Standardized uncertainty analysis for hydrometry: a review of relevant approaches and implementation examples

Abstract

The water-centric community has continuously made efforts to identify, assess and implement rigorous uncertainty analyses for routine hydrological measurements. This paper reviews some of the most relevant efforts and subsequently demonstrates that the Guide to the expression of uncertainty in measurement (GUM) is a good candidate for estimation of uncertainty intervals for hydrometry. The demonstration is made by implementing the GUM to typical hydrometric applications and comparing the analysis results with those obtained using the Monte Carlo method. The results show that hydrological measurements would benefit from the adoption of the GUM as the working standard, because of its soundness, the availability of software for practical implementation and potential for extending the GUM to hydrological/hydraulic numerical simulations.

Editor D. Koutsoyiannis

Citation Muste, M., Lee, K. and Bertrand-Krajewski, J.-L., 2012. Standardized uncertainty analysis for hydrometry: a review of relevant approaches and implementation examples. Hydrological Sciences Journal, 57 (4), 643–667.     

Bias and variance of some estimators of suspended sediment load

Abstract

Walling & Webb (1981) made empirical comparisons amongst a number of methods for the estimation of suspended sediment loads. This paper presents expressions for the means and variances of three of the estimates compared by Walling & Webb, on the assumption that suspended sediment concentration, c, and mean daily discharge, q, are bivariate lognormally distributed. With this assumption, not inconsistent with the literature and with data, one of the three estimates is unbiased and the other two are biassed, the bias being an exponential function of the correlation and standard deviations in the lognormal distribution. This confirms the empirical finding of Walling & Webb and supports theoretical results of Ferguson (1987) which were established without distributional assumptions. Expressions for the variances of the three estimators of suspended sediment loads also confirm the empirical result of Walling & Webb, the unbiassed estimate having variance of order 1/n whilst the biassed estimates have variances of order 1/n².     

Improved fitting of the Bartlett-Lewis Rectangular Pulse Model for hourly rainfall

Abstract

This paper examines the applicability of the Bartlett-Lewis Rectangular Pulse Model to rainfall taken from a site in Elmdon, Birmingham, UK. The approach used is to assess the performance of the model in terms of characteristics of the precipitation process, incorporating monthly seasonality. Analytical expressions are derived to complement those presented in Rodriguez-Iturbe et al. (1987). As in that paper, the shortcomings of the simple Poisson process are reduced by the use of a Bartlett-Lewis process. Different methods of parameter estimation are examined. The characteristic features of the time distribution of rainfall events, however, can be well approximated only by optimization and this enables an improved identification of the model parameters.     

Nonlinear wave propagation on a sphere: Interaction between Rossby waves and gravity waves; stability of the Rossby waves

Abstract

An analytical spectral model of the barotropic divergent equations on a sphere is developed using the potential-stream function formulation and the normal modes as basic functions. Explicit expressions of the coefficients of nonlinear interaction are obtained in the asymptotic case of a slowly rotating sphere, i.e. when the normal modes can be expressed as single spherical harmonics.     

Evaporation of grass under non-restricted soil moisture conditions

Abstract

The behaviour of various formulas for evapotranspiration of grass in Nonrestricted soil water conditions is considered. These are the expressions based on the Penman formula, i.e. “old” Penman, Penman-Monteith, Thorn-Oliver and the version recommended more recently by the FAO. Moreover, the Priestley-Taylor and the Makkink formulas are considered, which are radiation-based. Comparisons are made between daily mean values estimated with these formulas and direct measurements. The latter were collected over grass in the period 1979–1982 in the catchment area of the Hupselse Beek (The Netherlands). It was found that if all required input data were measured, the Priestley-Taylor and the “old” Penman formula yielded the best results. The assumption that soil heat flux can be neglected introduces a systematic and a random error of roughly 5%. The empirical estimates for net radiation from sunshine duration, temperature and humidity appear to perform rather poorly. These estimates improved significantly if solar radiation was measured directly. The empirical expression proposed by Slob (unpublished) that requires incoming solar radiation only as input, provided better results than the other more complicated expressions. Moreover, this study reveals that evaporation of unstressed grass is primarily determined by the available energy, i.e. good evaporation estimates can be obtained by using simply λE = 0.86(R_n ? G). The Makkink method appears to be attractive for practical applications. These findings support the use of Makkink's formula for routine calculations of crop-reference evapotranspiration as has been done by the Royal Netherlands Meteorological Institute since 1987.     

