Estimation of Parameter Uncertainty

EGIM14: Computational Case Study

“Estimation of Parameter Uncertainty”

Module Lecturer: Dr. Yunqing Xaun

Student Name: Mmeka Felix Uchenna

Student Number: 663327

EGIM14: Computational Case Study Mmeka Felix (663327)

Dr. Yunqing Xaun Page 1

Contents

1. Introduction…………………………………………………………………………...…..2

2. Literature Review…………………………………………………………………………5

3. Methods for accounting for Parameter Uncertainty………………………………...…….7

3.1) Mathematical Representation of Parameter Uncertainty

3.2) Generalized Likelihood Uncertainty Estimation (GLUE)

3.3) Differential Evolution Adaptive Metropolis (DREAM)

3.4) Major Differences Between the GLUE and DREAM Estimation Methods

4. THE CATCHMENT: Description……………………………………………………...18

4.1) Work done in 3rd

year thesis

5. Hydrological Predictions for the Environment (HYPE)……………………………......21

6. Applications of Parameter Estimation Methods to Case Study……………….......…....25

6.1) GLUE method

6.2) DREAM method

7. Conclusion…………………………………………………………………………..…..26

7. References…………………………………………………………………………..…..27

http://www.smhi.se/en/Research/Research-departments/Hydrology/hype-1.7994



1) Introduction

Hydrological modeling is the creation of simulations though mathematical

analogues/simplifications with the use of given data and computing techniques that best

represents complex physical, spatially distributed, and highly interrelated water, energy, and

vegetation processes in a watershed/catchment thereby predicting future data of the given

catchment. A hydrological model consists of 2 parts; the hydrological core and technological

shell. The core is a set of mathematical equations that produces an output with a given input

while the shell is the programming, user interface and pre and post processing facilities.

Rainfall runoff model can be divided into various stages of production and they are

perceptual model, conceptual model, procedural model, model calibration and validation.

Model Calibration is the setting of parameter values through the process of adjusting

parameters to fit a real catchment data while Model Validation/Verification is the checking of

the model flow against that of a predicted flow to model a similar hydrograph therefore

predicting future data [1]. Although model calibration attempts to replicate observed

hydrograph by making predicted simulations, due to the presence of significant

errors/uncertainty in the simulated hydrograph, this makes the model’s prediction not to be a

true representation of the observed. These predictions could be questioned in decision making

thereby hindering accurate judgment to be made from the model’s predictions; to what degree

the simulated hydrograph represents the observed one and this dilemma is a major problem.

The uncertainty estimation techniques have been developed to quantify these

uncertainties/errors in the model simulation, thereby aid better decisions made with respect to

the model. Uncertainty in hydrological modeling in recent years has been an increasing

consensus in hydrologic literatures; it can be defined as the presence of errors in hydrological

models that forces a derivation in the model’s predicted hydrograph from the observed

hydrograph. Uncertainty appears in models could be as a result of errors in measurement of

observed data, model structure errors (Over-parameterization), errors in initial and boundary

conditions of the model, output measurement errors, errors form potential evapotranspiration

or temperature and model parameters errors, conceptually limitations or errors in the model,

etc. These reasons for uncertainty can be further classified into various types of uncertainty

like structural, Input (forcing/state) and model parameter uncertainty (Model

parameterization) [2].



As previously stated, to aid judgments with model simulations, these uncertainties

have to be in some way quantified and the research of Uncertainty quantification has received

a very significant surge in recent decades. Over the years, various methods to quantify

uncertainty in models have been researched and they can be broadly split into parameter,

structural and Input/forcing data uncertainty estimations. Methods for parameter uncertainty

estimation is broadly divided into deterministic model calibration methods and stochastic

methods, while examples of deterministic model calibration methods (these methods are

normally used for models with many parameters calibration) are dynamic identifiability

analysis framework (DYNIA), multiobjective complex optimization method (MOCOM), etc.

and examples of stochastic methods bayesian model averaging (BMA), generalized

likelihood uncertainty estimation (GLUE), sequential uncertainty fitting (SUFI-2), first-order

second moment (FOSM), uncertainty estimation based on local errors and clustering

(UNEEC), differential evolution adaptive metropolis (DREAM), Shuffled Complex

Evolution Metropolis algorithm (SCEM), maximum likelihood bayesian averaging method

(MLBMA), bayesian recursive estimation technique (BaRE), all other markov chain monte

carlo (MCMC) sampling approaches, etc. Most of these models consider any errors between

the simulated and observed to be down to parameter uncertainty without any considering to

other sources of error. Structural and Input uncertainty are a bit difficult to estimate in

practice and therefore there are no established schemes for their estimation. Although this is

the case, Multi-model approach are used for Structural uncertainties because they are better

than single-model approach due to Equifinality. The concept of Equifinality is an assumption

of multiple possibilities for acceptable predictions for a particular simulation. While

examples of methods of Input (forcing) estimation developed over the years are state-space

filtering, kalman filtering, particle filtering, Variational data assimilation (VDA), dual state-

parameter estimation methods, etc [3]. Finally, some techniques have also been developed to

investigate the combination of Input and structural errors and Input and model parameters

errors and they include Bayesian model averaging (BMA), the integrated Bayesian

uncertainty estimator (IBUNE), the Bayesian total error analysis (BATEA) and closely

related developments, Framework for Understanding Structural Errors (FUSE), etc. and

Simultaneous Optimization and Data Assimilation (SODA), state augmentation, etc.

respectively. However, these schemes are not been commonly applied in practice besides

from each of the techniques’ limitations, is partly because of their computational

requirements [6].



