Shuffled Complex Evolution Metropolis Algorithm Research Articles

Analysing trade-offs in resource and labour allocation by smallholder farmers using inverse modelling techniques: A case-study from Kakamega district, western Kenya

Smallholder farms in sub-Saharan Africa face multiple trade-offs when deciding on the allocation of their financial, labour and nutrient resources. Day-to-day decisions have implications for the sustainability of their farming system, implying multiple trade-offs between short- and long-term objectives that have biophysical and socio-economic dimensions. We show that inverse modelling techniques can be used effectively for optimisation and trade-offs analysis of farming systems. By combining the multi-objective shuffled complex evolution metropolis algorithm and a crop/soil dynamic simulation model we were able to select farming strategies that resulted in the best possible trade-offs between different farming objectives. This integrated analytical tool allows optimisation of farmers’ goals similar to linear programming, but an advantage over linear programming is that the proposed method takes into account a wider spectrum of biophysical processes including their interactions and feedbacks. Tradeoffs between resource productivity, use efficiency and conservation in relation to different patterns of resource allocation were analysed for a maize-based, simplified case study farm from western Kenya (2.2 ha – comprising fields of poor, medium and high soil fertility), under three scenarios of financial liquidity to invest in labour and inputs (2000, 5000 and 10,000 KSh ha −1; 75 KSh = 1 US$). The maximum farm-scale maize production achieved was larger when financial resources increased. However, increasing maize yields above a certain threshold by applying mineral fertilisers was associated with larger N losses by leaching, runoff and soil erosion; such threshold was 2.7 t grain ha −1 for the scenario of no financial limitations (10,000 KSh ha −1). N losses at farm scale fluctuated between 36 and 54 kg N ha −1 season −1, while the maximum maize yields achieved were around 3.4 t grain ha −1. Soil losses by erosion increased abruptly beyond a certain maize yield (e.g. 1.8 t grain ha −1 for the 2000 KSh ha −1 scenario), while the minimum rate of soil loss differed between financial scenarios. Investments in hiring labour were prioritised over fertiliser use to obtain the greatest yields and the allocation of available resources favoured the more fertile fields. This inverse modelling exercise allowed us to analyse trade-offs between different farmers’ objectives and to compare potential resource allocation strategies to achieve them. The set of strategies to achieve different goals was more numerous and variable when the conditions were less conducive for farming. This questions the validity of the prevailing model of extension/communication, based on generalised recommendations for resource-poor farmers in Africa.

How effective and efficient are multiobjective evolutionary algorithms at hydrologic model calibration?

Abstract. This study provides a comprehensive assessment of state-of-the-art evolutionary multiobjective optimization (EMO) tools' relative effectiveness in calibrating hydrologic models. The relative computational efficiency, accuracy, and ease-of-use of the following EMO algorithms are tested: Epsilon Dominance Nondominated Sorted Genetic Algorithm-II (ε-NSGAII), the Multiobjective Shuffled Complex Evolution Metropolis algorithm (MOSCEM-UA), and the Strength Pareto Evolutionary Algorithm 2 (SPEA2). This study uses three test cases to compare the algorithms' performances: (1) a standardized test function suite from the computer science literature, (2) a benchmark hydrologic calibration test case for the Leaf River near Collins, Mississippi, and (3) a computationally intensive integrated surface-subsurface model application in the Shale Hills watershed in Pennsylvania. One challenge and contribution of this work is the development of a methodology for comprehensively comparing EMO algorithms that have different search operators and randomization techniques. Overall, SPEA2 attained competitive to superior results for most of the problems tested in this study. The primary strengths of the SPEA2 algorithm lie in its search reliability and its diversity preservation operator. The biggest challenge in maximizing the performance of SPEA2 lies in specifying an effective archive size without a priori knowledge of the Pareto set. In practice, this would require significant trial-and-error analysis, which is problematic for more complex, computationally intensive calibration applications. ε-NSGAII appears to be superior to MOSCEM-UA and competitive with SPEA2 for hydrologic model calibration. ε-NSGAII's primary strength lies in its ease-of-use due to its dynamic population sizing and archiving which lead to rapid convergence to very high quality solutions with minimal user input. MOSCEM-UA is best suited for hydrologic model calibration applications that have small parameter sets and small model evaluation times. In general, it would be expected that MOSCEM-UA's performance would be met or exceeded by either SPEA2 or ε-NSGAII.

