Modelling soil carbon development in Swedish coniferous forest soils—An uncertainty analysis of parameters and model estimates using the GLUE method

Boreal forest soils such as those in Sweden contain a large active carbon stock. Hence, a relatively small change in this stock can have a major impact on the Swedish national CO₂ balance. Understanding of the uncertainties in the estimations of soil carbon pools is critical for accurately assessing changes in carbon stocks in the national reports to UNFCCC and the Kyoto Protocol. Our objective was to analyse the parameter uncertainties of simulated estimates of the soil organic carbon (SOC) development between 1994 and 2002 in Swedish coniferous forests with the Q model. Both the sensitivity of model parameters and the uncertainties in simulations were assessed. Data of forests with Norway spruce, Scots pine and Lodgepole pine, from the Swedish Forest Soil Inventory (SFSI) were used. Data of 12 Swedish counties were used to calibrate parameter settings; and data from another 11 counties to validate. The “limits of acceptability” within GLUE were set at the 95% confidence interval for the annual, mean measured SOC at county scale. The calibration procedure reduced the parameter uncertainties and reshaped the distributions of the parameters county-specific. The average measured and simulated SOC amounts varied from 60 t C ha⁻¹ in northern to 140 t C ha⁻¹ in the southern Sweden. The calibrated model simulated the soil carbon pool within the limits of acceptability for all calibration counties except for one county during one year. The efficiency of the calibrated model varied strongly; for five out of 12 counties the model estimates agreed well with measurements, for two counties agreement was moderate and for five counties the agreement was poor. The lack of agreement can be explained with the high inter-annual variability of the down-scaled measured SOC estimates and changes in forest areas over time. We conclude that, although we succeed in reducing the uncertainty in the model estimates, calibrating of a regional scale process-oriented model using a national scale dataset is a sensitive balance between introducing and reducing uncertainties. Parameter distributions showed to be scale sensitive and county specific. Further analysis of uncertainties in the methods used for reporting SOC changes to the UNFCCC and Kyoto protocol is recommended.     

Impact of time series data on calibration and prediction uncertainty for a deterministic hydrogeochemical model

Model calibration is fundamental in applications of deterministic process-based models. Uncertainty in model predictions depends much on the input data and observations available for model calibration. Here we explored how model predictions (forecasts) and their uncertainties vary with the length of time series data used in calibration. As an example we used the hydrogeochemical model MAGIC and data from Birkenes, a small catchment in southern Norway, to simulate future water chemistry under a scenario of reduced acid deposition. A Bayesian approach with a Markov Chain Monte Carlo (MCMC) technique was used to calibrate the model to different lengths of observed data (4–29 years) and to estimate the prediction uncertainty each calibration. The results show that the difference between modelled and observed water chemistry (calibration goodness of fit) in general decreases with increasing length of the time series used in calibration. However, there are considerable differences for different time series of the same length. The results also show that the uncertainties in predicted future acid neutralizing capacity were lowest (i.e. the distribution peak narrowest) when using the longest time series for calibration. As for calibration success, there were considerable differences between the future distributions (prediction uncertainty) for the different calibrations.     

A comparison of sensitivity analysis and error analysis based on a stream ecosystem model

When individual model parameters must be measured in field or laboratory experiments, the provision of feedback information for allocation of research efforts is an important function of modeling. Both sensitivity analysis and Monte Carlo error analysis can be used to determine which parameters require intensified measurement effort. When both methods are applied to a stream ecosystem model, the assumptions of sensitivity analysis are violated if reasonable estimates of measurement errors on parameters are used. Sensitivity analysis estimates a linear relationship between a state variable and a parameter and largely ignores higher order effects.In the model investigated in this study, higher-order effects dominate prediction error, and the results of sensitivity analysis are misleading. It is suggested that the simple correlation coefficient derived from analysis of Monte Carlo simulations is a more reasonable way to rank model parameters according to their contribution to prediction uncertainty. For the stream model used in this study, halving variance on the four parameters, indicated as most important by sensitivity analysis, reduces prediction errors by only 2–6%. Halving variance on the four, completely different, parameters with the largest simple correlation coefficients reduces prediction errors by 17–31%.     

