Derivation of bivariate probability density functions with exponential marginals

A vivariate probability density function (pdf),f(x ₁,x ₂), admissible for two random variables (X ₁,X ₂), is of the form

A bivariate analysis of the volume and duration of low-flow events

: The knowledge of the volume and duration of low-flow events in river channels is essential for water management and the design of hydraulics structures. In this study, both preceding characteristics, X ₁ and X ₂, are considered simultaneously via two types of bivariate distributions whose marginals are exponential. One of these bivariate distributions has been presented by Nagao and Kadoya (1971) and the other has been used by Singh and Singh (1991) to the study of rainfall intensity and rainfall depth. The results are applied to the low-flow series (“peaks-below-threshold”) of Lepreau River (station 01AQ001) in New Brunswick, Canada. These results show that the model that was successfully employed by Singh and Singh (1991) to study rainfall, presents certain difficulties when a very strong correlation, ρ, between the two random variables X ₁ and X ₂, exists. The model by Nagao and Kadoya (1971) seems to be more satisfactory for such situations, although this model seems also to be quite sensitive to variations in ρ.     

A bivariate analysis of the volume and duration of low-flow events

: The knowledge of the volume and duration of low-flow events in river channels is essential for water management and the design of hydraulics structures. In this study, both preceding characteristics, X ₁ and X ₂, are considered simultaneously via two types of bivariate distributions whose marginals are exponential. One of these bivariate distributions has been presented by Nagao and Kadoya (1971) and the other has been used by Singh and Singh (1991) to the study of rainfall intensity and rainfall depth. The results are applied to the low-flow series (“peaks-below-threshold”) of Lepreau River (station 01AQ001) in New Brunswick, Canada. These results show that the model that was successfully employed by Singh and Singh (1991) to study rainfall, presents certain difficulties when a very strong correlation, ρ, between the two random variables X ₁ and X ₂, exists. The model by Nagao and Kadoya (1971) seems to be more satisfactory for such situations, although this model seems also to be quite sensitive to variations in ρ.     

Analytical procedures for weekly hydrological droughts: a case of Canadian rivers

Abstract

Two entities of importance in hydrological droughts, viz. the longest duration, L_T , and the largest magnitude, M_T (in standardized terms) over a desired time period (which could also correspond to a specific return period) T, have been analysed for weekly flow sequences of Canadian rivers. Analysis has been carried out in terms of week-by-week standardized values of flow sequences, designated as SHI (standardized hydrological index). The SHI sequence is truncated at the median level for identification and evaluation of expected values of the above random variables, E(L_T ) and E(M_T ). SHI sequences tended to be strongly autocorrelated and are modelled as autoregressive order-1, order-2 or autoregressive moving average order-1,1. The drought model built on the theorem of extremes of random numbers of random variables was found to be less satisfactory for the prediction of E(L_T ) and E(M_T ) on a weekly basis. However, the model has worked well on a monthly (weakly Markovian) and an annual (random) basis. An alternative procedure based on a second-order Markov chain model provided satisfactory prediction of E(L_T ). Parameters such as the mean, standard deviation (or coefficient of variation), and lag-1 serial correlation of the original weekly flow sequences (obeying a gamma probability distribution function) were used to estimate the simple and first-order drought probabilities through closed-form equations. Second-order probabilities have been estimated based on the original flow sequences as well as SHI sequences, utilizing a counting method. The E(M_T ) can be predicted as a product of drought intensity (which obeys the truncated normal distribution) and E(L_T ) (which is based on a mixture of first- and second-order Markov chains).

Citation Sharma, T. C. & Panu, U. S. (2010) Analytical procedures for weekly hydrological droughts: a case of Canadian rivers. Hydrol. Sci. J. 55(1), 79–92.     