Application of stochastic parameter optimization to the Sacramento Soil Moisture Accounting model

Hydrological models generally contain parameters that cannot be measured directly, but can only be meaningfully inferred by calibration against a historical record of input–output data. While considerable progress has been made in the development and application of automatic procedures for model calibration, such methods have received criticism for their lack of rigor in treating uncertainty in the parameter estimates. In this paper, we apply the recently developed Shuffled Complex Evolution Metropolis algorithm (SCEM-UA) to stochastic calibration of the parameters in the Sacramento Soil Moisture Accounting model (SAC-SMA) model using historical data from the Leaf River in Mississippi. The SCEM-UA algorithm is a Markov Chain Monte Carlo sampler that provides an estimate of the most likely parameter set and underlying posterior distribution within a single optimization run. In particular, we explore the relationship between the length and variability of the streamflow data and the Bayesian uncertainty associated with the SAC-SMA model parameters and compare SCEM-UA derived parameter values with those obtained using deterministic SCE-UA calibrations. Most significantly, for the Leaf River catchments under study our results demonstrate that most of the 13 SAC-SMA parameters are well identified by calibration to daily streamflow data suggesting that this data contains more information than has previously been reported in the literature.

Obtaining the Spatial Distribution of Water Content along a TDR Probe Using the SCEM‐UA Bayesian Inverse Modeling Scheme

Time domain reflectometry (TDR) has become one of the standard methods for the measurement of the temporal and spatial distribution of water saturation in soils. Current waveform analysis methodology gives a measurement of the average water content along the length of the TDR probe. Close inspection of TDR waveforms shows that heterogeneity in water content along the probe can be seen in the TDR waveform. We present a comprehensive approach to TDR waveform analysis that gives a quantitative estimate of the dielectric permittivity profile along the length of the probe and, therefore, the distribution of water content. The approach is based on the combination of a multisection scatter function model for the TDR measurement system with the shuffled complex evolution Metropolis algorithm (SCEM‐UA). This combined approach allows for the estimation of the 40 parameters in the transmission line model using a series of simple calibration measurements. The proof of concept is given with measurements in a layered system consisting of air and water. Finally, TDR waveforms from layered soil samples were analyzed to estimate the distribution of the water content along the length of the probe. Results show that the proposed method provides much more reproducible results than obtained with the traditional travel time method. Because the proposed method can be fully automated, it increases the applicability of the TDR method, especially in applications where detailed (real‐time) data are required on heterogeneous infiltration.

Obtaining the Spatial Distribution of Water Content along a TDR Probe Using the SCEM-UA Bayesian Inverse Modeling Scheme

Time domain reflectometry (TDR) has become one of the standard methods for the measurement of the temporal and spatial distribution of water saturation in soils. Current waveform analysis methodology gives a measurement of the average water content along the length of the TDR probe. Close inspection of TDR waveforms shows that heterogeneity in water content along the probe can be seen in the TDR waveform. We present a comprehensive approach to TDR waveform analysis that gives a quantitative estimate of the dielectric permittivity profile along the length of the probe and, therefore, the distribution of water content. The approach is based on the combination of a multisection scatter function model for the TDR measurement system with the shuffled complex evolution Metropolis algorithm (SCEM-UA). This combined approach allows for the estimation of the 40 parameters in the transmission line model using a series of simple calibration measurements. The proof of concept is given with measurements in a layered system consisting of air and water. Finally, TDR waveforms from layered soil samples were analyzed to estimate the distribution of the water content along the length of the probe. Results show that the proposed method provides much more reproducible results than obtained with the traditional travel time method. Because the proposed method can be fully automated, it increases the applicability of the TDR method, especially in applications where detailed (real-time) data are required on heterogeneous infiltration.