Uncertainty and sensitivity analysis of runoff and sediment yield in a small agricultural watershed with KINEROS2 / Analyse d'incertitude et de sensibilité des simulations d'écoulement et de transport solide de KINEROS2 dans un petit bassin versant agricole

Abstract

Using the Monte Carlo (MC) method, this paper derives arithmetic and geometric means and associated variances of the net capillary drive parameter, G, that appears in the Parlange infiltration model, as a function of soil texture and antecedent soil moisture content. Approximate expressions for the arithmetic and geometric statistics of G are also obtained, which compare favourably with MC generated ones. This paper also applies the MC method to evaluate parameter sensitivity and predictive uncertainty of the distributed runoff and erosion model KINEROS2 in a small experimental watershed. The MC simulations of flow and sediment related variables show that those parameters which impart the greatest uncertainty to KINEROS2 model outputs are not necessarily the most sensitive ones. Soil hydraulic conductivity and wetting front net capillary drive, followed by initial effective relative saturation, dominated uncertainties of flow and sediment discharge model outputs at the watershed outlet. Model predictive uncertainty measured by the coefficient of variation decreased with rainfall intensity, thus implying improved model reliability for larger rainfall events. The antecedent relative saturation was the most sensitive parameter in all but the peak arrival times, followed by the overland plane roughness coefficient. Among the sediment related parameters, the median particle size and hydraulic erosion parameters dominated sediment model output uncertainty and sensitivity. Effect of rain splash erosion coefficient was negligible. Comparison of medians from MC simulations and simulations by direct substitution of average parameters with observed flow rates and sediment discharges indicates that KINEROS2 can be applied to ungauged watersheds and still produce runoff and sediment yield predictions within order of magnitude of accuracy.     

An asymptotic model for resistive instability in the Earth's outer core

Abstract

Numerical work indicates that resistive instability may be the dominant mode of instability in the Earth's outer core for realistic core parameter regimes. In this paper, we assume that the Elsasser number is large in order to obtain an asymptotic analysis of resistive instability in an electrically conducting fluid confined to a rotating cylindrical shell of infinite extent in the axial direction. The dimensionless equations of motion are linearized about an ambient magnetic field which is purely azimuthal and depends only on the cylindrical radial variable. Applying the theory of ordinary differential equations with a large parameter, we obtain an asymptotic approximation to the solution. Relatively simple analytic expressions for the complex frequencies are obtained by applying the boundary conditions for insulating boundaries at the cylindrical sidewalls and then assuming that the ambient magnetic field vanishes at one or both of those sidewalls. The results appear to be consistent with previous numerical work.     

An exact solution for a constant-strength line-sink satisfying the modified Helmholtz equation for groundwater flow

We obtained an exact solution in terms of the discharge potential for a constant-strength line-sink that satisfies the modified Helmholtz equation for groundwater flow, for example for semi-confined flow and transient flow. The solution is obtained by integrating the potential for a point sink (well) along a straight line element. The potential for the point-sink is the modified Bessel function of the second kind and zero order K₀. Since K₀ cannot be integrated directly (in closed form) along a line-element, earlier solutions for a line-sink have been obtained by integrating polynomial approximations to K₀. These approximations, however, are only valid up to a certain distance from the well and consequently impose a limit on the length of the line-sink. In this paper we integrate an exact series representation for K₀ that is valid at any distance from the well, thus allowing integration along line-elements of any length, at least in theory. Numerical difficulties arise when evaluating our expressions at large distances from the line-sink, but these are shown to be of little consequence in practice. We made use of Wirtinger calculus to facilitate integration and also to allow us to arrive at exact expressions for the integrated flux over a poly-line and the total leakage over a domain. These properties are essential when using the solution in the context of the Analytic Element Method (AEM). We demonstrate our solution for the case of semi-confined flow (with leakage) and for the case of transient flow in the context of the Laplace Transform Analytic Element Method (LT-AEM).     