Although the ideal case in uncertainty estimation and thereby aiding decision making

with hydrological modeling, would be the ability to quantify all the various uncertainties in a

particular model, however, one could argue that a full uncertainty analyses would require a

very comprehensive knowledge on the processes involved and presence of these uncertainties

show deficiencies in the current models (conceptually, assumptions made and limitations of

the model in general) and data (limitations in observations, measurements techniques and

performance criterions), also practically due to time constraint on this research and level of

experience in the field needed, this is not the goal for this Case study and Dissertation. This

case study for my Dissertation will take an in-depth look at Model parameter uncertainty

using the generalized likelihood uncertainty estimation (GLUE) uncertainty estimation

method and will use the recently developed differential evolution adaptive metropolis

(DREAM) uncertainty estimation method to serve as a contrast to the GLUE method due to

their different philosophies in uncertainty estimation analysis and this analysis will be applied

to the Upper Medway catchment. Model parameterization is the representation of

hydrological processes in Hydrological models with parameters; this is extremely hard due

lack of accurate knowledge about the processes (can be due to poor measurement techniques,

etc.) and these differences between Model simulations and Observed data bring about

errors/uncertainties. Model Parameter uncertainty was chosen was chosen in the uncertainty

analysis because it typically dominates in many catchment analysis due to model

parameterization and as stated above, an analysis for the entire uncertainty distribution will

be too complex. While stochastic models (GLUE and DREAM) were chosen over

deterministic model calibration methods (Manual and Automatic calibration) for this analysis

because they not only make predication by estimating errors but also state how uncertain their

predictions are, in general deterministic methods are not really referred to as an uncertainty

analysis but a form of calibration for model with many parameters. Furthermore, this study

discusses GLUE and DREAM method in the context of both methods having different

philosophies but producing similar predictions.

The paper structure is organized as follows. Section 2 is the Literature review which

describes some previous work done in uncertainty estimation. Section 3 will give a detailed

general description of the GLUE and DREAM uncertainty estimation methods. Section 4

introduces the Upper Medway catchment and previous case study in calibration techniques.

Section 5 gives a brief introduction on the Hydrological Predictions for the Environment



(HYPE) hydrological model. Finally, section 6 talks about the applications of GLUE and

DREAM model uncertainty estimation methods to the HYPE model.

2) Literature review

So much work has been done on the topic of uncertainty estimation and there are a

couple of them that are relevant to this research. K. Bevan’s 2000 paper titled “Equifinality,

data assimilation and uncertainty estimation in mechanistic modeling of complex

environmental systems using the GLUE methodology”, this paper was focused on the

investigation of effect of data length on calibration of model (Topmodel) parameters within

GLUE methodology. The paper started by the explanation of the concept of equifinality,

which is the acceptance of many predictions as acceptable prediction and its application to

GLUE method, which it refer to as the basis of the Method. Then the paper explained the

concept of a likelihood function and how it distinctly shows behavioral models with a

particularly threshold limit, how this pertains to GLUE and finally, GLUE method and the

problem was applied to the Maimai M8 catchment in New Zealand. The paper concluded by

saying that an increase in behavioral threshold would reduce the number of behavioral

models, prediction limits are dependent on model, initial conditions and measurement of

observations and likelihood measure and that the likelihood weight assigned to each model

will implicitly reflect on the complex interactions between parameter values [4].

Another paper related to this case study is the one by L. Xiong titled “An empirical

method to improve the prediction limits of the GLUE methodology in rainfall-runoff

modeling”. The paper focuses on the implementation of GLUE method to the King and

Qingjiang catchment using the SMAR conceptual model, more specifically; the paper

primarily focused on increasing the number of behavioral models without any change to the

threshold frequency by a modification to the GLUE method and this was then applied to the

both catchments. The paper started by introducing the problem, which is that the number of

observations that fall within the prediction limits are normally much less than the prediction

limit used, which was majorly as a result of error in the simulation series produced by the

model and the purposed solution was a modification of the GLUE method by the addition of

a simulating-correcting module to produce better predictions. Then the paper described the

step-wise implementation of the GLUE method and the Modified GLUE method for any




problem and stated that the both step-wise differences between the both methods was that in

the modified method, a simulation bias curve is plotted for every behavioral parameter set

and at every time interval of the simulated hydrograph, all prediction values of all intervals

for the same prediction are corrected by dividing them with their median bias value before

the derivation of the prediction limits. The SMAR model was then talked about and finally

the entire process was applied to the both catchments. At the end of the analysis, it was

concluded the Modified GLUE method substantially improve in the percentage of the

observed runoff data within the prediction limits than the GLUE method [5].

Another paper related to this case study is the one by J. A. Vrugt’s 2008 paper titled

“Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic

modeling?”. The paper focuses the application of the GLUE and DREAM method to the Leaf

river and French Broad watersheds using the HYMOD conceptual model, more specifically;

the paper primarily focused in comparing and establishing common grounds between the two

different approaches. The paper began by stating that uncertainty is the reason for the both

models and move on to a brief introduction on GLUE method. Then it then discussed the

concept of inverse problem (likelihood function) and then derives a likelihood function. Then

the paper goes on to define the likelihood mostly used that was developed by K. Bevan, the

implementation of GLUE method and then it moved MCMC with DREAM where it

discussed the implementation and predictive inference. Finally, the paper ran analysis on the

both catchments and concluded with the following: although the both methods have different

philosophies, they still produce roughly same estimates, DREAM method has better coverage

of observations but smaller spread of predictions than GLUE, GLUE method can reveal when

the observation given the available input data without compensation by statistical error model

or input changes, can’t be reproduced by any model, Parameter uncertainty is usually large in

GLUE implementations because of the inclusion of implicit representation of model error and

finally, GLUE method has an inability to separate various individual error sources [3].