Towards reduced uncertainty in catchment nitrogen modelling: quantifying the effect of field observation uncertainty on model calibration

Abstract. The value of nitrogen (N) field measurements for the calibration of parameters of the INCA nitrogen in catchment model is explored and quantified. A virtual catchment was designed by running INCA with a known set of parameters, and field "measurements" were selected from the model run output. Then, using these measurements and the Shuffled Complex Evolution Metropolis algorithm (SCEM-UA), four of the INCA model parameters describing N transformations in the soil were optimised, while the measurement uncertainty was increased in subsequent steps. Considering measurement uncertainty typical for N field studies, none of the synthesised datasets contained sufficient information to identify the model parameters with a reasonable degree of confidence. Parameter equifinality occurred, leading to considerable uncertainty in model parameter values and in modelled N concentrations and fluxes. Fortunately, combining the datasets in a multi-objective calibration was found to be effective in dealing with these equifinality problems. With the right choice of calibration measurements, multi-objective calibrations resulted in lower parameter uncertainty. The methodology applied in this study, using a virtual catchment free of model errors, is proposed as a useful tool foregoing the application of a N model or the design of a N monitoring program. For an already gauged catchment, a virtual study can provide a point of reference for the minimum uncertainty associated with a model application. When setting up a monitoring program, it can help to decide what and when to measure. Numerical experiments indicate that for a forested, N-saturated catchment, a fortnightly sampling of NO3 and NH4 concentrations in stream water may be the most cost-effective monitoring strategy. Keywords: INCA, nitrogen model, parameter uncertainty, multi-objective calibration, virtual catchment, experimental design

Accuracy of frequency domain analysis scenarios for the determination of complex dielectric permittivity

Frequency domain analysis of time domain reflectometry waveforms has been shown to be useful for more accurate water content determination, water content determination in saline soils, and determination of such difficult to measure soil properties as specific surface area and soil solution conductivity. Earlier frequency domain analysis approaches to determine frequency‐dependent dielectric properties of soils have used a variety of methods. In this paper, these methods for the determination of dielectric permittivity were compared using the Shuffled Complex Evolution Metropolis algorithm (SCEM‐UA). SCEM‐UA is a global optimization method that allows the simultaneous determination of optimal Debye parameters, which describe the dielectric permittivity as a function of frequency, and their confidence intervals. The analysis of numerically generated measurements with added instrumental noise showed that analysis of network analyzer measurements in the frequency domain potentially has the highest accuracy for determination of dielectric permittivity. Furthermore, the analysis of time domain reflectometry waveforms in the time domain was found to be more accurate than analysis of these waveforms in the frequency domain. Analysis of real network analyzer measurements in the time and frequency domain showed that both analysis scenarios allowed reasonably accurate estimates of the Debye parameters with the SCEM‐UA algorithm, even when the true value of a parameter falls beyond the limits of the frequency bandwidth. However, frequency domain analysis of ethanol measurements showed that results were susceptible to model errors caused by nonideal probe behavior. These errors were larger for three‐wire probes than for seven‐wire probes. This study shows that the accuracy of the dielectric permittivity determination can be improved by reducing the model error. This can be achieved by the use of more accurate models, such as multiscatter functions, and by using more advanced probes, such as coaxial cells. The results also imply that future research on dielectric properties of soils should focus more on the use of network analyzers instead of cable testers, since model errors are more obvious in the frequency domain. The SCEM‐UA algorithm proved to be a valuable tool in frequency domain analysis because reported problems with parameter identification and initialization of the optimization are circumvented with this global optimization algorithm.

(Un)certainty of overall binding constants of Al with dissolved organic matter determined by the Scatchard approach