Azimuthal variation in AVO response for fractured gas sands

Natural fractures in reservoirs play an important role in determining fluid flow during production, and hence the density and orientation of fractures is of great interest. In the presence of aligned vertical fractures, the reflection amplitude at finite offset varies with azimuth. The effect of natural fractures on the azimuthal AVO response from a gas-sandstone reservoir encased within shale is investigated. A simple expression for the difference in P-wave reflection coefficient from the top of the reservoir parallel and perpendicular to the strike of the fractures is obtained in terms of the normal and tangential compliances, Z_N and Z_T, of the fractures. This expression is valid for small anisotropy and material contrasts and is compared with the results of numerical modelling. For a given value of Z_T, the azimuthal variation in reflection coefficient at moderate offsets is found to increase with decreasing Z_N/Z_T. For gas-filled open fractures Z_N/Z_T ≈ 1, but a lower ratio of Z_N/Z_T may result from the presence of cement or clay within the fractures, or from the presence of a fluid with non-zero bulk modulus. For Z_N/Z_T = 1 and moderate offsets, the variation with offset of the reflection coefficient from the top of the fractured unit is dominated by the contrast in Poisson's ratio between the gas sand and the overlying shale, the effect of fractures only becoming noticeable as the critical angle for the unfractured sandstone is approached. However, for reflections from the base of the fractured unit, the variation in reflection amplitude with azimuth is much greater at conventional seismic offsets than for the reflection from the top. Azimuthal variations in the strength of the reflection from the top of the reservoir depend only on the variation in reflection coefficient, whereas the raypath is also a function of azimuth for reflections from the base of the fractured unit, leading to stronger, more visible, variations of AVO with azimuth. It follows that an azimuthal variation in AVO due to fractures in the overburden may be misinterpreted as due to the presence of aligned fractures in the reservoir.     

Heavy tailed probability distributions for non-Gaussian simulations with higher-order cumulant parameters predicted from sample data

Forecasting of extreme events and phenomena that respond to non-Gaussian heavy-tailed distributions (e.g., extreme environmental events, rock permeability, rock fracture intensity, earthquake magnitudes) is essential to environmental and geoscience risk analysis. In this paper, new parametric heavy-tailed distributions are devised starting from the exponential power probability density function (pdf) which is modified by explicitly including higher-order “cumulant parameters” into the pdf. Instead of dealing with whole power random variables, novel “residual” random variables are proposed to reconstruct the cumulant generating function. The expected value of a residual random variable with the corresponding pdf for order G, gives the input higher-order cumulant parameter. Thus, each parametric pdf is used to simulate a random variable containing residuals that yield, in average, the expected cumulant parameter. The cumulant parameters allow the formulation of heavy-tailed skewed pdfs beyond the lognormal to handle extreme events. Monte Carlo simulation of heavy-tailed distributions with higher-order parameters is demonstrated with a simple example for permeability.     

Mathematical model for evaluation of the effect of soil erosion on soil productivity

This paper is concerned with the development of a stochastic model for evaluating the long-term effect of soil erosion on soil productivity. Due to random variations in annual crop yield, the effect of erosion on crop production is not easily detectable in the short run, but becomes gradually evident over a sufficiently long time period. Under these circumstances, it seems that an experimental approach to this problem may be very difficult. The long period of time over which such an experiment has to be conducted may result in prohibitively high costs. In addition, it also means that eventual resolution of this problem must be postponed until a distant future time. The stochastic model formulated here provides us with a useful tool to assess the trend in quantitative changes in crop production due to erosion and to project future crop losses. The model is a discrete parameter stochastic process. Its derivation is based on a single assumption that the annual loss rates form a sequence of independent random variables {Z_i}₁∞ (in this paper, we consider only two particular cases: (a) {Z_i}₁∞ is a sequence of constants; (b) {Z_i}₁∞ is a sequence of independent identically distributed random variables). For these particular cases, we obtained its marginal n-dimensional distribution function and correlation function. One of the principal model features is its simple structure and remarkable lack of restrictive and unrealistic assumptions.     

Conditioned ranges for independent symmetric three valued inflows

The paper considers, as inputs to a storage system, independent random variablesX ₁,X ₂,... which have a symmetric three-valued distribution. Expressions are obtained for the conditional expected values of (i) the adjusted range ofX ₁,...X _n, conditioned on the eventX ₁+...+X _n=0; (ii) the Hurst range ofX ₁,...,X _n , conditioned on essentially the same event. Numerical comparisons of these two conditioned ranges show that they are close in value unless the inputs have high kurtosis.     

