# Models and Modeling in FEWS Part 2 by wulinqing

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```									Models and Modelling in FEWS
Part II

Micha Werner
Deltares & UNESCO-IHE
Error correction

In this section we will discuss two methods used for correcting the
outputs of a hydrological model. The method used widely in the
NWS is ADJUST-Q, which typically requires manual interaction
during the forecast run. The second is the ARMA method in FEWS.
This is a statistical error model that is widely used in forecasting.
This does not need interaction during the forecast process
Improving the Forecast

A: Input correction
B: State Updating (data assimilation)
C: Parameter Updating
D: Postprocessing (including Error Correction)

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Output Processing

This can be done using very simple approaches as well as with more
complex methods that canb also provide an estimate of uncertainty

Simple methods:
• Adjust Q (correction at start forecast)
• AR or ARMA type error correction

More “complex” methods:
• Quantile regression
• Bayesian Output Processor (HUP)

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Overview of error correction models/methods

Available methods for error correction in FEWS

Internal
• ARMA Error correction method

External (models) – run using the adapter approach
• MCRM/DODO Error Correction approach
• CEH ARMA Module
• PDM Error correction/State updating
• Implementations of Quantile Regression & HUP

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Overview of available error correction methods

• Parameter steps determines convergence speed
• steps may be changed interactively during forecast

Example: simple model with constant bias

steps

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Overview of available error correction methods

Statistical model of error
• Time series modeling
• ARMA: Auto Regressive – Moving Average

Concept
• Error is typically highly correlated in time
• Establish model of error – predict future error
• Correct model simulation in forecast period with predicted error

Model Order
Model Parameters

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ARMA module Delft-FEWS - 1

Autoregressive Moving Average Models used for forecasting of stationary
timeseries – in this case applied to modelling the time evolution of the
model error

AR: This part of the model describes how each observation (error) is a
function of the previous k observations (errors). For example, if k = 1,
then each observation is a function of only one previous observation. That
is,

where Qres(t) represents the observed residual (error) value at time t,
Qres(t−1) represents the previous observed residual (error) at time t − 1,
e(t) represents some random error and c and a1 are constants. Other
observed values of the series can be included in the right-hand side of the
equation if k > 1:

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ARMA module Delft-FEWS - 2

MA: This part of the model describes how each observation is a
function of the previous y errors. For example, if y = 1, then each
observation is a function of only one previous error. That is,

Here e(t) represents the random error at time t and e(t−1)
represents the previous random error at time t − 1. Other errors
can be included in the right-hand side of the equation if y > 1.

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ARMA Model

Example of error correction using ARMA. Corrected time series (red)
will converge to uncorrected time series (pink) as lead time
increases
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ARMA Model

Simple example of ARMA model

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AR module Delft-FEWS - 3

What is required for setting up an ARMA Model
• Simulated trace (typically SQIN)
• Observed trace (typically QIN)

Parameterisation of error model
- Model Order –
- Model parameters

Three ways of defining error model in FEWS
- Automatic:     Establish both order & parameters dynamically (AR only)
- Defined order: Order defined by user; Dynamic parameter identification
- Define all:    Order & parameters defined by user

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Establishing ARMA model order and parameters

Window length

Statistical behavior of error in window of defined length used to
identify order and/or parameters of error model.
Rule of thumb: Window should be > 50 x order of AR model

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Establishing ARMA model order and parameters

Length of window will influence the estimation of AR parameters.
As window increases autocorrelation of errors will decrease for
most hydrological time series

Window length

When estimating order of model: Define maximum order
Typical AR orders vary in range 1-3   14
Error Correction using FEWS ARMA model

FEWS ARMA Error model

• Error correction using AR and MA
• pre-processing methods to normalize errors (Log, Box-Cox etc)

• Interpolation of observed data to remove “small’ gaps
• Data hierarchy for simulated inputs
• Constraints on outputs
• Constraints on inputs

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Error Correction using FEWS ARMA model

Options for ARMA model

Free order & Free                   Fixed order & Free
parameters                          parameters

This allows the error model to      Order is now fixed – but parameters
establish both order & parameters   dynamically - order
dynamically                         established/calibrated offline