Quantifying parameter uncertainty in a coral reef model using Metropolis-Coupled Markov Chain Monte Carlo

Coral reefs are threatened ecosystems, so it is important to have predictive models of their dynamics. Most current models of coral reefs fall into two categories. The first is simple heuristic models which provide an abstract understanding of the possible behaviour of reefs in general, but do not describe real reefs. The second is complex simulations whose parameters are obtained from a range of sources such as literature estimates. We cannot estimate the parameters of these models from a single data set, and we have little idea of the uncertainty in their predictions.We have developed a compromise between these two extremes, which is complex enough to describe real reef data, but simple enough that we can estimate parameters for a specific reef from a time series. In previous work, we fitted this model to a long-term data set from Heron Island, Australia, using maximum likelihood methods. To evaluate predictions from this model, we need estimates of the uncertainty in our parameters. Here, we obtain such estimates using Bayesian Metropolis-Coupled Markov Chain Monte Carlo. We do this for versions of the model in which corals are aggregated into a single state variable (the three-state model), and in which corals are separated into four state variables (the six-state model), in order to determine the appropriate level of aggregation. We also estimate the posterior distribution of predicted trajectories in each case.In both cases, the fitted trajectories were close to the observed data, but we had doubts about the biological plausibility of some parameter estimates. We suggest that informative prior distributions incorporating expert knowledge may resolve this problem. In the six-state model, the posterior distribution of state frequencies after 40 years contained two divergent community types, one dominated by free space and soft corals, and one dominated by acroporid, pocilloporid, and massive corals. The three-state model predicts only a single community type. We conclude that the three-state model hides too much biological heterogeneity, but we need more data if we are to obtain reliable predictions from the six-state model. It is likely that there will be similarly large, but currently unevaluated, uncertainty in the predictions of other coral reef models, many of which are much more complex and harder to fit to real data.     

Multiple sources of predictive uncertainty in modeled estimates of net ecosystem CO2 exchange

Net ecosystem CO₂ exchange (NEE) is typically measured directly by eddy covariance towers or is estimated by ecosystem process models, yet comparisons between the data obtained by these two methods can show poor correspondence. There are three potential explanations for this discrepancy. First, estimates of NEE as measured by the eddy-covariance technique are laden with uncertainty and can potentially provide a poor baseline for models to be tested against. Second, there could be fundamental problems in model structure that prevent an accurate simulation of NEE. Third, ecosystem process models are dependent on ecophysiological parameter sets derived from field measurements in which a single parameter for a given species can vary considerably. The latter problem suggests that with such broad variation among multiple inputs, any ecosystem modeling scheme must account for the possibility that many combinations of apparently feasible parameter values might not allow the model to emulate the observed NEE dynamics of a terrestrial ecosystem, as well as the possibility that there may be many parameter sets within a particular model structure that can successfully reproduce the observed data. We examined the extent to which these three issues influence estimates of NEE in a widely used ecosystem process model, Biome-BGC, by adapting the generalized likelihood uncertainty estimation (GLUE) methodology. This procedure involved 400,000 model runs, each with randomly generated parameter values from a uniform distribution based on published parameter ranges, resulting in estimates of NEE that were compared to daily NEE data from young and mature Ponderosa pine stands at Metolius, Oregon. Of the 400,000 simulations run with different parameter sets for each age class (800,000 total), over 99% of the simulations underestimated the magnitude of net ecosystem CO₂ exchange, with only 4.07% and 0.045% of all simulations providing satisfactory simulations of the field data for the young and mature stands, even when uncertainties in eddy-covariance measurements are accounted for. Results indicate fundamental shortcomings in the ability of this model to produce realistic carbon flux data over the course of forest development, and we suspect that much of the mismatch derives from an inability to realistically model ecosystem respiration. However, difficulties in estimating historic climate data are also a cause for model-data mismatch, particularly in a highly ecotonal region such as central Oregon. This latter difficulty may be less prevalent in other ecosystems, but it nonetheless highlights a challenge in trying to develop a dynamic representation of the terrestrial biosphere.     