Dependence of the power consumed by the magnetosphere on the solar wind parameters

The results of the previous studies, where the expressions were obtained for the electric current, which is generated at the bow shock front and is closed through the magnetosphere, and for the magneto-pause potential as a function of such solar wind parameters as the plasma density and velocity and the IMF intensity, are used. The power (W) consumed by the magnetosphere is equal to the Poynting vector flux S through the magnetopause. According to the special case of the Poynting theorem applied by Heikkila to the-magnetosphere, the energy flux can be expressed in terms of the electric potential (the integration is carried out over the entire surface of the magnetosphere). As a result, the required dependence, which is quadratic with respect to the IMF B _z component, has been obtained for W. It is discussed why the magnetosphere is energy-isolated at the northward IMF B _z component despite this.     

Stochastic spatial analysis of earthquake sequence

In this paper, by means of the statistical analysis method of stochastic spatial point process, statistical analysis of spatial distribution of earthquakes in the large northern region of China is made. Emphasis is on the test and analysis of the complete spatial randomness, correlation of earthquake distribution in the different magnitude interval and random labeling. It is shown by the analysis that the spatial distribution of earthquakes in the large northern region is “clustered”, the distributions of earthquakes in different magnitude interval are positively correlated and can be modeled by a two-dimensional process. The results obtained in the paper can be used for the establishment of a reasonable spatial distribution model and have some application in the reasonable estimation of seismic hazard. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,15, 129–135, 1993.     

Properties of a nonlinear solar dynamo model

Abstract

A simple nonlinear model is developed for the solar dynamo, in which the real convective spherical shell is approximated by a thin flat slab, and only the back-reaction of the field B on the helicity is taken into account by choosing the simple law α = α(1-ζB ²), where α and ζ are constants, to represent the decrease in generation coefficient ζ with increasing field strength. Analytic expressions are obtained for the amplitude of the field oscillation and its period, T, as functions of the deviation d - d_CT of a dynamo number d from its critical value d_cr for regeneration. A symmetry is found for the case of oscillations of small constant amplitude: B(t+½T)= -B(t). A Landau equation is obtained that describes the transition to such oscillations.     

Application of the honey-bees mating programming (HBMP) algorithm to sediment concentration modelling

Abstract

The honey-bees mating programming (HBMP) algorithm is introduced as a novel tool for predicting suspended sediment concentration for the Mad River catchment near Arcata, USA. The paper also applies gene expression programming (GEP) as a comparison and shows that these two approaches can the produce transparent, nonlinear relationships between the independent and dependent variables. Some modifications have been made to the HBMP algorithm to improve its capability and efficiency. The results achieved from this method and GEP are compared with two different sediment rating curves based on regression techniques. The findings show that the results from both the HBMP and GEP methods are promising and outperform the results obtained from the sediment rating curves.

Editor D. Koutsoyiannis; Associate editor L. See     

Hydromagnetic dynamo model with spiral convection

Summary The present paper deals with a hydromagnetic dynamo model of the generation mechanism of the Earth's magnetic field. An attempt has been made at selecting a flow-velocity field in the Earth's core which would satisfy the condition 0 for regenerating the field according to [2], and which would yield a velocity field pattern on the core surface as given in the papers by Kahle et al. [9]. These conditions are satisfied by the velocityv=V ₁+U ₂ ^c –V ₂ ^c and, geometrically, this velocity field is represented in space by a spiral convective motion. On the core surface two downflows and two upflows with the corresponding rotating cells may then be found. Only the axisymmetric harmonic component regeneration of the magnetic field has been considered. Adequate regeneration equations have been obtained by means of Braginski's method of quantity estimates in order of magnitude.     