Free order & Fixed                  Fixed order & Fixed
parameters                          parameters

not applicable                      Everything is now fixed –
parameters & order
established/calibrated offline

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Error Correction using FEWS ARMA model

Options for ARMA model – pro’s and con’s
Free order & Free                     Fixed order & Free
parameters                            parameters

Pro: may utilize full potential       Pro: utilize potential of dynamic
Con: statistical optimization with    orders
many degrees of freedom –small        Con: very small risk of coming
risk of coming unstuck                unstuck
model unpredictable                   model unpredictable
Con: need to establish - order. 3 is
good working max.
Free order & Fixed                    Fixed order & Fixed
parameters                            parameters

not applicable                        Pro: controlled, predictable,
behavior
Con: need to establish order &
parameters. Calibration required
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Error Correction using FEWS ARMA model

Notes on inputs to Error model
• 2 Traces are required
• Simulated trace – shoud cover historical & forecast period
• Observed trace – normally ends at T0
•   When there is missing data in simulated time series – failure L

•   Error correction module allows multiple simulated time series to be
allocated
• Simulated – Forecast
• Simulated – Historical
• Simulated – Backup (use in case problems with cold start!)

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Error Correction using FEWS ARMA model

Range check on input can be defined (min/max)
• This is like validation – values beyond range become Missing
• Better to apply a more stingent validation berfore going in to error model
(e.g rate change checks etc)

•   Interpolation of input data
• Avoid spurious results due to small gaps
• Same function as in InterpolationModule: Linear Interpolation for defined
gap length

•   Ignore Doubtfull. Doubtfull data can be set to be ignored
• Be very careful – as rated flows often doubtful beyond range of rating –
but we do want these to be used

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Error Correction using FEWS ARMA model

•   Range check on outputs can be defined (min/max)
• This is NOT a validation – values are constrained to min-max
• Typically used for constraining discharge values to zero (or a
minimum flow, e.g. as input to HD model)

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Application of Error correction

General notes
• Error correction is a form of modeling!

•   Careful thought of the nature of errors being corrected

•   Calibration & validation
• Calibration required if orders are fixed
• Validation required in both cases !

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Application

Typical application of error model
• Rainfall-runoff model calculates flow to catchment outlet (C)
• Error correction applied to flow at C
• Routing-model calculates propagation of flow in steep river
• Uses error corrected flow as input
• Error correction applied to flow at B
• HD model calculates levels & flows in
reach from B to A
• Uses error corrected flow as input        C
B

A                     Legend:
Main River
Small River
Sub-Catchment

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Application

Error model cannot be applied to tidal signal as is!
• Periodic signal requires different approach

•   Approach 1: Correction of surge residuals
• Possible – but…
• Forecast surge may be very different from observed surge (bias)

•   Approach 2: Correction Frequency domain (Prosymfo2)
• Significant training periods (several months data)
• If to be considered – integrate as external module

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Setting up the ARMA Model in FEWS

Configuration when using automatic estimation methods is very easy
• Identify inputs and outputs
• If fixing order - set order of AR to e.g. 3 (typically maximum order)
• Typically MA can be ignored – as AR dominates

If fixing both order AND parameters: Recommended approach
• Set up models & ARMA in UpdateStates workflow
• Configure ARMA to estimate parameters
• Run UpdateStates for extended period (e.g. 1 year)
• Run ARMA in DEBUG mode for 1 year of data (through e.g. cold
state selection).

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Configuration

ARMA model run in DEBUG mode – allow parameters to be
estimated
• Read AR (and MA values if relevant from DEBUG message
• Copy values as fixed

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1
12
Blending steps = 100

26

30
12
1
Blending steps = 100

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Calibrating and Validating ARMA models

Calibration of ARMA models using e.g. FEWS inernal routines, or other statistical
packages
Validation
• Run series of hindcast runs
• Plots of lead time accuracy

Fig. 3. (a) Lead time accuracy of the discharge forecast expressed as RMSE at the gauging stations of Cochem on the
Mosel River, and Maxau on the River Rhine. Both the accuracy with and without error correction are shown. (b) Shows
an example of the corrected and simulated flows at the gauge of Maxau in the Rhine for the forecast of 24th of
December 2002
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Calibrating and Validating ARMA models