Uncertainty in future soil carbon trends at a central U.S. site under an ensemble of GCM scenario climates

This study uses DAYCENT model to investigate the sensitivity of soil organic carbon (SOC) at an intensely cultivated site in the U.S. Midwest under an ensemble of scenario climates predicted by IPCC models. The model ensemble includes three IPCC models (Canadian, French, German), three emission scenarios (B1, A1B, A2) and three time periods (late 20th, mid-21st, late 21st century). DAYCENT shows that SOC at the site would decline by 0.3-2.6 kg m⁻² (5-35%) depending on the models and scenarios from late 20th to mid-21st century despite a larger increase of future net primary production (NPP) than respiration. The future SOC decrease is mostly attributable to harvest loss. The wide spread in future SOC decline rates are in part because SOC decrease (by respiration) is directly proportional to SOC itself. Any uncertainty in absolute SOC in DAYCENT would translate directly into its trend, unlike other variables such as temperature whose trends are independent of their values themselves, contrasting the reliability of SOC trend with temperature change.     

Sensitivity Analyses of Spatial Population Viability Analysis Models for Species at Risk and Habitat Conservation Planning

Abstract:  Population viability analysis (PVA) is an effective framework for modeling species- and habitat-recovery efforts, but uncertainty in parameter estimates and model structure can lead to unreliable predictions. Integrating complex and often uncertain information into spatial PVA models requires that comprehensive sensitivity analyses be applied to explore the influence of spatial and nonspatial parameters on model predictions. We reviewed 87 analyses of spatial demographic PVA models of plants and animals to identify common approaches to sensitivity analysis in recent publications. In contrast to best practices recommended in the broader modeling community, sensitivity analyses of spatial PVAs were typically ad hoc, inconsistent, and difficult to compare. Most studies applied local approaches to sensitivity analyses, but few varied multiple parameters simultaneously. A lack of standards for sensitivity analysis and reporting in spatial PVAs has the potential to compromise the ability to learn collectively from PVA results, accurately interpret results in cases where model relationships include nonlinearities and interactions, prioritize monitoring and management actions, and ensure conservation-planning decisions are robust to uncertainties in spatial and nonspatial parameters. Our review underscores the need to develop tools for global sensitivity analysis and apply these to spatial PVA.     

Ensemble Bayesian model averaging using Markov Chain Monte Carlo sampling

Bayesian model averaging (BMA) has recently been proposed as a statistical method to calibrate forecast ensembles from numerical weather models. Successful implementation of BMA however, requires accurate estimates of the weights and variances of the individual competing models in the ensemble. In their seminal paper (Raftery et al. Mon Weather Rev 133:1155–1174, 2005) has recommended the Expectation–Maximization (EM) algorithm for BMA model training, even though global convergence of this algorithm cannot be guaranteed. In this paper, we compare the performance of the EM algorithm and the recently developed DiffeRential Evolution Adaptive Metropolis (DREAM) Markov Chain Monte Carlo (MCMC) algorithm for estimating the BMA weights and variances. Simulation experiments using 48-hour ensemble data of surface temperature and multi-model streamflow forecasts show that both methods produce similar results, and that their performance is unaffected by the length of the training data set. However, MCMC simulation with DREAM is capable of efficiently handling a wide variety of BMA predictive distributions, and provides useful information about the uncertainty associated with the estimated BMA weights and variances.     

Using global sensitivity analysis of demographic models for ecological impact assessment

Population viability analysis (PVA) is widely used to assess population‐level impacts of environmental changes on species. When combined with sensitivity analysis, PVA yields insights into the effects of parameter and model structure uncertainty. This helps researchers prioritize efforts for further data collection so that model improvements are efficient and helps managers prioritize conservation and management actions. Usually, sensitivity is analyzed by varying one input parameter at a time and observing the influence that variation has over model outcomes. This approach does not account for interactions among parameters. Global sensitivity analysis (GSA) overcomes this limitation by varying several model inputs simultaneously. Then, regression techniques allow measuring the importance of input‐parameter uncertainties. In many conservation applications, the goal of demographic modeling is to assess how different scenarios of impact or management cause changes in a population. This is challenging because the uncertainty of input‐parameter values can be confounded with the effect of impacts and management actions. We developed a GSA method that separates model outcome uncertainty resulting from parameter uncertainty from that resulting from projected ecological impacts or simulated management actions, effectively separating the 2 main questions that sensitivity analysis asks. We applied this method to assess the effects of predicted sea‐level rise on Snowy Plover (Charadrius nivosus). A relatively small number of replicate models (approximately 100) resulted in consistent measures of variable importance when not trying to separate the effects of ecological impacts from parameter uncertainty. However, many more replicate models (approximately 500) were required to separate these effects. These differences are important to consider when using demographic models to estimate ecological impacts of management actions.     