Finally, another paper related to this case study is the one by J. A. Vrugt’s 2008 paper

titled “Treatment of input uncertainty in hydrologic modelling: Doing hydrology backward

with Markov chain Monte Carlo simulation”. This paper focuses on the application of the

DREAM method to the Leaf river and French Broad watersheds using the HYMOD

conceptual model, more specifically; the effect of rainfall multipliers on model performance.

The paper began by explaining why just calibration is not enough instead of DREAM method



due to the presence of errors, then describes the rainfall forcing data error, where an option of

rainfall multipliers can be used for more realistic simulations of the data, then the paper

talked about the MCMC methods and DREAM method theorem and finally the case studies

of the experiment. In the Case study, the paper tried to compare estimation of parameters and

hydrograph without the use of rainfall multipliers against estimation of parameters and

hydrograph with them and from the analysis, the second experiment (with multipliers) as

predicted had better values. The paper concluded by stated that explicit treatment of forcing

error during model calibrations significantly alters the posterior distribution of model

parameters, The DREAM algorithm provides an accurate estimate of the Posterior probability

density function (PDF) of model parameters, Rainfall multipliers help give better estimation

for hydrograph and Rainfall multipliers provide important diagnostic information to quantify

rainfall error [10].

3) Methods for accounting for Parameter Uncertainty.

As stated above, this paper will take an in-depth study of model uncertainty using

GLUE and DREAM parameter estimation methods applied to the Upper Medway catchment.

Also as stated previously, deterministic model calibration methods validity has been

questioned due to the presence of significant errors/uncertainty in the output hydrograph

(which could be improved by better understanding of the physical processes and high

resolution simulations) and because of this problem, the uncertainty estimation techniques

have been developed to help quantify errors in the model simulation. Below further explains

concept of uncertainty estimation;

3.1) Mathematical Representation of Parameter Uncertainty

Mathematically, a model is normally composed on 7 different components and they

are system boundary (B), inputs (u), initial states (xo), system properties (θ), structure (M),

states (x), and outputs (y). The input and output (u and y respectively) are considered as

fluxes of mass/energy in the system across the system boundary B while the system

properties (θ) is considered to be time-invariant. As shown in the Fig. 1 below;



Fig. 1: a diagram showing the mathematical components of a model

Using a set of model parameters (will be refer to as Model/parameter set from now on), the

model simulates a response Y (which is a function of all its components) while the observed

flow for the catchment is �� . The error will be denoted as;

∈𝐢= ��𝒊 − 𝐘𝒊 Where i = 1,……No of time steps

It can also be observed that 5 of the 7 components (system boundary (B), inputs (u), initial

states (xo), system properties (θ) and structure (M)) are needed to be specified, estimated or

defined before the model can run while the other two (states (x) and outputs (y)) are obtained

by running the model. The five assumed components may be uncertain and it would lead to

errors in the other 2 dependent components and that is why the 5 components are the major

sources of uncertainty in hydrological modelling. The point of an uncertainty measure is to

estimate accurately this error distribution, to ideally be able to make an error-free model for

analysis. The above point is also the basic concept of likelihood measure (will be later

discussed in the paper) [3].

As previously stated, this error is can be quantified by various methods but this

research aims to use the GLUE method and then compare with results obtained from

DREAM method in the Upper Medway catchment.

3.2) Generalized Likelihood Uncertainty Estimation (GLUE)

The GLUE method is an informal bayesian method that was developed Keith Bevan

and it is inspired by the Hornberger and Spear method of sensitivity analysis; it operates with

the Monto Carlo simulations with bayesian estimation and propagation of uncertainty. The

method has found a widespread because it’s conceptually simple and easy to implementation.

The fundamental assumption of the method is that it rejects the concept of an optimal method



because of the presence of uncertainties and adopts concept of Equifinality, which believes in

multiple models as an acceptable predictions due to imperfect knowledge about the model.

GLUE rejects typical statistical principle for likelihood functions (measure of how well a

model fit with the observed data) in favor of finding behavioral models. For GLUE to be

implemented, many different models have to be run and these models are assigned likelihood

function values. Based on the assign threshold value (this could be a certain value or a

percentage of the total number of models), the behavioral models are then selected from the

all the models ran. The behavioral models are then used to provide predictions of the system

behavior by obtaining the cumulative distribution (CDF) for each prediction. To access

uncertainties in predictions, weights of behavioral models are calculated by normalizing their

corresponding likelihood values so that all weights sum to 1. Finally for any comparisons, the

deterministic model prediction is practically assumes to be the median of output prediction

and upper and lower limits for the distribution could be found (usually 5% - 95 %). The main

improvement of this method over deterministic calibration methods is the ability to account

for uncertainties in model parameter (structural uncertainties could also be argued) by using

multiple models. With this all said, the 4 key areas are choice of a likelihood measure, choice

of hydrological model, method of generating parameter set (models) and the threshold value

definition [4].

Choice of Hydrological model: The choice is a model for the GLUE analysis is completely

subjective on the modeler in question due to the availability of various models that are

capable for the analysis. Some of the reasons for choice may be; number of model parameters

(due to model sampling), good user interface (due to numerous runs of the Hydrological

model), control of the model (convenient for modelers), etc. This decision is total up to

choice of modeler choice and application the model is needed for.