One of the best approaches to date to obtain overall binding constants ( K o) for Al and dissolved organic matter (DOM) from acidic soil solutions is to collect ‘free’ Al data with diffusive gradients in thin films (DGT) and to infer the K o values by fitting a continuous distribution model based on Scatchard plots. Although there is clear established literature demonstrating the usefulness of the Scatchard approach, relatively little attention has been given to a realistic assessment of the uncertainties associated with the final fitted K o values. In this study we present an uncertainty analysis of the fitted K o values using a synthetic dataset with different levels of random noise and a real data set using DGT data from an acidic soil solution. The parameters in the continuous distribution model and their corresponding upper and lower 95% uncertainty bounds were determined using the Shuffled Complex Evolution Metropolis (SCEM) algorithm. Although reasonable fits of the distribution model to the experimental data were obtained in all cases, an appreciable uncertainty in the resulting K o values was found due to three main reasons. Firstly, obtaining ‘free’ Al data even with the DGT method is relatively difficult, leading to uncertainty in the data. Secondly, before Scatchard plots can be constructed, the maximum binding capacity (MBC) must be estimated. Any uncertainty in this MBC propagates into uncertainty associated with the final plots. Thirdly, as the final fitted K o values are largely based on extrapolation, a small uncertainty in the fit of the binding data results in an appreciable uncertainty in the obtained K o. Therefore, while trends in K o for Al and DOM could easily be discerned and compared, the uncertainty in the K o values hinders the application in quantitative speciation calculation. More comprehensive speciation models that avoid the use of K o seem to fit better for this purpose.

A Shuffled Complex Evolution Metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters

Markov Chain Monte Carlo (MCMC) methods have become increasingly popular for estimating the posterior probability distribution of parameters in hydrologic models. However, MCMC methods require the a priori definition of a proposal or sampling distribution, which determines the explorative capabilities and efficiency of the sampler and therefore the statistical properties of the Markov Chain and its rate of convergence. In this paper we present an MCMC sampler entitled the Shuffled Complex Evolution Metropolis algorithm (SCEM‐UA), which is well suited to infer the posterior distribution of hydrologic model parameters. The SCEM‐UA algorithm is a modified version of the original SCE‐UA global optimization algorithm developed by Duan et al. [1992]. The SCEM‐UA algorithm operates by merging the strengths of the Metropolis algorithm, controlled random search, competitive evolution, and complex shuffling in order to continuously update the proposal distribution and evolve the sampler to the posterior target distribution. Three case studies demonstrate that the adaptive capability of the SCEM‐UA algorithm significantly reduces the number of model simulations needed to infer the posterior distribution of the parameters when compared with the traditional Metropolis‐Hastings samplers.

Effective and efficient algorithm for multiobjective optimization of hydrologic models

Practical experience with the calibration of hydrologic models suggests that any single‐objective function, no matter how carefully chosen, is often inadequate to properly measure all of the characteristics of the observed data deemed to be important. One strategy to circumvent this problem is to define several optimization criteria (objective functions) that measure different (complementary) aspects of the system behavior and to use multicriteria optimization to identify the set of nondominated, efficient, or Pareto optimal solutions. In this paper, we present an efficient and effective Markov Chain Monte Carlo sampler, entitled the Multiobjective Shuffled Complex Evolution Metropolis (MOSCEM) algorithm, which is capable of solving the multiobjective optimization problem for hydrologic models. MOSCEM is an improvement over the Shuffled Complex Evolution Metropolis (SCEM‐UA) global optimization algorithm, using the concept of Pareto dominance (rather than direct single‐objective function evaluation) to evolve the initial population of points toward a set of solutions stemming from a stable distribution (Pareto set). The efficacy of the MOSCEM‐UA algorithm is compared with the original MOCOM‐UA algorithm for three hydrologic modeling case studies of increasing complexity.

Toward Improved Identifiability of Soil Hydraulic Parameters: On the Selection of a Suitable Parametric Model

We present a thorough identifiability analysis of the soil hydraulic parameters in the parametric models of Brooks and Corey (BC; Brooks and Corey, 1964), Mualem–van Genuchten (VG; van Genuchten, 1980), and Kosugi (KC; Kosugi, 1996, 1999) using the recently developed Shuffled Complex Evolution Metropolis (SCEM‐UA) algorithm (Vrugt et al., 2002b, and unpublished data). Because the SCEM‐UA algorithm globally thoroughly exploits the parameter space and therefore explicitly accounts for parameter interdependence and nonlinearity of the employed parametric models, the algorithm is suited to generate a useful description of parameter uncertainty and its antithesis, parameter identifiability. A set of measured water retention characteristics of the UNSODA database (Leij et al., 1996) spanning a wide range of soil textures and three transient laboratory outflow experiments with decreasing flow rates were used to illustrate that a parameter identifiability analysis facilitates the selection of an adequate parametric model structure and provides useful information about the limitations of a model. Moreover, results suggest that one should be especially careful in establishing pedotransfer functions without knowledge of the underlying posterior uncertainty associated with the soil hydraulic parameters using direct estimation methods.