Analysis of binary data via a centered spatial-temporal autologistic regression model

A centered spatial-temporal autologistic model is developed for analyzing spatial-temporal binary data observed on a lattice over time. We propose expectation-maximization pseudolikelihood and Monte Carlo expectation-maximization likelihood as well as consider Bayesian inference to obtain the estimates of model parameters. Further, we compare the statistical efficiency of the three approaches for various sizes of sampling lattices and numbers of sampling time points. Regarding prediction, we use Monte Carlo to obtain predictive distributions at future time points and compare the performance of the model with the uncentered spatial-temporal autologistic regression model. The methodology is demonstrated via simulation studies and a real data example concerning southern pine beetle outbreak in North Carolina.     

Assessing plausible rates of population growth in humpback whales from life-history data

The rate of growth of any population is a quantity of interest in conservation and management and is constrained by biological factors. In this study, recent data on life-history parameters influencing rates of population growth in humpback whales, including survival, age at first parturition and calving rate are reviewed. Monte Carlo simulations are used to compute a distribution of rates of increase (ROIs) taking into account uncertainty in biological parameter estimates. Two approaches for computing juvenile survival are proposed, which taken into account along with other life-history data, resulted in the following estimates of the rate of population growth: Approach A: mean of 7.3%/year (95% CI = 3.5–10.5%/year) and Approach B: mean of 8.6%/year (95% CI = 5.0–11.4%/year). It is proposed that the upper 99% quantile of the resulting distribution of the ROI for Approach B (11.8%/year) be established as the maximum plausible ROI for humpback whales and be used in population assessment of the species. Possible sources of positive and negative biases in the present estimates are presented and include measurement error in estimation of life-history parameters, changes in the environment within the period these quantities are measured, density dependence or other natural factors. However, it is difficult to evaluate potential biases without additional data. The methods presented in this study can be applied to other species for which life-history parameters are available and are useful in assessing plausibility in the estimation of population growth rates from time series of abundance estimates.     

密云水库上游地区农田土壤有机碳储量及变化模拟

农田土壤有机碳(SOC)库对粮食安全和全球气候变化具有重要影响,因此,开展农田土壤有机碳储量及其动态变化研究在政治经济和生态环境层面具有重要意义.采用农业生物地球化学模型--DNDC对密云水库上游地区农田土壤有机碳储量及其变化进行模拟研究,首先应用当地实测结果进行模型验证,然后根据当地气候条件、土壤性质和现行农业耕作管理特点等建立GIS区域数据库,并在数据库的支持下进行区域模拟和分析.结果表明:2006年密云水库上游地区214 920 hm~2农田土壤(0～25 cm)的总有机碳储量为7 646×10~6 kg,其中位于河北省境内的该地区63.1%的农田储存了全区68.1%的SOC;平均每公顷农田SOC储量为35 576.1 kg,低于全国平均水平;由于化肥和有机肥投入的增加,经过1 a耕种后,2006年该地区农田SOC储量增加142.5×10~6 kg,整个地区及各区县农田土壤碳收支均为正,是大气CO_2的一个汇.情景分析表明,气温升高对该地区农田SOC积累具有显著的负效应;而提高秸秆还田比例、适量施用化肥、增施有机肥、增加灌溉和采取免耕方式等措施均能有效增加土壤有机碳的积累.     

Uncertainty analysis of a pharmacokinetic modeling for inorganic arsenic

This paper presents an uncertainty and sensitivity analysis of a pharmacokinetic modeling of inorganic arsenic deposition in rodents for a short‐term exposure. Efforts to develop the pharmacokinetic model are directed towards predicting the kinetic behavior of inorganic arsenic in the body, including tissue and blood concentrations, and especially, the urinary excretion of arsenic and its methylated metabolites. However, the use of the model raises an important question when fixed values of model parameters are used: how is the uncertainty in the model prediction based on the collective uncertainties in the model inputs? This study focuses on an “epistemic”; uncertainty in order to handle this problem. In this case, the uncertainty refers to an input that has a single value which cannot be known with precision due to a lack of knowledge about items or its measurement. The combination of the pharmacokinetic model and the uncertainty analysis would help understand the uncertainties in risk assessment associated with inorganic arsenic.     