FEWS can be easily applied in setting up such hindcast runs

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Pros;
• ARMA allows for an automated approach to adjusting errors –
reduces need for interactivity
• ARMA makes statistical sense – errors typically have structure
• ARMA provides an objective method – can be verified using
hindcasts
• ADJUST-Q supports changing interactively when not behaving
properly
Cons;
• ARMA is a statistical model – not a hydrological model – statistical
sanity is not always hydrologically correct
• ADJUST-Q is subjective – difficult to apply in verification

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Questions…
Routing models in FEWS
Hydrodynamic models

In this section we will discuss the application of routing models in
FEWS – focusing primarily on the use of hydrodynamic models such
as HEC-RAS. Some of the particular aspects of using HD models in
real time are discussed.
Routing models

Objective: Calculate propagation of flood wave through river system
• Simple Hydrological Routing (KW, Lag-K, Muskingum, …)
• Complex with 1-D hydrodynamic model (ISIS, Mike11, SOBEK,
HEC)
• Potentially more complex – 2D models (Delft3D, Telemac, Flow2D
etc)

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Routing models linked to FEWS (Examples)

Hydrological   LAG-K                 NWS                      US
TATUM                 NWS                      US
Kinematic Wave (KW)   CEH-Wallingford          England & Wales, Scotland
2-Lyr Muskingum       Deltares                 -

Hydrodynami    SOBEK-1D              Deltares                 Rhine basin, Waterboards
c              ISIS                  HR Wallingford/Halcrow   England & Wales, Scotland
Mike-11               DHI                      England & Wales, Italy, Spain
HEC-RAS               USACE                    US, Italy, Sudan
Delft3D               Deltares                 Scotland
SOBEK-1D2D            Deltares                 Thailand

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Differences between model approaches

Kinematic Wave

Diffusive Wave

Dynamic Wave (all other cases) Full Equations

Most models are derivations of the shallow water equations – ignoring
different terms that are insignificant: Depends on the hydraulic situation

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Hydrodynamic vs. Hydrological Models

Typical set-up
Simple routing – often in hydrological model
e.g. UNIT-HG

Hydrological routing
e.g. LAG-K, Kinematic Wave

C
Hydrodynamic routing
e.g. HEC-RAS                              B

A                      Legend:
Main River
Small River
Sub-Catchment

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Hydrodynamic vs. Hydrological Models

Pros;
• Hydrodynamic routing provides more realistic simulation of flood
wave propagation
• Deals well with backwater effects, change in flood wave
propagation when flow goes out of bank
• Allows incorporation of structures and control of structures
• Allows outputs at intermediate locations (not gage à gage)

Cons;
• More complex models, data intensive
• Computationally more demanding
• Risk of instability

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Hydrodynamic vs. Hydrological Models

Apply HD models only when really required
• Extensive floodplains
• Reaches with structures
• Tidal Reaches
• Confluences

Mixing models
• Hydrological à Hydrodynamic
• Hydrodynamic à Hydrological

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HD model in a forecast workflow

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Exchange between HD models & FEWS

•   All HD models integrated with FEWS using standard adapter approach
•   Inputs (typical)
• Flows at upstream boundary and tributary inflows
• Level at downstream boundary – may be a tidal boundary
(not required when internal rating curve boundary is used)
•   Inputs (less common)
• Gate settings
• Temperature
•   Outputs (typical)
• Water Level & Flow
(point – or – longitudinal)

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Exchange between HD models & FEWS

•   Location of boundaries needs careful thought to avoid “reading” a
defined boundary as the result of a HD model

Reach influenced by d/s boundary condition

Ignore results
from this point

Upstream boundary – Q(t)

Downstream boundary – Q-h

Flow direction

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Exchange between HD models & FEWS

Tidal boundaries offer a specific problems
• Astronomical constants to derive astronomical tide
• Difficult to work with harmonic constants
• Work with surge residuals (interpolate, ARMA modeling etc) – then add back to
astronomical tide
• Other option – link with coastal shelf model (see case study…)

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Hydrodynamic models & Error correction
•   Hydrodynamic models typically cover long reaches of river, which means
that intermediate gages are not utilized for error correction

Options
• State updating: e.g. Ensemble/Extended Kalman Filter; Particle Filter)
• Particle filter applied in Rhine for updating
• Simple “nudging” techniques
• Available in Mike11 & ISIS
à These are computationally intensive

•   Splitting model in sections – use error correction at each gage
• Assumes rating curve is reliable!