Generating parameter sets: The GLUE method separates the sampling of parameters to

ensure independence of model selection process been used to select models for the analysis.

There are various techniques that have been developed for this, such as Monte Carlo

methods, Latin-Hypercube simulations (LHS), Uniform Distribution method, Variance-based

methods, etc. Monte Carlo sampling methods aim to randomly sample models due to their

likelihood function values with hope to saving computer time by the sampling of the most

probable models first (highest likelihood function). The sampling method can be very

effective in a well-defined surface but not so great for surfaces with multi local peak values.



The Latin-Hypercube simulations (LHS) is very robust and may require a lot of sampling, it

could be described as a stratified sampling approach Monte Carlo method because it allows

adequate estimation of output statistics. Some of LHS advantages are robustness and

efficiency while a major drawback is its assumption of linearity when sampling parameters.

Uniform distribution method is a methodical way of sampling, where every possible

combination for all the parameters is modeled to make various models. The major advantage

is that the method leaves no room for subjective assumptions but the major drawbacks are

that the method is time consuming and not feasible at times. The Variance-based methods

aim to decompose the variance, when all inputs are varying into partial variances. With all

the above choices, one method will be chosen for the analysis [8].

Choice of likelihood function: Likelihood functions as previously discussed tries to estimate

errors and furthermore assign a value of likelihood (trust) to each model which is obtained by

comparing the simulated against observed data. There are many likelihood functions that

have been developed such as Mean absolute error (MAE), Root mean square error (RMSE),

Correlation (R2), Nash-Sutcliffe efficiency criterion, Auto correlated Gaussian error model,

Inverse error variance, Exponential transformation of error variance, etc. All these likelihood

functions have their various ways to relate the likelihood of a simulated hydrograph against

their observed counterparts [9].

Threshold value definition: This value is chosen with respect to the likelihood function

chosen for the analysis and it can be very subjective to the modeler except if otherwise stated

from an earlier requirement or something.

Detailed Procedure for GLUE method analysis

This is a step-wise method of the GLUE method;

Step 1: Select a hydrological model for the analysis and determine the range of every

parameter of the model.

Step 2: Select a likelihood measure, likelihood threshold value and total sampling

time or number of behavioral models needed.

Step 3: Repeatedly run the hydrological model for the sampling time or till the

number of behavioral models are achieved. Behavioral models are indexed

by r (1 ≤ 𝑟 ≤ 𝑀)and each model is associated with its corresponding

likelihood function.



Step 4: This is the core of the GLUE method

a) run the hydrological model M times with the behavioral models to obtain

M different predictions [Qs(t,r)], then each prediction is assigned a weight

which is normally the likelihood value of the model.

b) The weighted predictions are then ranked in ascending order denoted by

{Q[t, r(i)], l[r(i)]}and then a cumulative distributed distribution (CDF) is

made with the formula below;

Pr{𝑄 ≤ Q[t, r(i)]} =∑ l[r(j)]𝑖

𝑗=1

∑ l[r(j)]𝑀𝑗=1

c) For each certainty level, find the two quartiles and Median value.

The GLUE method normally has a low efficiency in its prediction limits capturing more

observed discharge was applied to a catchment due to 2 groups of errors. The first are those

erors that are independent of the method like uncertainties in the model structure, input data,

etc. while the second group are based on GLUE induced subjective decisions such as the

choice of threshold value, assumption of the unknown probability distribution of model

parameters, etc. The improved/proposed method is aims to treat the errors in the first group

because these errors are passed down into the prediction limits therefore affecting the

percentage of observed flow. The above stated step-wise can be improved as stated in the L.

Xiong paper [5] by the addition simulation-correcting module to reduce the errors in the

simulation output of models. The first 3 steps of the modified GLUE method are same to the

GLUE method while the steps differ slightly from then on and the steps are stated below;

Detailed Procedure for the Modified GLUE method analysis.

Step 1: Select a hydrological model for the analysis and determine the range of every

parameter of the model.

Step 2: Select a likelihood measure, likelihood threshold value and total sampling

time or number of behavioral models needed.

Step 3: Repeatedly run the hydrological model for the sampling time or till the

number of behavioral models are achieved. Behavioral models are indexed

by r (1 ≤ 𝑟 ≤ 𝑀)and each model is associated with its corresponding

likelihood function.

Step 4: The aim is to plot a simulation bias curve that compares the simulated against

observed data



a) For the rth

(each) behavioral model, divide the predicted runoff into equal

intervals and denoted the number of intervals as ψ .

b) For the Kth interval [between Vk and Vk+1] (1 ≤ 𝑘 ≤ 𝛙), identify all the

simulated runoff values that fall within the interval and calculate the sum

(SSk), then do the same for the corresponding observed values (SOk: observed)

c) For the interval, a simulation bias curve can be achieved by plotting br(k)

against their corresponding interval for the each model

𝑏𝑟 (𝑘) =𝑆𝑆𝑘

𝑆𝑂𝑘

Step 5: a) The hydrological model is run M times with the M behavioral parameter

sets with constant inputs and observations, thereby producing M predictions

[Qs(t,r)], each prediction is then assigned a weight which is normally its

corresponding likelihood function.

b) For each prediction Qs(t,r) (1 ≤ 𝑟 ≤ 𝑀) , estimate the prediction’s Br(t)

value by looking at the rth

parameter set’s constructed simulation bias curve

c) Rank all value of Br(t) (1 ≤ 𝑟 ≤ 𝑀) in ascending order and find the

median value Bmedian(t).

d) Correct all the weighted prediction values with the formula below;

Qs(t, r) ←Qs(t, r)

𝐵𝑚𝑒𝑑𝑖𝑎𝑛(𝑡)

e) Rank the M corrected predictions Qs(t,r) in ascending order, then construct

a cumulative probability distribution (CDF) for the corrected runoff

prediction

f) For a given certainty level, find the two quartiles and Median value

As previously stated, the major difference between the GLUE model and the Modified GLUE

model addition simulation-correcting module which could be seen in the step-wise

application of the both methods; it could be observed that in the modified method, a

simulation bias curve is plotted for every behavioral parameter set and at every time interval

of the simulated hydrograph, then all prediction values of all intervals for the same prediction

are corrected by dividing them with their median bias value before the derivation of the

prediction limits.