Topographic effects on the wind‐driven ocean circulation

Abstract

This is a study of the influence of bottom topography of an ocean basin on the wind‐driven, barotropic ocean circulation. A detailed investigation is made of the role of vorticity transfer to the ocean bottom in the presence of varying topography. It is shown that the wind‐driven gyre over the topography of the North Atlantic has a transport in the western part of the basin only half of that obtained in an ocean of constant depth.     

Determination of Q S, spreading factor and S-wave acceleration by inversion of seismic intensity data

In this paper, on the basis of intensity data from 85 earthquakes occurred in China, the method of resolving overdetermined equations by using the damped least squares method is applied to inversing for the mean value of 1 s. S wave quality factor, the spreading factor and the S-wave acceleration at the foci of earthquakes in 7 regions of China. The relation between S-wave acceleration at the foci of earthquakes and magnitude is discussed. As an example, 2-DQ _S distribution in Taiwan Province is obtained by inversion. It is found that there exists some corresponding relation between this distribution and Bouguer gravitational anomaly and seismicity. Preliminary analysis and discussion on results of the inversion, and estimation of error inQ _S are made. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,13, 202–211, 1991. The English version of this paper is improved by Professor Yushou Xie.     

F) Personalia

Abstract

Knowledge of the natural resources of a country is essential for the assessment of its development possibilities. This is particularly true for the evaluation of groundwater resources in a country like Uruguay, where less than 5 per cent of the potential is exploited and this in an inefficient way, and where no drastic future increase in demand is predicted.

In the present paper the generalized hydrogeological model of the country was reconstructed, and recharge values, obtained by an approach adapted from nearby areas, were introduced.     

Simultaneous hydrological prediction at multiple gauging stations using the NARX network for Kemaman catchment,Terengganu, Malaysia

ABSTRACT

This paper presents a neural network model capable of catchment-wide simultaneous prediction of river stages at multiple gauging stations. Thirteen meteorological parameters are considered in the input, which includes rainfall, temperature, mean relative humidity and evaporation. The NARX model is trained with a representative set of hourly data, with optimal time delay for both the input and output. The network trained using 120-day data is able to produce simulations that are in excellent agreement with field observations. We show that for application with one-step-ahead predictions, the loss in network performance is marginal. Inclusion of additional tidal observations does not improve predictions, suggesting that the river stage stations under consideration are not sensitive to tidal backwater effects despite the claim commonly made.

EDITOR D. Koutsoyiannis ASSOCIATE EDITOR F. Pappenberger     

Recent changes in level of Lake Victoria / Récents changements enregistrés dans le niveau du Lac Victoria

ABSTRACT

The level of Lake Victoria rose by over 2.5 m between October 1959 and May 1964. Following a slight fall the lake began to rise again in 1978 and by mid 1979 had again reached almost to the level of 1964. Because these recent rises are in contrast to the previous long period (60 years), of relatively stable levels they aroused considerable interest. This paper examines possible manmade and natural causes for the rises and opts for an increase in over-lake precipitation as the most likely cause. Comparison is also made with similar rises in other East African lakes.     

Relating hydrograph components to rainfall and streamflow: a case study from northern Taiwan

Abstract

This study investigates the characteristics of hydrograph components from a watershed in Taiwan. Hydrograph components were modelled by using a model of three serial reservoirs with one parallel reservoir. Mean rainfall was calculated by using the block kriging method. The model parameters for 38 events were calibrated by using the shuffled complex evolution optimization algorithm. The model verification was made using 18 events. Based on the study results, the following findings were obtained: (1) for single-peak events, times to peak of hydrograph components are an increasing power function of the peak time of rainfall; (2) peak discharges of hydrograph components are linearly proportional to that of total runoff, and the ratios of quick and slow runoff are approximately 83% and 17% of total runoff, respectively; and (3) the total volume of quick runoff component is 52% of total runoff and that of slow runoff is 27%.

Editor D. Koutsoyiannis

Citation Li, Y.-J., Cheng, S.-J. Pao, T.-L. and Bi, Y.-J., 2012. Relating hydrograph components to rainfall and streamflow: a case study from northern Taiwan. Hydrological Sciences Journal, 57 (5), 861–877.