The Development of Rating Curve Considering Variance Function Using Pseudo-likelihood Estimation Method

The aim of this article is to propose a multilevel combined model for repeated, hierarchical, and overdispersed time-to-event outcomes, extending the so-called combined model proposed by Molenberghs et al. (2010), and using three different estimation strategies: full likelihood, pseudo-likelihood, and Bayesian estimation. For the first two estimation methods, we implemented the alternating imputation posterior (AIP) algorithm (Clayton and Rasbash, 1999). It is shown that the multilevel combined model can be fitted nicely using all three estimation methods. In addition, the multilevel combined model has the advantage that it not only can capture the hierarchical structure of the data but also can accommodate overdispersion within the data set. From our simulation results, it follows that the multilevel combined model performs well in terms of point estimation and its precision, fitted with the three different estimation methods. We also observed that pairwise likelihood estimation, a particular form of pseudo-likelihood, is more time-intensive than full likelihood and Bayesian estimation. However, pseudo-likelihood estimation is less sensitive to starting values.

Correlated responses have been widely analyzed since Liang and Zeger (1986) introduced the famous Generalized Estimating Equations(GEE). However, their variance functions were restricted to known quantifies multiplied by scale parameter. In so many industries and academic/research fields, power-of-the-mean variance function is one of the common variance function. We suggest GEE-type pseudolikelihood estimation based on the power-of-the-mean variance using existing software and investigate it's efficiency for different working correlation matrices.

Hydrometric data poverty compounds the challenge of accounting for uncertainties in non-stationary stage–discharge relationships. This paper builds on three methods to explore the integration of a dynamic approach to rating curve assessment and a physically based Bayesian framework for quantifying discharge amid geomorphologically induced rating shifts in a sparsely gauged alluvial river. The Modified GesDyn–FlowAM–BaRatin method entails sequentially segmenting gaugings according to residual indicators of riverbed instability and channel conveyance variability, leveraging cross-sectional surveys to augment calibration data, and eliciting hydraulic priors for probabilistic rating curve estimation. This method is applied to a Philippine watershed, where quarrying near the gauging station has ostensibly caused morphodynamic adjustments. Time-variable credible intervals for discharge are computed. The optimal estimates root mean square error (RMSE = 2.96 m3/s) from maximum a posteriori rating curves outperform the hydrographer’s benchmark (RMSE = 5.00 m3/s), whose systematic errors from the gauged flows arise from lapses in shift detection.

This thesis presents the results of new and original research on suspended sediment (SS) and nutrient concentrations and loads on the Rivers Bandon and Owenabue in the south of Ireland. It addresses the knowledge gap in SS and nutrient concentration data and in particular SS load/yield estimation methods. A comprehensive database of SS and nutrient concentrations has been established. An investigation into the relationship between water discharge and sediment and nutrient transport is presented and new original relationships between turbidity and suspended sediment concentration (SSC) and nutrients are developed. Turbidity-based suspended sediment loads (SSL) are estimated from the continuous turbidity record using the developed relationships with annual loads of 9.7 tonnes per square kilometre per year (t km‘‘ year’') and 18 t km'^ year ' estimated for the Rivers Bandon and Owenabue, respectively. Rating curves are developed to assess their accuracy compared to the turbidity-based SSL for the first time in an Irish context. Generally, rising stage and winter rating curves explain the variation in sediment concentration well (r‘ > 0.4), but winter and falling stage curves are not a good fit. Rating curves are shown to provide acceptable estimates of SSL at the large (annual) scale on both rivers, without the need for stage or seasonal rating curve separation. The variation in SSC between and within storm events is found to be large and over 73% of the total annual load is transported when flows are above the 10% exceedance flow rate on both rivers, indicating that stonn-based events are critical to load delivery. The sediment associated nutrient transport in the two rivers is also examined, with brief surges in particulate forms of nitrogen and phosphorus observed at the onset of the events analysed. These are accompanied by decreases in the dissolved forms of the monitored nutrients. Turbidity is used as a surrogate for phosphorus concentrations, as it correlates well with most of phosphorus species concentrations. Nitrogen does not correlate well with turbidity or SSC, and loads are estimated using an averaging estimator (interpolation). The particulate fraction of nutrient transport is found to be 25% and 35% of the respective total nitrogen and total phosphorus yields on the River Bandon, and 41% and 52% on the River Owenabue. The total portion of SSL consisting of particulate nutrients is estimated to be in excess of 10% on both the River Bandon and the River Owenabue.