Example of the GLUE method used in previous studies: GLUE method has been

implemented in many papers/case studies but particularly in the L. Xiong paper, it showed

comparisons between the GLUE method and the Modified GLUE method with their

application to the King and Qingjiang catchment using the SMAR conceptual model. The

paper concluded that the Modified GLUE method increased the number of behavioral models

without any change to the threshold value over the GLUE method.

This paper firstly shows the implementation of GLUE method in general then

secondly showed the superiority of the Modified version to the normal version [5]. The data

below show the implementation of the above points;

In the King catchment:

Table 1: Showing the both versions in the King catchment.

Where Containing ratio (CR) means the proportion of the observed data contained within

the prediction limits.

Fig. 2: Showing the CR values of the both versions in the King catchment.

From the Table 1 and Fig. 2 above, it can be observed that they both agree with the paper’s

conclusion for the King catchment.



In the Qingjing catchment:

Table 2: Showing the both versions in the Qingjing catchment.

Fig. 3: Showing the CR values of the both versions in the Qingjing catchment.

From the Table 2 and Fig. 3 above, it can be observed that they both agree with the paper’s

conclusion for the Qingjing catchment.

3.3) Differential Evolution Adaptive Metropolis (DREAM)

The DREAM method is a formal Bayesian method that is an adaptive from the

MCMC sampling method. The formal Bayesian method is a purely statistical method that

offers a simple way to combine several probability distributions using Bayes theorem, in

hydrological terms, the method has the ability to systematically estimate uncertainty within a

single cohesive and consolidated method. To implement the method, sampling methods are

needed to produce the posterior density function (PDF) of parameters which combines the

model likelihood with the assumed parameter distribution. Unfortunately, the PDF in most

cannot be analytical achieved so therefore Monte Carlo (MCMC) sampling is used to

generate a sample of it. The heart of an MCMC method is the Markov chain that generates a



random stepping throughout the parameter with their limits with a stable frequency. After

experiment and analysis, it is observed that the convergence to a posterior distribution is

often very slow which normally results from the inappropriate initial choice parameter

distribution used to generate trial moves in the Markov chain. To improve this phenomenon

and make the model more efficient, the proposed distribution’s orientation and scale is tuned

during the iterations to make the sample PDF to its target form by the help of information

from the previous step and this self-adaptive randomized subspace sampling is what makes

DREAM an adaptive MCMC algorithm. In practice, the MCMC evolution is run/repeated

until the procedure shows convergence to a stationary posterior distribution (target PDF). The

DREAM method has received a surge of popularity over other MCMC methods (e.g. STEM,

RWM) before of the adaptive form that gives it efficiency in solving highly non-linear and

multimodal target distributions, advantage of maintaining detailed balance and ergodicity.

With all this said, DREAM and MCMC methods in general assume that error are

addictive and fundamentally random and this may not be the case in reality but will instead

be a lack of knowledge and understanding of the processes [10].

Detailed Procedure for DREAM MCMC method analysis

Step 1: Draw an initial population X of size N, typically N = d or 2d (where

represents the number of parameters to be estimated) using the specified

distribution.

Step 2: Compute the density p(Xi|Y, Φ)of each point of X, where i=1,….,N using the

antilog of the likelihood function.

FOR i = 1,….,N DO (CHAIN EVOLUTION)

Step 3: Generate a candidate point, zi in chain i,

𝒛𝒊 = 𝒙𝒊 + 𝜸(𝜹). (∑ 𝒙𝒓(𝒋)

𝜹

𝒋=𝟏

− ∑ 𝒙𝒓(𝒏)

𝜹

𝒏=𝟏

)

Where δ represents the number of pairs used to achieve the candidate point

r(j) ≠ r(n) ≠ i.

γ value depends on the number of pairs to create proposal, a good choice is

𝛾 = 2.38/√2𝛿𝑑𝑒𝑓𝑓 , with deff = d, but potentially decreased in next step.



Step 4: Replace each element, j = 1,….,d of the proposal Zij with x

ij using the

binominal scheme with crossover probability CR,

𝑧𝑖𝑗 = {

𝑥𝑖𝑗 𝑖𝑓 𝑈 ≤ 1 − 𝐶𝑅, 𝑑𝑒𝑓𝑓 = 𝑑𝑒𝑓𝑓 − 1

𝑧𝑖𝑗 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

j = 1,…,d

Where U є [0,1] is a draw from the uniform distribution

Step 5: Compute p(Xi|Y, Φ) and accept the candidate point with Metropolis

acceptance probability, ∝ (𝑥𝑖 , 𝑧𝑖)

∝ (𝑥𝑖 , 𝑧𝑖) = {𝑚𝑖𝑛 (

p(zi|Y, Φ)

p(xi|Y, Φ), 1) 𝑖𝑓 p(Xi|Y, Φ) > 0

1 𝑖𝑓 p(Xi|Y, Φ) = 0

Step 6: If the candidate point is accepted, move the chain, xi = z

i; otherwise remain at

the old location, xi.