A semi-empirical method for predicting hydrological drought magnitudes in the Canadian prairies

Abstract

A hydrological drought magnitude (M _T ) expressed in standardized terms is predicted on annual, monthly and weekly time scales for a sampling period of T years in streamflow data from the Canadian prairies. The drought episodes are considered to follow the Poisson law of probability and, when coupled with the gamma probability distribution function (pdf) of drought magnitude (M) in the extreme number theorem, culminate in a relationship capable of evaluating the expected value, E(M _T ). The parameters of the underlying pdf of M are determined based on the assumption that the drought intensity follows a truncated normal pdf. The E(M _T ) can be evaluated using only standard deviation (σ), lag-1 autocorrelation (ρ) of the standardized hydrological index (SHI) sequence, and a weighting parameter Φ (ranging from 0 to 1) to account for the extreme drought duration (L _T ), as well as the mean drought duration (L_m ), in a characteristic drought length (L_c ). The SHI is treated as standard normal variate, equivalent to the commonly-used standardized precipitation index. A closed-form relationship can be used for the estimation of first-order conditional probabilities, which can also be estimated from historical streamflow records. For all rivers, at the annual time scale, the value of Φ was found equal to 0.5, but it tends to vary (in the range 0 to 1) from river to river at monthly and weekly time scales. However, for a particular river, the Φ value was nearly constant at monthly and weekly time scales. The proposed method estimates E(M _T ) satisfactorily comparable to the observed counterpart. At the annual time scale, the assumption of a normal pdf for drought magnitude tends to yield results in close proximity to that of a gamma pdf. The M _T , when transformed into deficit-volume, can form a basis for designing water storage facilities and for planning water management strategies during drought periods.

Editor D. Koutsoyiannis; Associate editor C. Onof

Citation Sharma, T.C. and Panu, U.S., 2013. A semi-empirical method for predicting hydrological drought magnitudes in the Canadian prairies. Hydrological Sciences Journal, 58 (3), 549–569.     

Generous statistical tests

A common statistical problem is deciding which of two possible sources, A and B, of a contaminant is most likely the actual source. The situation considered here, based on an actual problem of polychlorinated biphenyl contamination discussed below, is one in which the data strongly supports the hypothesis that source A is responsible. The problem approach here is twofold: One, accurately estimating this extreme probability. Two, since the statistics involved will be used in a legal setting, estimating the extreme probability in such a way as to be as generous as is possible toward the defendant’s claim that the other site B could be responsible; thereby leaving little room for argument when this assertion is shown to be highly unlikely. The statistical testing for this problem is modeled by random variables {X _i } and the corresponding sample mean the problem considered is providing a bound ɛ for which for a given number a ₀. Under the hypothesis that the random variables {X _i } satisfy E(X _i ) ≤ μ, for some 0  < μ < 1, statistical tests are given, described as “generous”, because ɛ is maximized. The intent is to be able to reject the hypothesis that a ₀ is a value of the sample mean while eliminating any possible objections to the model distributions chosen for the {X _i } by choosing those distributions which maximize the value of ɛ for the test used.     

Hydrocarbon source systems and formation of gas fields in Sichuan Basin

The formation of large and middle gas fields in Sichuan Basin is investigated based on source wntrolling theory and hydrocarbon source systems. It is indicated that Є₁, S_l, P₁, P₂ and T₃ are the main source beds and Є₁/Z₂ d, C₂ h/S₁, P₁/P₂, P₂ ch/P₂, T_1,2/P,T3 x /T3 x are important hydrocarbon source systems in the basin. All these source systems are the prospective formations and exploration spaces of large and middle gas fields. It is also emphasized that hydrocarbon generation intensity is the most important geochemical factor to estimate large and middle gas fields. Project supported by the “85–102” Chinese National Key Science and Technology Project.     

Predicting return periods of hydrological droughts using the Pearson 3 distribution: a case from rivers in the Canadian prairies

Abstract

The standardized series of monthly and weekly flow sequences, referred to as standardized hydrological index (SHI) series, from five rivers in the Canadian prairies were subjected to return period (T_r) analysis of drought length (L). The SHI series were truncated at drought probability levels q ranging from 0.5 to 0.05 with the intention of deducing drought events and corresponding drought lengths. The values of L were fitted to the Pearson 3, the gamma (2-parameter), the exponential (1-parameter), the Weibull 3 and the Weibull (2-parameter) probability density functions (pdfs). A priori assignment of one week or one month for the location parameter in the Pearson 3 pdf proved logical and also facilitated the rapid estimation of other parameters using either the method of moments or the method of maximum likelihood. The Pearson 3 turns out to be the most suitable pdf to describe and to estimate return periods of drought lengths. At the monthly and weekly time scales, it was inferred that the sample size (T, months or weeks) of SHI series could be treated equivalent to the return period of the largest recorded drought length. At the annual time scale, however, the sample size (T, years) should be modified using either the Hazen or the Gringorten plotting position formula to reflect the actual return period of the largest recorded drought length in years.