Landslides are of wide interest because they shape topography and influence fluvial sediment transport. To investigate the duration of landsliding's influence on fluvial sediment discharge, we analyzed suspended sediment concentration measurements at 77 gauging stations across Taiwan over an 11-year period after Typhoon Morakot in 2009, the wettest typhoon on record in Taiwan which generated nearly 20,000 landslides. At each gauging station, we computed annual rating curves for suspended sediment concentration as a power-law function of the centered water discharge, which isolates the efficiency of suspended sediment discharge from temporal variations in water discharge. Among the 40 stations in basins that were strongly impacted by landsliding, the discharge-normalized rating curve coefficient ã increased by a mean factor of 4.89 within 1-2 years of Morakot, while the rating curve exponent b exhibited no systematic response to Morakot (mean factor of 1.21). Elevated values of ã declined exponentially at 26 of the 40 stations with a median characteristic timescale of 13.2 years (interquartile range: 7.6-20.7) and tended to respond faster in basins with more intense landsliding. In contrast, at the 37 stations that were not impacted by landsliding, neither ã nor b exhibited a systematic response to Morakot. To quantify the effect of landsliding on sediment discharge, we compared the measured sediment discharges after Morakot to the hypothetical sediment discharges that would have occurred if no change in sediment transport efficiency had occurred at the time of Morakot, calculated by applying the post-Morakot water discharge history to pre-Morakot rating curves. This analysis suggests that Morakot-induced landsliding increased sediment discharge by as much as >10-fold in some basins in southeast Taiwan in the 1-2 years after Morakot. Post-Morakot changes in ã were positively correlated with landslide intensity for approximately seven years after Morakot, and changes in b were negatively correlated from 2011 to 2014, indicating that the influence of landsliding on rating curves diminished within a few years. Together, these results suggest that Morakot-induced landsliding amplified fluvial sediment fluxes for a relatively short time (< 10 years). To the extent that these results are applicable to rivers in other landscapes, this suggests that rivers may more efficiently transport landslide-derived sediment for relatively short times.

Benchmarking the quality of river discharge data and understanding its information content for hydrological analyses is an important task for hydrologic science. There is a wide variety of techniques to assess discharge uncertainty. However, few studies have developed generalized approaches to quantify discharge uncertainty. This study presents a generalized framework for estimating discharge uncertainty at many gauging stations with different errors in the stage‐discharge relationship. The methodology utilizes a nonparametric LOWESS regression within a novel framework that accounts for uncertainty in the stage‐discharge measurements, scatter in the stage‐discharge data and multisection rating curves. The framework was applied to 500 gauging stations in England and Wales and we evaluated the magnitude of discharge uncertainty at low, mean and high flow points on the rating curve. The framework was shown to be robust, versatile and able to capture place‐specific uncertainties for a number of different examples. Our study revealed a wide range of discharge uncertainties (10–397% discharge uncertainty interval widths), but the majority of the gauging stations (over 80%) had mean and high flow uncertainty intervals of less than 40%. We identified some regional differences in the stage‐discharge relationships, however the results show that local conditions dominated in determining the magnitude of discharge uncertainty at a gauging station. This highlights the importance of estimating discharge uncertainty for each gauging station prior to using those data in hydrological analyses.

Objective segmentation in compound rating curves

Abstract. When estimating discharges through rating curves, temporal data consistency is a critical issue. In this research, consistency in stage–discharge data is investigated using a methodology called Bidirectional Reach (BReach), which departs from a (in operational hydrology) commonly used definition of consistency. A period is considered to be consistent if no consecutive and systematic deviations from a current situation occur that exceed observational uncertainty. Therefore, the capability of a rating curve model to describe a subset of the (chronologically sorted) data is assessed in each observation by indicating the outermost data points for which the rating curve model behaves satisfactorily. These points are called the maximum left or right reach, depending on the direction of the investigation. This temporal reach should not be confused with a spatial reach (indicating a part of a river). Changes in these reaches throughout the data series indicate possible changes in data consistency and if not resolved could introduce additional errors and biases. In this research, various measurement stations in the UK, New Zealand and Belgium are selected based on their significant historical ratings information and their specific characteristics related to data consistency. For each country, regional information is maximally used to estimate observational uncertainty. Based on this uncertainty, a BReach analysis is performed and, subsequently, results are validated against available knowledge about the history and behavior of the site. For all investigated cases, the methodology provides results that appear to be consistent with this knowledge of historical changes and thus facilitates a reliable assessment of (in)consistent periods in stage–discharge measurements. This assessment is not only useful for the analysis and determination of discharge time series, but also to enhance applications based on these data (e.g., by informing hydrological and hydraulic model evaluation design about consistent time periods to analyze).

Recently, Markov processes for the evolution of coding DNA with neighbor dependence in the instantaneous substitution rates have been considered. The neighbor dependency makes the models analytically intractable, and previously Markov chain Monte Carlo methods have been used for statistical inference. Using a pseudo-likelihood idea, we introduce in this paper an approximative estimation method which is fast to compute. The pseudo-likelihood estimates are shown to be very accurate, and from analyzing 348 human-mouse coding sequences we conclude that the incorporation of a CpG effect improves the fit of the model considerably.