END FOR (CHAIN EVOLUTION)

Step 7: Remove potential outlier chains using the inter-quartile range.

Step 8: Compute the Gelman-Rubin (Rstat) convergence diagnostic.

Step 9: If Rstat ≤ 1.2, stop, otherwise go to CHAIN EVOLUTION.

As discussed in the J. A. Vrugt’s 2008 paper, The DREAM method could be

improved by the use of a single rainfall multiplier for each storm event to describe the

Rainfall forcing (input) data. By the use of these multipliers to vary in hydrologic period

ranges, system based errors in the rainfall input data can be corrected and parameter inference

and stream flow predictions can also be improved [10].

Example of the DREAM method used in previous studies: The DREAM method has been

applied in many papers but the J. A. Vrugt’s 2008 paper, it compared the DREAM method

with and without the use of rainfall multipliers in the Leaf river and French Broad watersheds

and the paper concluded by stating that the analysis with the multipliers produced better

results [10].

This paper firstly shows the implementation of DREAM method in general then

secondly showed the superiority of the Method with multipliers (Modified) version to the

normal version [10]. The table below shows the implementation of the above points;



Table 2: Showing the both versions in the both catchments.

From the table above, it can be observed that the results support the conclusion made by the

paper because the Case study 2 (DREAM with rainfall multipliers) have better values in the

both catchment than Case study 1 (DREAM without rainfall multipliers).

3.4) Major Differences Between the GLUE and DREAM Estimation Methods.

Although the both methods could be used to quantify parameter errors, as previously

stated, they both have their differences. With the obvious being that the GLUE method takes

an informal Bayesian approach while the DREAM takes a formal Bayesian approach, there

are still some more in-depth differences. The table below states some in-depth differences

from research from other papers (Literature review) and Note that this list will be improved

upon after more research with the both models in the Dissertation;

Table 1: Showing the differences between the DREAM and GLUE method.

Properties GLUE method DREAM method

Concept GLUE results are based on weighted

simulations from the set of behavioral

models.

DREAM simulations uses a first-order

auto correlated error model with explicit

information form the stream flow

observations.

Separation of errors

sources

GLUE can’t individually separate errors

sources

DREAM can individually separate

errors sources

Errors This is not the case for GLUE Assumes erors are addictive and

fundamentally random.

Ease of

Implementation

Easier of Implementation. Harder of Implementation.

Efficiency Not that good for finding behavioral

models.

Much better at finding behavioral

models due to adaptive updating

component.

User casualties The method could be very subjective with

the choice of the threshold value.

This is not the case for DREAM.



4) THE CATCHMENT: Description

The study catchment is the Upper Medway catchment which is located on the south of

London, 50km from the Thurnham radar which has an area of around 220km2 .The Medway

river flows for about 112 km in the county of Kent, starting flow from East Grinstead in

Sussex and flowing into the Thames estuary which is a main tributary to the lower basin. The

catchment contains complex topography and a combination of permeable (chalk) and

impermeable (clay) geologies. The average annual rainfall and annual potential

evapotranspiration is about 729mm and 633mm respectively. Predicting and managing fluvial

flooding has been historically referred to as mostly difficult in the Medway catchment and

this is due to the convergence of sub catchments with very different responses to rainfall and

tidal influences from the wide Thames Estuary. The sluice gates at Leigh control the high

flows of the River Medway which is operational during high rainfalls on the Upper Medway

catchment. These gates control the flow before the Chafford water level gauge but without

consistent functional regulations, it is hard to calculate its influence on the Chafford

discharge. The presence of many natural springs in the catchment, soft and muddy soils,

increases the speed of the rainfall-runoff procedure as the soil is almost always fully saturated

during autumn and winter and this brings an assumption that the ground water flow is

relatively high and these ground water levels are very close to the surface during both

periods. There is a water storage area that was confirmed by Google maps and window live

map, near the upstream of the River Medway. The structure is located close to Forest Row

with a 1.5 mile long stretch of water and a full capacity of 1,237 gallons of water and area of

2.8km2

[9].

It was found by Dr D.Zhr [9] during his conducted visit to the catchment that it had

accessible hydrological gauges including 9 tipping buckets rain gauges, 4 water level gauges

and 3 stream-flow gauges. Rainfall in the catchment is measured by the tipping bucket rain

gauge (TBR), telemeters and storage gauges while rainfall data was obtained automatically

from the TBR but it is calibrated using data from rainfall storage gauges which is next to it,

there is normally about 8% discrepancy in the calibration. Manual measurements of water

level are taken once a month and are compared to that of the logger and these values are

plotted on a graph. The diagram below gives a visual explanation to the Catchment.



Fig. 2: Showing the Upper Medway catchment.

4.1) Work done in 3rd

year thesis

Last year during my final project with the same catchment and data, I did my research

on Hydrological forecasting and Model Optimization where my research was primarily

focused on the impact of data length in model calibration and effect of selective calibration

on model performance. The Case study used the Probability Distributed Model (PDM) to

analyze the data from Upper Medway catchment. The rainfall data is gotten from the tipping

bucket rain gauge and calibrated with the storage gauges while the water level or outflow of

catchment is gotten from the water level gauges and stream-flow gauges. It can be noticed

that limited amount of recording devices were present during the readings, therefore data was

lumped data [5c]. The model data for the catchment was aimed at Collier’s land bridge and

the readings were done for a period from 1 July 2006 to 8 December 2007 (about 1.5 years).

