improving distributed hydrologocal model simulation accuracy using polynomial chaos expansion

M2 - Putika Ashfar Khoiri

Water Engineering LaboratoryDepartment of Civil Engineering

24th Cross-Boundary SeminarInternational Program of Maritime and Urban EngineeringOsaka University

Improving Distributed Hydrological Model Simulation Accuracy using Polynomial Chaos Expansion (PCE)

*tentative title

December 21st, 2017

1

Background of study

(Data from Japan Meteorological Agency)

There is a change in precipitation

pattern due to climate change

It is necessary to analyse rainfall-

runoff relationship to predict the

risk of flood and drought

increases due to climate change

Perform hydrological model

Input

Hydrological

Model Watershed

characteristics

Output river discharge

grid input set

2

Background of study

based on parameter complexity conceptHydrological

Model

Lumped Model

same parameter (𝜭)in the sub-basin

Semi-distributed Model

parameters assigned in each grid cell but cells with the same parameters are grouped

𝜭

𝜭1

𝜭2

𝜭3

Fully-distributed Model

parameters assigned in each grid cell

𝜭1

𝜭2𝜭3

𝜭4

3

Background of study

based on parameter complexity conceptHydrological

Model

Fully-distributed Model

parameters assigned in each grid cell

Advantages

1. Can consider the spatial distribution of input

2. Can predict output discharge at any point

Disadvantage

1. Require many parameters so the setting and

determination of parameter is difficult

We need to assess the effectiveness of distributed

parameter including the characteristics of every

parameter

Parameter optimization is required to decrease the uncertainty

Approach:

𝜭1

𝜭2

𝜭3

𝜭4

Approach

In order to optimize the poorly known parameters and improve the model forecast

ability, data assimilation is required

Input and/or parameters

Uncertain characterisation𝑥 = [𝑥1,𝑥2, …… . . 𝑥𝑛]

System simulation

• Process• Equations• Code

𝑦 = 𝑓(𝑥)

Outputs𝑦 = [𝑦1,𝑦2, …… . . 𝑦𝑛]

- Parameter optimization- Sensitivity analysis

- Distribution statistics- Performance measures

(variance, RMSE)

Characterization of uncertainty in hydrologic models is often critical for many water

resources applications (drought/ flood management, water supply utilities, reservoir

operation, sustainable water management, etc.).

4

Previous study

Previous study about parameter estimation of hydrological model in Japan

Time-dependent effect

(Tachikawa, 2014) Investigation of rainfall-runoff model in dependent flood scale

When large scale flood occurs, only roughness coefficient become a dominant

parameter because soil layer is saturated and other parameter may change over time

Spatial distributions of parameters effect

(Miyamoto, 2015) Estimation of optimum parameters sets of distributed runoff

model for multiple flood events

- Nash coefficient show the

efficiency measure of the model,

which show not good result for

medium and small floods.

5

Common method use for data assimilation :

- Variation Method : 3DVar, 4DVar- EnKF

Previous study

Improving Hydrological Model response based on the soil types, land-uses and slope classes

Example : Soil and Water Assessment Tool (SWAT) model

The turning of parameters is difficult because so many parameter need to be considered -> sensitivity analysis is needed

𝑆𝑊𝑡 = 𝑆𝑊0 +

𝑖=1

𝑡

(𝑅𝑑𝑎𝑦 + 𝑄𝑠𝑢𝑟𝑓 + 𝐸𝑎 − 𝑤𝑠𝑒𝑒𝑝 − 𝑄𝑔𝑤)

Uncertainty analysis methods

e.g. GLUE (Generalized Likelihood Method Uncertainty Estimation) , based on Monte-Carlo simulationProblem -> Need large number of parameter sets sample

SWt = final soil water content

SW0 = initial soil water content on day i

Rday = amount of precipitation on day i

Qsurf = amount of surface runoff on day i

Ea = amount of evatranspiration on day i

Wseep = amount of water entering the vadose zone from the soil

Qgw = amount of water return flow on day i

6

Previous study

Improving Distributed Hydrological Model for the flood forecasting accuracy

Spatial distribution view Necessary to reduce grid spatial resolution (not objective)

Land-use correlated parameter

Most-considered parameter:C

Soil related parameters

-evaporation coefficient-roughness coefficient

-tank storage constant-hydraulic conductivity of layer-soil thickness-slope gradient-permeability coefficient

Therefore, I want to focus on land-use correlated parameter and soil related parameter in my study

7

Objective and Method

Polynomial Chaos Expansion (PCE)

𝑓 𝑥, 𝑡, 𝜃 =

𝑘=0

𝑘𝑚𝑎𝑥

𝑎𝑘(𝑥, 𝑡) 𝜙𝑘(𝜃)

𝑏𝑎𝑠𝑖𝑐 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙

Any model output Expansion coefficient

Polynomial of order-k in parameter space determined by 𝜃

𝑎𝑘 𝑥, 𝑡 =1

𝑁𝑘 𝑓 𝑥, 𝑡, 𝜃 𝜙𝑘 𝜃 𝑝 𝜃 𝑑(𝜃)

Density function of parameter 𝜃approximate by GaussianQuadrature

(Mattern, 2012)