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Complex interaction
• State in d/s hydrodynamic model
affects state in u/s hydrodyanic model
I
• Overlap models                    Model 1                  III

II

Model 2                               IV
VI

VII

Region of influence of d/s boundary

Model 1                              Model 2
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Connecting two hydrodynamic models

Error correction on flow from u/s model
• Note that this does assume rating curve is reliable!! May not
include hysterisis

Gauge u/s of model transition
Calculate error ε
Add error to flow at d/s
Q                      boundary Model 1
u/s boundary Model 2
Q+ε

Model 1                          Model 2
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Burn-in profiles

Avoid “abrupt” shock on startup
Mainly relevant to HD modules (stability)
Only applied when starting from a cold state
• Identify start value in cold state
• Gradual “climb” to actual value

Burn-in section

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Inundation Mapping

Inundation maps provide spatial view of extent of inundation

Two main approaches in integrating these maps in FEWS

•   Running external (2D) hydrodynamic model – importing resulting grid data
to view dynamic inundation profile
• HEC-RAS (ID + Interpolation)
• TUFlow
• SOBEK-1D2D
•   Running a 1D hydrodynamic model
• Export levels at cross sections to FEWS Flood Mapping Module
• Interpolate water surface profile in GIS (PCRaster)
• Import dynamic flood map to FEWS

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Inundation Mapping using a 2D model

•   Model runs through General
Adapter – as does any model
•   Time series of grid data
returned – map stack
•   Imported to FEWS database –
displayed as any other grid

Example:
SOBEK 1D2D model of the Barotse Floodplain
Zambezi River, Zambia

DEM extent 303 x 541 cells; 720m resolution (resampled from 90m SRTM data)
SOBEK model using 1D for main stem rivers

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Inundation Mapping using a 1D model + Interpolation

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Example: Modeling of bifurcation/Confluence
1D: Modeler decides division
2D: Division depends on water level

?

Pannerdensche Kop
Forecasting using 1D & 2D HD models in the Firth of Clyde,
Scotland

Low Pressure

increased tide
levels

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Forecasting using 1D & 2D HD models in the Firth of
Clyde, Scotland

Firth of Clyde (FoC) Flood Forecasting
Model setup in Delft3D-FLOW
• Hydrodynamics module of Delft3D
framework, applied for the modelling of
surface water systems

FoC Model provides
• Tidal surge forecasts at locations
distributed in Firth of Forth
• Downstream boundary to 1D river models

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Firth of Clyde model development

Model setup - computational grid
• Orthogonal curvilinear grid, aligned with local
geometric features
• Spatially varying resolution (1 km – 100 m)
• Run in 2D, 3D effects are secondary
• Based on a time step of 1 minute, a 1 day
simulation takes approximately 6 minutes

•Model does not run often (4x per day) when
forcings are updated. Provides d/s boundary for
river models

•Runs on dedicated server to avoid conflicting with
other resources

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Firth of Clyde model development

Model setup - boundary forcing
• Tidal boundary conditions (harmonic constituents) for 50 tidal components
• External surge conditions by time-varying, spatially uniform water level elevation
• Meteorological forcing by time-varying, spatially uniform wind speed and direction
• Assuming one-way coupling at rivers (model provides d/s level boundary), no river
discharge taken into account

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Questions…
Wrap-up
Wrap up of models in FEWS

•   Variety of different types of models available for running in FEWS
• All integrated using the same “adapter” concept
• Models can be mixed in a single workflow – extremely useful
for creating “integrated modeling structures”

•   Increasing use of distributed & physically based models in
forecasting
• Issues: speed, database sizes, complexity, …
•   Variety of models & adapters available and used operationally
• Actual availability depends on model & supplier (licences)