The input data was initially ran in PDM to get the simulated flow data and this is compared to

the real data and the comparison criterions are then used to analyze how the simulated data is

similar to the real data. When this was done for the entire data set, The R2 value was 0.3966

(40% correlation) and RSME was 3.0025 (7.7% error), as it can be noticed that according to

the criterions, the simulated data is a poor comparison to the real data.

This previous case study is related to the present GLUE method analysis because

although, last year’s case study did not estimate errors, the both case studies are still model

optimization techniques. Model calibration (Manual and Automatic calibration) as previously

defined in this paper, is the adjusting of model parameters to fit a real catchment while Model

uncertainty estimation measures (GLUE and DREAM) try fit the real data through the

removal of errors. This shows that the both techniques although try to achieve the same goal,



model uncertainty estimation are better because they consider errors while calibration

techniques don’t [Please refer to Section 2]. The uncertainty estimation techniques have been

developed to quantify these uncertainties/errors in the model simulation, thereby aid better

decisions made with respect to just model calibration methods in a model.

Firstly in the research, the 2 year long data was divided into 2 non-equal parts for

calibration and verification (Experiment 1) and then the model was then selectively calibrated

(Experiment 2) where the data length included for calibration and verification were just

High/peak flow (since flooding was the point of the case study) and in this calibration data

range, the data was strategically divided to change length at every calibration. The

experiment was achieved as shown in the Fig 3 below;

Figure 3: the calibration and verification data for Experiment 2.

The paper concluded by saying that Experiment 2 showed better results than

Experiment 1 which meant that selective calibration worked while The 3 calibrations in

Experiment 2 showed a contrary results to the expected result which was the longer the data,

the better the model. Experiment 2 was concluded that the reason for this subjective of

automatic calibration, longer data is needed and presence of errors in data and results errors.

At this point of the analysis, Uncertainty estimation techniques like GLUE method should

have been implemented to check the basis of the conclusion of this case study [11].

0

10

20

30

40

50

1 10001 20001 30001 40001 50001

Flo

w (

m^3

/s)

Time step

CALIBRATION 3

VERIFICATION CALIBRATION 2

CALIBRATION 3



5) Hydrological Predictions for the Environment (HYPE)

The proposed model for this analysis is the Hydrological Predictions for the

Environment (HYPE) model. It is a dynamic, semi-distributed, process-based and integrated

catchment model recently developed by the Swedish Meteorological and Hydrological

Institute (SMHI). It can be both used for small and large (E-HYPE, India-HYPE, etc.) scale

assessments in hydrological, nutrient and water transport applications. The model code is

written in FORTRAN computer language which runs on either Windows or Linux and is

open sourced under the GNU Lesser General Public license. The code is divided into the

hydrological simulation system (HYSS) and the actual hydrological model (HYPE). HYSS is

the computational framework component that runs the input and output to the model and also

its time stepping procedures while The HYPE part is the main hydrological model that

contains defined parameters, output variables and procedures of simulating various

hydrological sections [12].

HYPE model has a very simplified file structure which is a text file format for all

input and outputs which contain 203 various variables. The input are grouped into time series

input data (which is forcing data and observation series) and geographical input data while

the Output is grouped into time series of water and nutrients, and performance criteria (can be

likelihood function). The following are the five mandatory input files for the a successful run

of the model; Pobs.txt, Tobs.txt, par.txt, GeoData.txt and GeoClass.txt. The Pobs.txt

and Tobs.txt files contain forcing time series of perception and temperature, the GeoData.txt

file contain the geographical input data for each subbasin while GeoClass.txt files contain

their classification information (calculation units) and the par.txt file contain the model

parameters that determine the function of the model (which can be land use type, soil type or

a general value). The output file with not special request is the log-file with information about

running time and files used but the model can also provide the performance criteria (such as

Nash-Sutcliffe Efficiency, Correlation Coefficient, Mean Absolute Error, etc.) for every

subbasin in the subass1.txt file [13].

The basic concept of HYPE model which describes the flow and transformation of

water, nutrients and organic carbon in the soil, lake and rivers can be broken down into the

following; Processes above the ground, Land routines, Rivers and lake, Nitrogen and

phosphorus in land routines, Nitrogen and phosphorus in rivers and lakes, water management,

deep processes and organic carbon. All these hydrological processes are the basic concepts of



http://www.smhi.net/hype/wiki/doku.php?id=start:hype_file_reference:pobs.txt

http://www.smhi.net/hype/wiki/doku.php?id=start:hype_file_reference:tobs.txt

http://www.smhi.net/hype/wiki/doku.php?id=start:hype_file_reference:par.txt

http://www.smhi.net/hype/wiki/doku.php?id=start:hype_file_reference:geodata.txt

http://www.smhi.net/hype/wiki/doku.php?id=start:hype_file_reference:geoclass.txt

http://www.smhi.net/hype/wiki/doku.php?id=start:hype_file_reference:pobs.txt

http://www.smhi.net/hype/wiki/doku.php?id=start:hype_file_reference:tobs.txt

http://www.smhi.net/hype/wiki/doku.php?id=start:hype_file_reference:geodata.txt

http://www.smhi.net/hype/wiki/doku.php?id=start:hype_file_reference:geoclass.txt

http://www.smhi.net/hype/wiki/doku.php?id=start:hype_file_reference:par.txt



the HYPE model which are explained in detail in the reference [13]. These processes are

shown in great detail in the Fig. 4 and Fig. 5 below;