The Reliability of Using Population Viability Analysis for Risk Classification of Species

I examine whether or not it is appropriate to use extinction probabilities generated by population viability analyses, based on best estimates for model parameters, as criteria for listing species in Red Data Book categories as recently proposed by the World Conservation Union. Such extinction probabilities are influenced by how accurately model parameters are estimated and by how accurately the models depict actual population dynamics. I evaluate the effect of uncertainty in parameter estimation through simulations. Simulations based on Steller sea lions were used to evaluate bias and precision in estimates of probability of extinction and to consider the performance of two proposed classification schemes. Extinction time estimates were biased (because of violation of the assumption of stable age distribution) and underestimated the variability of probability of extinction for a given time (primarily because of uncertainty in parameter estimation). Bias and precision in extinction probabilities are important when these probabilities are used to compare the risk of extinction between species. Suggestions are given for population viability analysis techniques that incorporate parameter uncertainty. I conclude that testing classification schemes with simulations using quantitative performance objectives should precede adoption of quantitative listing criteria.     

Making conservation decisions under uncertainty for the persistence of multiple species.

Population models for multiple species provide one of the few means of assessing the impact of alternative management options on the persistence of biodiversity, but they are inevitably uncertain. Is it possible to use population models in multiple-species conservation planning given the associated uncertainties? We use information-gap decision theory to explore the impact of parameter uncertainty on the conservation decision when planning for the persistence of multiple species. An information-gap approach seeks robust outcomes that are most immune from error. We assess the impact of uncertainty in key model parameters for three species, whose extinction risks under four alternative management scenarios are estimated using a metapopulation model. Three methods are described for making conservation decisions across the species, taking into account uncertainty. We find that decisions based on single species are relatively robust to uncertainty in parameters, although the estimates of extinction risk increase rapidly with uncertainty. When identifying the best conservation decision for the persistence of all species, the methods that rely on the rankings of the management options by each species result in decisions that are similarly robust to uncertainty. Methods that depend on absolute values of extinction risk are sensitive to uncertainty, as small changes in extinction risk can alter the ranking of the alternative scenarios. We discover that it is possible to make robust conservation decisions even when the uncertainties of the multiple-species problem appear overwhelming. However, the decision most robust to uncertainty is likely to differ from the best decision when uncertainty is ignored, illustrating the importance of incorporating uncertainty into the decision-making process.     

Estimating parameters of hidden Markov models based on marked individuals: use of robust design data

Development and use of multistate mark-recapture models, which provide estimates of parameters of Markov processes in the face of imperfect detection, have become common over the last 20 years. Recently, estimating parameters of hidden Markov models, where the state of an individual can be uncertain even when it is detected, has received attention. Previous work has shown that ignoring state uncertainty biases estimates of survival and state transition probabilities, thereby reducing the power to detect effects. Efforts to adjust for state uncertainty have included special cases and a general framework for a single sample per period of interest. We provide a flexible framework for adjusting for state uncertainty in multistate models, while utilizing multiple sampling occasions per period of interest to increase precision and remove parameter redundancy. These models also produce direct estimates of state structure for each primary period, even for the case where there is just one sampling occasion. We apply our model to expected-value data, and to data from a study of Florida manatees, to provide examples of the improvement in precision due to secondary capture occasions. We have also implemented these models in program MARK. This general framework could also be used by practitioners to consider constrained models of particular interest, or to model the relationship between within-primary-period parameters (e.g., state structure) and between-primary-period parameters (e.g., state transition probabilities).     

Assessment of uncertainties in expert knowledge, illustrated in fuzzy rule-based models

The coherence between different aspects in the environmental system leads to a demand for comprehensive models of this system to explore the effects of different management alternatives. Fuzzy logic has been suggested as a means to extend the application domain of environmental modelling from physical relations to expert knowledge. In such applications the expert describes the system in terms of fuzzy variables and inference rules. The result of the fuzzy reasoning process is a numerical output value. In such a model, as in any other, the model context, structure, technical aspects, parameters and inputs may contribute uncertainties to the model output. Analysis of these contributions in a simplified model for agriculture suitability shows how important information about the accuracy of the expert knowledge in relation to the other uncertainties can be provided. A method for the extensive assessment of uncertainties in compositional fuzzy rule-based models is proposed, combining the evaluation of model structure, input and parameter uncertainties. In an example model, each of these three appear to have the potential to dominate aggregated uncertainty, supporting the relevance of an ample uncertainty approach.     

