Models and Modelling in FEWS Part II Micha

Models and Modelling in FEWS Part II Micha Werner Deltares & UNESCO-IHE

Error correction ARMA & ADJUST-Q 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) 3

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) 4

Overview of error correction models/methods Available methods for error correction in FEWS Internal • Adjust. Q type operation • 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 5

Overview of available error correction methods ADJUST-Q: Empirical error correction • Parameter steps determines convergence speed • steps may be changed interactively during forecast Example: simple model with constant bias steps 6

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 7

ARMA module Delft-FEWS - 1 Autoregressive Moving Average Models used 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 a 1 are constants. Other observed values of the series can be included in the right-hand side of the equation if k > 1: 8

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. 9

ARMA Model Example of error correction using ARMA. Corrected time series (red) will converge to uncorrected time series (pink) as lead time increases 10

ARMA Model Simple example of ARMA model See also spreadsheet… 11

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 12

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 13

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 14 Typical AR orders vary in range 1 -3

Error Correction using FEWS ARMA model FEWS ARMA Error model Additional Features • Error correction using AR and MA • pre-processing methods to normalize errors (Log, Box-Cox etc) • Additional options • Interpolation of observed data to remove “small’ gaps • Data hierarchy for simulated inputs • Constraints on outputs • Constraints on inputs 15

Error Correction using FEWS ARMA model Options for ARMA model Free order & Free parameters Fixed order & Free parameters This allows the error model to establish both order & parameters dynamically Order is now fixed – but parameters dynamically - order established/calibrated offline Free order & Fixed parameters Fixed order & Fixed parameters not applicable Everything is now fixed – parameters & order established/calibrated offline 16

Error Correction using FEWS ARMA model Options for ARMA model – pro’s and con’s Free order & Free parameters Fixed order & Free parameters Pro: may utilize full potential Con: statistical optimization with many degrees of freedom –small risk of coming unstuck Con: behavior with strange data/bad model unpredictable Pro: utilize potential of dynamic orders Con: very small risk of coming unstuck Con: behavior with strange data/bad model unpredictable Con: need to establish - order. 3 is good working max. Free order & Fixed parameters Fixed order & Fixed parameters not applicable Pro: controlled, predictable, behavior Con: need to establish order & parameters. Calibration required 17

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 T 0 • When there is missing data in simulated time series – failure • Error correction module allows multiple simulated time series to be allocated • Simulated – Forecast • Simulated – Historical • Simulated – Backup (use in case problems with cold start!) 18

Error Correction using FEWS ARMA model Additional options – manipulating inputs 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 Interpolation. Module: Linear Interpolation for defined gap length • Ignore 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 19

Error Correction using FEWS ARMA model Additional options – manipulating outputs • 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) 20

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 ! 21

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 22

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 (Prosymfo 2) • Significant training periods (several months data) • If to be considered – integrate as external module 23

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 Update. States workflow • Configure ARMA to estimate parameters • Run Update. States for extended period (e. g. 1 year) • Run ARMA in DEBUG mode for 1 year of data (through e. g. cold state selection). 24

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 25

Comparison of ADJUSTQ to AR Blending steps = 100 12 1 26

Comparison of ADJUSTQ to AR Blending steps = 100 1 30 12 27

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 24 th of December 2002 28

Calibrating and Validating ARMA models FEWS can be easily applied in setting up such hindcast runs 29

ARMA versus ADJUST-Q 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 30

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, Mike 11, SOBEK, HEC) • Potentially more complex – 2 D models (Delft 3 D, Telemac, Flow 2 D etc) 33

Routing models linked to FEWS (Examples) Hydrological LAG-K TATUM Kinematic Wave (KW) 2 -Lyr Muskingum NWS CEH-Wallingford Deltares US US England & Wales, Scotland - Hydrodynamic SOBEK-1 D ISIS Mike-11 HEC-RAS Delft 3 D SOBEK-1 D 2 D Deltares HR Wallingford/Halcrow DHI USACE Deltares Rhine basin, Waterboards England & Wales, Scotland England & Wales, Italy, Spain US, Italy, Sudan Scotland Thailand 34

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 35

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 36

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) Cons; • More complex models, data intensive • Computationally more demanding • Risk of instability 37

Hydrodynamic vs. Hydrological Models Apply HD models only when really required • Extensive floodplains • Reaches with structures • Tidal Reaches • Confluences Mixing models • Hydrological Hydrodynamic • Hydrodynamic Hydrological 38

HD model in a forecast workflow 39

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) 40

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 41

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 • ADJUST-T (NWS operation addresses similar issue) • Other option – link with coastal shelf model (see case study…) 42

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 Mike 11 & ISIS These are computationally intensive • Splitting model in sections – use error correction at each gage • Assumes rating curve is reliable! 43

Model cascades Hydrodynamic – Hydrodynamic model cascade Complex interaction • State in d/s hydrodynamic model affects state in u/s hydrodyanic model Model 1 • Overlap models I II Model 2 VI IV VII Region of influence of d/s boundary Model 1 Model 2 44

Model cascades 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 boundary Model 1 u/s boundary Model 2 Read Levels from d/s model!!! Q Q+ε Model 1 Model 2 45

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 46

Inundation Mapping Inundation maps provide spatial view of extent of inundation Two main approaches in integrating these maps in FEWS • Running external (2 D) hydrodynamic model – importing resulting grid data to view dynamic inundation profile • HEC-RAS (ID + Interpolation) • TUFlow • SOBEK-1 D 2 D • Running a 1 D 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 47

Inundation Mapping using a 2 D 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 1 D 2 D model of the Barotse Floodplain Zambezi River, Zambia DEM extent 303 x 541 cells; 720 m resolution (resampled from 90 m SRTM data) SOBEK model using 1 D for main stem rivers 48

Inundation Mapping using a 1 D model + Interpolation 49

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

Forecasting using 1 D & 2 D HD models in the Firth of Clyde, Scotland Low Pressure increased tide levels 51

Forecasting using 1 D & 2 D HD models in the Firth of Clyde, Scotland Firth of Clyde (Fo. C) Flood Forecasting Model setup in Delft 3 D-FLOW • Hydrodynamics module of Delft 3 D framework, applied for the modelling of surface water systems Fo. C Model provides • Tidal surge forecasts at locations distributed in Firth of Forth • Downstream boundary to 1 D river models 52

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 2 D, 3 D effects are secondary • Based on a time step of 1 minute, a 1 day simulation takes approximately 6 minutes • Model does not run often (4 x per day) when forcings are updated. Provides d/s boundary for river models • Runs on dedicated server to avoid conflicting with other resources 53

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 54

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) • Adapters to new models can be readily developed 57