Editor D. Koutsoyiannis; Associate editor E. Gargouri     

Impact of climate change on runoff from a mid‐latitude mountainous catchment in central Japan

Hydrologic balance in high‐altitude, mid‐latitude mountain areas is important in terms of the water resources available to associated lowlands. This study examined how current and historical shifts in precipitation (P) patterns and concurrent increases in temperature (T) affected runoff (Q) and other hydrologic components in a mid‐latitude mountain catchment of central Japan, using a combination of long‐term data and a simplified hydrologic model, along with their stochastic treatment. The availability of intensive meteorological and hydrological data from the period 1997–2001 allowed the derivation of key relationships for the current climate that tie the forcing term to the parameters or state variables. By using the data recorded in the period 1965–2001, the force for driving the historical simulation was generated. Based on this model and historical shifts in P and T, the probability density functions of Q (pdf(Q)) was computed. A main novelty in this study is that such a stochastic representation, which is useful for considering the influence of projected shifts in environmental factors on the hydrologic budget, was provided. Despite the large increase in the rate of T in winter and spring, pdf(Q) in spring and summer varied appreciably during the time studied mainly because of an increase in snowmelt. An interannual change in whole‐year Q was robust to shifts in T because while Q in spring increased, in summer it decreased, implying a crucial effect of global warming on mountain hydrologic regimes is change in the timing of Q. Copyright © 2009 John Wiley & Sons, Ltd.     

Regionalization of floods in New Brunswick (Canada)

This study uses the method of peaks over threshold (P.O.T.) to estimate the flood flow quantiles for a number of hydrometric stations in the province of New Brunswick, Canada. The peak values exceeding the base level (threshold), or `exceedances', are fitted by a generalized Pareto distribution. It is known that under the assumption of Poisson process arrival for flood exceedances, the P.O.T. model leads to a generalized extreme value distribution (GEV) for yearly maximum discharge values. The P.O.T. model can then be applied to calculate the quantiles X <sub>T</sub> corresponding to different return periods T, in years. A regionalization of floods in New Brunswick, which consists of dividing the province into `homogeneous regions', is performed using the method of the `region of influence'. The 100-year flood is subsequently estimated using a regionally estimated value of the shape parameter of the generalized Pareto distribution and a regression of the 100-year flood on the drainage area. The jackknife sampling method is then used to contrast the regional results with the values estimated at site. The variability of these results is presented in box-plot form. Received: June 1, 1997     

Regionalization of floods in New Brunswick (Canada)

This study uses the method of peaks over threshold (P.O.T.) to estimate the flood flow quantiles for a number of hydrometric stations in the province of New Brunswick, Canada. The peak values exceeding the base level (threshold), or `exceedances', are fitted by a generalized Pareto distribution. It is known that under the assumption of Poisson process arrival for flood exceedances, the P.O.T. model leads to a generalized extreme value distribution (GEV) for yearly maximum discharge values. The P.O.T. model can then be applied to calculate the quantiles X <sub>T</sub> corresponding to different return periods T, in years. A regionalization of floods in New Brunswick, which consists of dividing the province into `homogeneous regions', is performed using the method of the `region of influence'. The 100-year flood is subsequently estimated using a regionally estimated value of the shape parameter of the generalized Pareto distribution and a regression of the 100-year flood on the drainage area. The jackknife sampling method is then used to contrast the regional results with the values estimated at site. The variability of these results is presented in box-plot form. Received: June 1, 1997     