祁连山中部土壤颗粒组分有机质碳含量及其与海拔和植被的关系

调查分析了祁连山中段不同海拔土壤颗粒有机碳及其与植被的关系.结果显示,土壤颗粒组分比例在0～15 cm和15～35cm土层随海拔升高而呈现下降趋势(P>0.2);土壤颗粒有机碳比例在0～15 cm土层随海拔升高也呈现下降趋势(P≤0.001).土壤颗粒组分比例0～15 cm土层在阴坡3 000 m～3 500 m、15～35 cm土层在阴坡3 200 m和3 500 m及半阴坡2 200和2 800 m处较高;土壤颗粒有机碳比例0～15 cm土层在阴坡3 000 m和3 200 m、半阴坡2 200 m和2 800 m,以及15～35 cm土层在阴坡3 200 m和3 500 m、阳坡3 300 m和3 500 m处较高(P<0.05).土壤颗粒有机碳和颗粒组分碳含量随海拔升高变化不显著(P<0.9).土壤颗粒有机碳含量0～15cm土层在阴坡3 000 m～3 500 m、15～35 cm土层在阴坡3 000 m～3 500 m及阳坡3 300m处较高;土壤颗粒组分碳含量0～15 cm土层在阴坡3 000 m～3 400 m和阳坡3 300 m,以及15～35 cm土层在阴坡3 200 m和3 400 m及阳坡3 300 m处较高.土壤颗粒组分比例0～15 cm土层在森林和灌丛草甸中较高;15～35 cm土层在森林、灌丛草甸和干旱草原中较高(P<0.05).土壤颗粒有机碳比例0～15 cm土层在荒漠草原和干旱草原,以及15～30 cm土层在森林和灌丛草甸中较高(P<0.05).土壤颗粒组分碳含量0～15 cm和15～35 cm土层在森林和灌丛草甸中较高(P<0.05).土壤颗粒有机碳含量0～15cm和15～35cm土层在森林中最高(P<0.05).土壤颗粒组分碳含量和颗粒有机碳含量与土壤有机碳含量有显著的正相关性(P<0.001),土壤颗粒有机碳含量与颗粒组分碳含量也有显著的正相关性(P<0.001),土壤颗粒组分比例与有机碳含量相关性不显著(P=0.15),土壤颗粒有机碳含量与颗粒组分比例有显著正相关性(P<0.005).结果说明祁连山中部北坡土壤有机碳稳定性受植被和海拔共同影响,荒漠草原和干旱草原表层土壤有机碳稳定性较低,森林和灌丛草甸土壤中非保护性碳含量较高.     

Nitrogen source apportionment—a comparison between a dynamic and a statistical model

A dynamic model, HBV-N, and a statistical model, MESAW, for nitrogen source apportionment were compared regarding model performance, model uncertainty and user applicability. The HBV-N model simulates continuous series of nitrogen concentrations with meteorological data and sub-basin characteristics as input. Diffuse nitrogen emissions are defined as regional model parameters which are calibrated by comparison of observed and simulated nitrogen data. The MESAW model uses nitrogen loads for a fixed time interval at each monitoring site as response variable and sub-basin characteristics as explanatory variables to estimate diffuse nitrogen emissions through non-linear regression analysis. The two models were applied in the Matsalu Bay watershed (3640 km²) in Estonia and the same land use and point sources data were used as input. Both models gave similar levels of diffuse total nitrogen emissions and retention rates, which also fit well with previous estimates made in Estonia and Scandinavia. A sensitivity analysis of the model parameters also showed similar uncertainty levels, which indicated that the model uncertainty was more dependent on the availability of nitrogen data and land cover distribution than the choice of model. Furthermore, the sensitivity analysis showed a parameter interdependency in both models, which implied the risk of compensation between estimated diffuse emissions and retention. In conclusion, however, the study showed that both models were capable of estimating nitrogen leakage from the dominating land classes and giving reliable source apportionment from the available input data. The study indicated that the HBV-N model has its advantage in assessments where detailed outputs are needed and when run-off data are limited, while the statistical MESAW model has its advantage in extensive studies since it is easily applied to large watersheds that have dense monitoring networks.