated by leave‐one‐out cross test. The results showed that canopy interception rates ranged from less than 0.10 all the way to 0.80, affected by forest structure and precipitation, with interception rate decreasing logarithmically as precipitation increased. Forests with a larger canopy area (CA), leaf area index (LAI), and higher average height (H) had a narrow range of canopy interception rates, and forest with larger value of diameters at breath height (DBH), H, LAI, vertical heterogeneity coefficient (T), and structure complexity index (SCI) had higher interception rates. Forests with higher value of DBH, H, and horizontal heterogeneity coefficient (R) had higher shrub/litter interception rates on the forest floor. The run‐off coefficient was only significantly associated with LAI, T, and SCI. The validation test indicated that regression analysis of canopy interception rates and shrub interception are reliable and SCI is a key factor to influence the run‐off coefficient. However, the regression results of litter interception have a relatively large error. According to the results, to reduce the risks of the landslides and floods, forest managers should complicate the canopy and preserve trees with thicker stems and larger canopies. By contrast, to obtain more water resource from run‐off in arid regions, forest managers should harvest trees with large canopies and construct complex vertical structures by intermediate cutting.     

Linear combinations of order statistics to estimate the position and scale parameters of the Generalized Pareto distribution

 Ad hoc techniques for estimating the position and the scale parameters of the Generalized Pareto distribution are introduced. The estimators proposed are simple linear combinations of the order statistics: they provide valuable estimates of the parameters of interest, both when the shape parameter is known and when it is unknown (this latter case being of great relevance in practical applications), and show a good performance as well when the sample size is small. The procedures are tested on simulated data, and comparisons with other techniques are shown.