PCE has used to optimize parameter on biological ocean model

8

Objective and Method


Increasing Distributed Hydrological Model Simulation Accuracy

by using Polynomial Chaos Expansion (PCE) method

Objective

The method of simulating with the value of the quadrature points in the parameter space, and estimating the optimal parameters using the difference between the observations and models (Mattern, 2012)

𝑅𝑀𝑆𝐸が最小

parameter space

Calculation at all quadrature points and interpolate

𝑅𝑀𝑆𝐸

0 1

𝑅𝑀𝑆𝐸

2 4 6 8 10

x 10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

parameter: KDOMf

para

mete

r: r

atio

_n

0.5

1

1.5

2

2.5

3

𝑅𝑀𝑆𝐸

Relative parameter ranges 𝜭1

Rel

ativ

e p

aram

eter

ran

ges 𝜭

2

Global minimum of RMSE

(optimal parameter value)

Contour plot of distance function

(Hirose, 2015) 9

Method (PCE)

Emulator techniques


Advantages:

- More effective than Monte-Carlo

because of their random sampling

- PCE performs a polynomial interpolation

in parameter space so it can estimate

any model output for the parameter

value of choice

Challenges:

- We must know what kind of input

parameters that make large uncertainty

- Upper and lower limits of parameter is

difficult to set

- Only two parameters can be optimized

We need to carefully consider:

1. Uncertain model input (parameters)

2. The prior distributions assigned to these

input

3. kmax (max. order of polynomial) -> we

have to check which value of kmax is

applicable for DHM

10

Previous studyImproving parameter estimation in Hydrological by applying PCE

HYMOD model (Fan, 2014) is a uniformly distributed model

Parameter Description Value

Cmax Maximum soil moisture capacity within the catchment 150-500

bexp Spatial variability of soil moisture capacity 15-5

α Distribution factor of water flowing to the quick flow reservoir 0.46

Rs Fraction of water flowing into the river from the slow flow reservoir 0.11

RqFraction of water flowing into the river from the quick flow reservoir

0.82

Soil moisture capacity function

c= soil moisture capacity

𝑭 𝒄 = 1 − 1 −𝒄

𝑪𝒎𝒂𝒙

𝒃𝒆𝒙𝒑

0 ≤ 𝒄 ≤ 𝑪𝒎𝒂𝒙

The PCE are applied for those two parameter, because its uniformly distributed. While another parameter is assumed to be deterministic

The results indicated both 2- and 3- order PCE's could well reflect the uncertainty of streamflow result 11

Previous study

PCE method for HYMOD model (Fan, 2014)

12

Model

Distributed Hydrological Model of Ibo River (Ishizuka, 2010)

In order to know how the impact runoff mechanism and water penetration in soil due to soil capacity can be approached by :

Storage Function method

𝑑𝑆

𝑑𝑡= 𝑟 − 𝑄

𝑆 = 𝐾𝑄𝑃

𝑆 : storage height𝑟 : effective amount of rainfall𝑄: runoff height𝐾: storage constant 𝑃: storage power constant

Runoff and slope on river drainage effect

Kinematic wave method

𝑑ℎ

𝑑𝑡+𝑑𝑞

𝑑𝑥= 𝑟

𝑞 =𝑘 sin 𝜃

𝛾ℎ +

sin 𝜃

𝑛ℎ − 𝐷

5

3 （ ℎ＞𝐷）

𝑞 =sin 𝜃

𝑛ℎ5

3 （ ℎ＜𝐷）

𝑟 : effective amount of rainfallℎ : water depth 𝑞 : dischargen : roughness coefficient

Try to determine couple of optimal fixed parameter that I want to optimize (e.g. 𝐷 and k in Kinematic wave method)

𝑘 : permeability coefficient𝛾 : effective porosity𝐷 : A-layer thickness𝜃 : Slope gradient 13

DHM (Ishizuka, 2010)

Assumptions:

1. The storage function method in layer 1

is considered for underground

penetration after rain, because of

saturated/ unsaturated layer due to

high water depth.

2. Horizontal ground water flow is not

considered

3. Dam influence on the river way is not

considered

4. Artificial drainage system is not

considered

5. Evaporation from waterway is not

considered

6. Irrigation system is not included?

Magari

Yamazaki

Shiono

Kurisu

Tatsuno

Kamigawara

Kamae

Observation point (hourly discharge)

14

Method (Apply PCE to DHM)

Candidate for parameter optimization which are related to storage capacity of water in the soil (Ishizuka, 2010)

Runoff on slope and river channel

Description Parameter Valueslope gradient θ 0.01-13.5roughness coefficient n 0.01-2layer A thickness D 200effective porosity of layer A γ 0.2hydraulic conductivity of layer A k 0.3distance difference ∆x 20time interval ∆t 0.001storage constant of tank I K1 3.2storage constant of tank II K2 14

(Note : The range of value for each parameter should be discussed to avoid many trial and error)

15

1. Try to apply PCE to soil capacity related parameter within the catchment in

the first layer refer to Kinematic Wave equation

2。Therefore, it is necessary to check which value of polynomial order (k) is

applicable to this model

Future task

16

Test the D and k parameter with PCE to select suitable and optimum

value for the polynomial order (k) within single flood event

Determine time range for the single flood event

Study MATLAB to make modification on PCE code, apply on DHM model

Convert D and k parameter to Gaussian variables