Toward Improved Identifiability of Soil Hydraulic Parameters: On the Selection of a Suitable Parametric Model

We present a thorough identifiability analysis of the soil hydraulic parameters in the parametric models of Brooks and Corey (BC; Brooks and Corey, 1964), Mualem-van Genuchten (VG; van Genuchten, 1980), and Kosugi (KC; Kosugi, 1996, 1999) using the recently developed Shuffled Complex Evolution Metropolis (SCEM-UA) algorithm (Vrugt et al., 2002b, and unpublished data). Because the SCEM-UA algorithm globally thoroughly exploits the parameter space and therefore explicitly accounts for parameter interdependence and nonlinearity of the employed parametric models, the algorithm is suited to generate a useful description of parameter uncertainty and its antithesis, parameter identifiability. A set of measured water retention characteristics of the UNSODA database (Leij et al., 1996) spanning a wide range of soil textures and three transient laboratory outflow experiments with decreasing flow rates were used to illustrate that a parameter identifiability analysis facilitates the selection of an adequate parametric model structure and provides useful information about the limitations of a model. Moreover, results suggest that one should be especially careful in establishing pedotransfer functions without knowledge of the underlying posterior uncertainty associated with the soil hydraulic parameters using direct estimation methods.

Toward Improved Identifiability of Soil Hydraulic Parameters

We present a thorough identifiability analysis of the soil hydraulic parameters in the parametric models of Brooks and Corey (BC; Brooks and Corey, 1964), Mualem–van Genuchten (VG; van Genuchten, 1980), and Kosugi (KC; Kosugi, 1996, 1999) using the recently developed Shuffled Complex Evolution Metropolis (SCEM-UA) algorithm (Vrugt et al., 2002b, and unpublished data). Because the SCEM-UA algorithm globally thoroughly exploits the parameter space and therefore explicitly accounts for parameter interdependence and nonlinearity of the employed parametric models, the algorithm is suited to generate a useful description of parameter uncertainty and its antithesis, parameter identifiability. A set of measured water retention characteristics of the UNSODA database (Leij et al., 1996) spanning a wide range of soil textures and three transient laboratory outflow experiments with decreasing flow rates were used to illustrate that a parameter identifiability analysis facilitates the selection of an adequate parametric model structure and provides useful information about the limitations of a model. Moreover, results suggest that one should be especially careful in establishing pedotransfer functions without knowledge of the underlying posterior uncertainty associated with the soil hydraulic parameters using direct estimation methods.

Copper complexation by dissolved organic matter and uncertainty assessment of their stability constants.

The interaction of Cu with dissolved organic matter (DOM, extracted from an organic forest floor) was investigated and the resulting data was evaluated in terms of their uncertainty. The speciation of Cu over 'free' Cu (as analysed by diffusive gradients in thin films (DGT)), dissolved Cu-DOM complexes and precipitated Cu-DOM was determined as a function of pH (3.5, 4.0 and 4.5) and Cu/C ratio. The dissolved organically bound fraction was highest at pH 4.5, but this fraction decreased with increasing Cu/C ratio, which was observed for all pH levels. In the range of Cu/C = 7 x 10(-5) -2.3 x 10(-2) (mol/mol) the precipitated fraction was very small. The speciation of both Al and Fe was not affected by increasing Cu concentrations. From a continuous distribution model using the Scatchard approach, we calculated the optimal fit and corresponding upper and lower 95% uncertainty bounds of the overall stability constants (K(o)) with the shuffled complex evolution Metropolis (SCEM) algorithm. Although the optimal equation fitted the data very well, the uncertainty of the, according to literature, most reliable approach to establish stability constants, was still large. Accordingly, the usually reported intrinsic stability constants exhibited large uncertainty ranging from log K(i) = 6.0-7.1 (optimal 6.7) for pH 3.5, log K(i) = 6.5-7.1 (optimal 6.8) for pH 4.0, and log K(i) = 6.4-7.2 (optimal 6.8) for pH 4.5 and showed only little effect of pH.