Diffusion is a basic mathematical tool for modern financial engineering. The theory of the estimation methods for diffusion models is an important topic of the financial engineering. Many researches have been tried to apply the likelihood estimation method for estimating diffusion models. However, the likelihood estimation method for diffusion is complicated and needs much amount of computing. In this paper we develop the estimation methods which are simple enough to be compared to the Euler approximation method, and efficient enough statistically to be compared to the likelihood estimation method. We devise pseudo-likelihood and propose the maximum pseudo-likelihood estimation methods. The pseudo-likelihoods are obtained by approximating the transition density with normal distributions. The means and the variances of the distributions are obtained from the delta expansion suggested by Lee, Song and Lee (2012). We compare the newly suggested estimators with other existing estimators by simulation study. From the simulation study we find the maximum pseudo-likelihood estimator has very similar properties with the maximum likelihood estimator. Also the maximum pseudo-likelihood estimator is easy to apply to general diffusion models, and can be obtained by simple numerical steps.

We consider a simple yet flexible spline estimation method for quasi-likelihood models. We approximate the unknown function by B-splines and apply the Fisher scoring algorithm to compute the estimates. The spline estimate of the nonparametric component achieves the optimal rate of convergence under the smooth condition, and the estimate of the parametric part is shown to be asymptotically normal even if the variance function is misspecified. The semiparametric efficiency of the model can be established if the variance function is correctly specified. A direct and consistent variance estimation method based on the least-squares estimation is proposed. A simulation study is performed to evaluate the numerical performance of the spline estimate. The methodology is illustrated on a crab study.

This paper describes a computer-based approach named multiobjective stage–discharge evaluator (MOSDE), which employs the method of regression analysis for the development of stage–discharge rating curves. The regression is done recursively as the program steps through the data set, by adding one point at each iteration. Multiobjective analysis is used to evaluate the improvement or deterioration of the model using a set of statistical criteria. The overall task is to fit a proper curve, or curves, to any given stage–discharge data set. Also, the key transition points in the stage–discharge relation are identified. This process eliminates ambiguity or subjectivity associated with the current methods used for the development of stage–discharge relationships. The MOSDE system has been tested using two different data sets. Key words: operational hydrology, stage–discharge, multiobjective analysis, mathematical modelling.

Evaluation of the efficiency of streamflow data collection strategies for alluvial rivers

ABSTRACTEstimates of peak flood discharges are most often based on stage measurements, and the discharge is determined indirectly from the extrapolation of a rating-curve. It is possible to improve the extrapolation of the rating-curve by hydraulic modeling, such as scale or numerical models. In this paper, we discuss a case study for a streamflow gauging station in a mountainous river in Norway. A computational fluid dynamics model is used as part of a hybrid modeling study to study the flow at this gauging station with challenging geometry and flow and derive a reliable rating-curve. The numerical model produces stage-discharge data with comparable accuracy to the scale model. Analysis of the results of the hybrid modeling shows that the original location of stage measurements is not suitable for field measurements or for modeling the rating curve. Based on the modeling, an alternative point of measurement with better flow conditions at high flows is identified. The results and findings highlight the importance of 3-dimensional flow at the site during floods and shows how computational fluid dynamics can be used to provide stage-discharge data for rating curves and provide additional insight in hybrid modeling.

Calculating the volumes of water discharged by streams is becoming increasingly important in water accounting and deciding how much water to allocate to competing uses. Water accounting is par- ticularly important in Australia, as the driest inhabited continent and also in the face of potential impacts of a changing climate. Stream networks all over the world are littered with gauging stations, which take regular measurements of steam flow in order to help natural resource managers make decisions regard- ing water allocation. Estimating total discharge volumes is also of utmost importance when estimating pollutant loads from catchments. In order to calculate the total discharge volume, one must integrate the hydrograph (the graph of stream flow with time) over the period of interest. The simplest method to perform the integration is a trapezoidal scheme, however this fails to account for a number of sources of uncertainty inherent in the hydrograph, namely: (i) what happens between the discrete observations; (ii) gauging stations measure water height and flow is estimated using a rating curve between height and flow; and (iii) there are measurement er- rors associated with the height data recorded at gauging stations. We present a Monte Carlo method that employs: (i) nonparametric stochastic differential equations (SDEs) to bridge the gaps between discrete observations; and (ii) the Weighted Nadaraya-Watson estimator to estimate the conditional distribution of log-flow given water height. The output of the method is an ensemble of hydrographs that are faithful to the observed data, but incorporating these uncertainties/errors. Integrating the members of this ensem- ble gives rise to a distribution for the total discharge volumes and properly accounts for the imperfect information available. We demonstrate the methods using hydrographic data from Obi Obi Creek in the Mary River catchment, Queensland, Australia and examine the uncertainty inherent in total discharges when integrating over a single month and over an entire year. We also introduce an artificial gap of 375 days into the hydrograph and demonstrate how well our simulated diffusions replicate the dynamics of stream flow. Whilst our Monte Carlo method is useful for estimating total discharge volumes, the nonparametric SDEs used also appear to have good potential as stochastic rainfall-runoff models in their own right.