蒙新高原岱海夏季叶绿素a浓度空间分布及影响因子

基于2019年夏季(8月)对岱海水样的实测数据分析,通过运用克里金插值、相关性分析、多元线性逐步回归、主成分分析方法,探究了叶绿素a(Chl.a)的空间分布特征及其与水环境因子的相关关系,并讨论了相应的防治措施。研究显示:Chl.a空间分布呈现由岸边向湖心递减的趋势,总氮(TN)、总磷(TP)、氨氮(NH₃-N)、硝态氮(NO^-₃-N)、正磷酸盐(PO^3-₄-P)空间分布特征与Chl.a空间分布特征相近,采样期内岱海湖局部区域水质状况已达到富营养状态;Chl.a与浊度(Turbidity)、TP、TN、悬浮物(SS)、pH、NO^-₃-N、NH₃-N、PO^3-₄-P、蓝绿藻丰度(CYANO)呈极显著正相关,与溶解氧(DO)呈显著负相关,与电导率(Cond.)呈正相关、与氮磷比(TN/TP)呈负相关;各湖区Chl.a与环境因子相关关系不同,全湖逐步线性回归方程为Y_Chl.a=-21.42+8.658X_pH-0.865X_DO+0.779X_NH3-N+0.699X_Turbidity+0.502X_CYANO;岱海不同湖区因子对Chl.a浓度的影响存在差异,各湖区Chl.a与环境因子相关关系不同,通过岱海与我国其他湖泊Chl.a与环境因子的相关性关系对比分析,湖泊地理属性差异及营养物质输入浓度是影响Chl.a变化的重要因素;本研究岱海的TN/TP平均值为12.23,说明夏季岱海湖Chl.a变化为氮磷共同限制。     

Estimating time parameters of overland flow on impervious surfaces by the particle tracking method

Abstract

Travel time and time of concentration T_c are important time parameters in hydrological designs. Although T_c is the time for the runoff to travel to the outlet from the most remote part of the catchment, most researchers have used an indirect method such as hydrograph analysis to estimate T_c. A quasi two-dimensional diffusion wave model with particle tracking for overland flow was developed to determine the travel time, and validated for runoff discharges, velocities, and depths. Travel times for 85%, 95% and 100% of particles arrival at the outlet of impervious surfaces (i.e. T_t85, T_t95, and T_t100) were determined for 530 model runs. The correlations between these travel times and T_c estimated from hydrograph analysis showed a significant agreement between T_c and T_t85. All the travel times showed nonlinear relationships with the input variables (plot length, slope, roughness coefficient, and effective rainfall intensity) but showed linear relationships with each other.

Editor D. Koutsoyiannis; Associate editor S. Grimaldi     

Some models of the geomagnetic field in western Europe from 1960 to 1980

Achache et al.'s averages over 10 west European observatories of the annual mean magnetic components X, Y, Z are studied from 1953 to 1979. A 1970 jerk is fitted to the data by least squares, and the fit is quantitatively compared with that of a polynomial having the same number of free parameters as the jerk (a quintic). A crude correction for the 11-y sunspot cycle is also attempted. The jerk and quintic are equally good fits to X, the jerk is clearly better for Y, and the quintic is modestly better for Z. The jerk amplitudes of X and Z but not Y depend heavily on whether a sunspot correction is made, and such a correction may be required to estimate the energy of the internal part of the jerk in the manner of Malin and Hodder. The sunspot correction to Y probably cannot be brought above the noise because it is small and highly correlated with both the jerk and the quintic. The sunspot corrections to X and Z are statistically significant but depend on whether the core signal is taken as a jerk or a quintic. Courtillot and Le Mouël's physical model for the jerk predicts that the jerk amplitudes of X, Y, Z will be proportional to the longitudinal derivatives of X, Y and Z. This prediction is roughly verified for the magnitudes but not the signs; it is just possible that the signs in X and Z are not statistically significant.     

Spatial variability of the surface energy balance of Lake Kasumigaura and implications for flux measurements

ABSTRACT

The spatial variability of the lake surface energy balance and its causes are not well-understood. Energy balance maps (90 m resolution) of Lake Kasumigaura (172 km²), Japan, obtained by interpolating station data and bulk equations, allowed an investigation of these issues. Due to lake-scale variations in meteorological variables and small-scale fluctuations of surface temperature, T_s, surface heat fluxes differed horizontally at two distinct scales, while radiative fluxes were more uniform. As the key variable to surface flux T_s was only homogeneous for directions with a longer fetch or under calm wind conditions. Using these findings, the suitability of two flux station locations, one at the centre of the lake and another within a cove, was considered. Although both locations satisfied the fetch requirements, T_s was not always found to be homogeneous in the cove, making this location less suitable for flux measurements, an issue that, to date, has been overlooked.