Fig. 4: Showing the various components of the HYPE model

Fig. 5: Showing the entire concept of the HYPE model

Although the model has a wide range of applications, it needed for this case study for its

rainfall-runoff modelling application and to tackle runoff modelling, the 4 basic concepts of

the HYPE model used are Processes above the ground, Land routines, Rivers and lake and

Deep processes. The Processes above the ground concept is mostly divided into temperature,



precipitation and evaporation; the temperature and precipitation is adjusted through various

methods in the model (please refer to reference [13]) but the model assumes mixed (rain and

snow) precipitation occurs when the air temperature is around than threshold temperature and

purely liquid (rain) and solid (snow) when the air temperature is greater and lesser

respectively to the threshold temperature. While in evaporation, the potential evaporation

only occurs when the air temperature is greater than threshold value and evaporation form the

soil is assumed to occur only in the 2 upper layers (if more than 1) and it decreases with

depth. The land routines could be divided into soil water, snow routines and glaciers but with

respect to the case study, we focus mainly on Soil water (Rainfall). The model contains up to

3 soil layers with each layer having their various properties (such as water retention,

thickness, porosity, etc.). Groundwater runoff in the model only occurs when the soil water

surpasses the field capacity of the soil layers and this runoff occurs in all number of layers

while if infiltration is greater than zero, it is added to the upper layer but surface runoff

occurs when the high ground water table reaches the surface. The figure below shows the soil

classifications in the model;

Fig. 6: Showing the soil classifications in the HYPE model.

In the HYPE model, there are two types of rivers and they are local stream and main river

while lakes are also divided into local and outlet lakes and all these classification all have

threshold/percentage values that above which flow occurs. The model assumes that all local

runoff enters the local stream then some of the local runoff flows into the local flow while the

rest goes directly to the main river. The main river receives all the local runoff from the

combination of the local lakes and local stream and the inflow from upstream subbasins and

the main river empties in the outlet lakes which finally flow into downstream subbasins.

Either the river or lake could be represented to be 1-dimensional (no precipitation added) or

soil land-use/area. The diagram below shows the described process;



Fig. 7: Showing the Lake and River classifications in the HYPE model.

Finally, Deep processes below ground are divided into regional groundwater flow and

aquifers. The regional groundwater flow model simulates groundwater flow between

subbasins but no explicit storage of deep groundwater while the aquifer model simulates

storages and delay before regional groundwater reaches its destination [13].

As the modeller, the HYPE model could be applied to a lot of hydrological

applications river levels application, transport of pollutants, water quantity, etc. but in this

case study, it is needed to verify the Catchment river flow data. The HYPE model was chosen

for this analysis mainly because of the Open source GNU Lesser General Public license

which gives the modeller, the ability to modify the program to suit the particular problem.

Although this would be the case, a possible drawback will be the over 200 model parameters

(although most might not be used in a particular application) which might make sampling of

the model lengthy and the time for each run might be long thereby increase computational

cost.



6) Applications of Parameter Estimation Methods to Case Study

6.1) GLUE method

The Modified GLUE method will be used for this analysis as preferred to the

normally GLUE method for reasons previously stated in this paper and The application of the

method to the Upper Medway catchment using the HYPE model could be easily summarized

in the Detailed Procedure for GLUE method analysis in Section (3.2) previously discussed.

The Choices for the GLUE method application is as follows;

Choice of Hydrological model: As previously stated, the Hydrological Predictions for the

Environment (HYPE) model will be used for this analysis mainly because of Open source

GNU Lesser General Public license.

Generating parameter sets: The Monte Carlo methods and Latin-Hypercube simulations

(LHS) will be both applied in the GLUE method application because of their efficiency and

robustness, to compare which sampling would produce the best model.

Choice of likelihood function: The Nash-Sutcliffe efficiency likelihood coefficient will be

used for this analysis because it quantitatively describes the accuracy of model simulations to

the observed hydrograph. The coefficient varies from −∞ to 1 where an efficiency of 1

corresponds to a perfect match of modelled discharge to the observed flow, then an efficiency

of 0 indicates that the model simulations are as accurate as the mean of the observed data,

whereas an efficiency of less than zero occurs when the observed mean is better than the

model simulations. Essentially, the closer the model efficiency is to 1, the more accurate the

model.

Threshold value definition: During previous research, the maximum achieved likelihood

function was 0.82, so therefore, the analysis will have a likelihood value of 0.7 (this value is

totally subjective).

6.2) DREAM method

The application of the DREAM method with rainfall multipliers to the Upper

Medway catchment using the HYPE model could be easily summarized in the Detailed

Procedure for DREAM method analysis in Section (3.3) previously discussed.




7) Conclusion

At this point, this case study has introduced the concept of model parameter

uncertainty estimation, went though some literature that was very related to the topic, in great

detail explained the GLUE and DREAM uncertainty estimation methods (assumptions/basic

concepts, stet-wise method, etc.), introduced the catchment and discussed my previous case

study in model calibration in same catchment, gave a concise but detailed knowledge of the

Hydrological Predictions for the Environment (HYPE) model and finally, explained the

application of the both methods to the HYPE model and catchment in general.

With all this said, the research will continue in my Dissertation by the actual

application of the GLUE and DREAM methods to the Upper Medway catchment using the

HYPE model and then results obtained will be analyzed and then compared to results from

other case studies and a suitable conclusion of all the analysis will be done with respect to the

expected results.




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Estimation of Parameter Uncertainty

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