1 Applications of wavelet analysis in hydrological modelingA case study on PRACTICAL applications of Wavelet transform -Vinit Sehgal -Project Associate Under the guidance of Dr. Rakesh Khosa Professor Dr. R. Maheswaran Inspire Faculty Department of Civil Engineering Indian Institute of Technology (IIT) Delhi
2 OBJECTIVES OF STUDY This study discusses application of wavelet analysis in climatic downscaling and river discharge forecasting applications. The paper discusses practical considerations in application of wavelet analysis for hydrological modeling like: 1)Selection of suitable mother wavelet for decomposition 2)Selection of suitable level of decomposition 3) Consideration of boundary value problem for forecasting applications
3 What is wavelet analysis?Wavelet analysis- mathematical technique for decomposing the signal into different components at different scales. The features at different scales are captured using a low pass(H) and high pass filter(G). WT 16 months scale Multi scale decomposition 8 months scale 4 months scale 2 months scale
4 Wavelet transform A contracted and dilated version of wavelet is translated through the signal creating coefficients at different resolution level. These coefficients form basis of obtaining information about the decomposed signal at different time frequency scale. 1 1/2 This example illusrates that how the wavelet transforms works.
5 Dyadic scheme for wavelet transformationThe figure shows the dyadic scheme of generation of wavelet coefficients. The scheme follows the following formula: C=(2^m) , where m=1,2,3,4.. etc.
6 Practical issues in wavelet based model implementationSelection of suitable wavelet decomposition Optimal number of levels of decomposition required Treatment of boundary problems
7 BRIEF INTRODUCTION OF THE CASE STUDIES PRESENTEDA quick preview to our models for three hydrologic applications.
8 Case study 1 & 2: Downscaling of Precipitation and TemperatureWe have developed models for climatic downscaling for precipitation and temperature. Wavelet based Principal component – MLR approach was used for this purpose.
9 Scheme for climatic downscalingNCEP ( ) (Historical data) VARIABLE AVERAGING PCA & WAVELET DECOMPOSTION MODELING MLR STANDARDISATION GCM ( ) RCP4.5 (Simulated data) NCEP= National centre for environmental prediction -Obtained from observed data, hence more reliable * GCM= General circulation model -Developed from partial differential equations at a coarser scale assuming some specific initial conditions
10 Case 3: Monsoonal discharge forecasting at Mahanadi Basin, IndiaMonsoonal flood discharge at the downstream station Naraj is to be forecasted using information from the upstream stations and antecedent discharges at Naraj. Wavelet- Bootstrap-Multiple linear regression models (W-B-MLR) were proposed. Relative performance of the models w.r.t the vanishing moment of the mother wavelet used for DWT was studied. **“Wavelet Bootstrap Multiple Linear Regression based Hybrid Modeling for Daily River Discharge Forecasting”- V. Sehgal, M. K. Tiwari and C. Chatterjee (Water Resources Management, Springer, 28 (10), )
11 SELECTION OF MOTHER WAVELETObservations from all three case studies are studied and are presented to derive a common conclusion.
12 Station-wise Correlation of W-PCA-MLR model output with observed IMD data- PrecipitationPerformance (Correlation coefficient) of the top 5 best performing Wavelet based models at each station for Precipitation downscaling at Krishna river basin is shown here. We observe that wavelets with small vanishing moment like db1, db2 and db3 are performing good at all stations and feature in top 5 best performing models in almost all cases.
13 Station-wise Correlation of W-PCA-MLR model output with observed IMD data- TemperaturePerformance (Correlation coefficient) of the top 5 best performing Wavelet based models at each station for Temperature downscaling at Krishna river basin is shown here. We observe that wavelets with medium vanishing moment like db30 and db37 are performing good at all stations and feature in top 5 best performing models in almost all cases.
14 Performance of the models v/s Vanishing moment of mother wavelet used for DWTbior 1.1 1 92.65 Haar 93.53 coif1 2 bior 3.3 3 97.27 db 5 5 97.81 coif3 6 98.83 db 10 10 99.24 db 15 15 99.65 db 20 20 99.67 db 25 25 99.74 db 30 30 99.78 db 35 35 99.79 db 40 40 99.81 db 45 45 Table shows performance of the flood forecasting models w.r.t vanishing moment of the mother wavelet used for the DWT. It can be observed that models with mother wavelets with vanishing moment 3 explains more than 95% of the variance.
15 Relating selection of mother wavelet to inherent process featuresPRECIPITATION (2 years sample data) Short events and transient features Hence wavelets with small vanishing moments suitable. Suitable Wavelets: db1, db2, db3 TEMPERATURE (2 years sample data) Mix of both Short and long memory features Hence wavelets with medium vanishing moments suitable. Suitable Wavelets: db37, db30 etc. ACF ACF PRECIPITATION As the models are designed for monsoonal flood discharge which occurs for 3 months in a year namely July, August, September, a model having vanishing moment of the mother wavelet used equal or greater to monsoonal months would be able to explain the inherent process better. Suitable Wavelets: db3.
16 Suitable level of decompositionLevel of time series decomposition must be chosen after an investigation for the underlying dominant features in order to enable an unambiguous detection of these features.
17 Case: Precipitation downscalingWet season includes months from April to October which receives monsoonal rainfall in the basin. As the models have to capture properties of the precipitation, the level of multi- resolution transformation should match with the wet months. Hence, optimum level of decomposition was chosen to be 3 which corresponds to 8 time units. DRY SEASON WET SEASON
18 Case: Discharge forecastingIt takes slightly above two days for water released from Hirakud dam to reach downstream station of Naraj. Moreover, the flood forecasting models were developed for a lead time of maximum five days. Hence, information upto 8 days interval was considered to be important for a multi resolution model. 3 days
19 Dyadic scheme for wavelet transformationThe figure shows the dyadic scheme of generation of wavelet coefficients. The scheme follows the following formula: C=(2^m) , where m=1,2,3,4.. etc.
20 Treatment of boundary problems.
21 Understanding the boundary problemSignal Scale 1 Scale 2 Scale 3 Approx 3 Filter window
22 Selection of wavelet decompositionDWT a’ trou wavelet decomp.
23 Application of causal filtersSignal Scale 1 Scale 2 Scale 3 Approx 3 Filter window
24 Treatment of boundary problems,In general wavelet applications, various kinds of boundary conditions such as (i) periodic boundary, (ii) reflective boundary extension, and (iii) constant extension are usually used for extending the series from ‘t- p’ and ‘t+p’. However, in the case of forecasting models, these extensions cannot work.
25 Conclusion The paper discusses key practical issues in application of wavelet analysis in hydrological forecasting. The paper advocates selection of mother wavelet selection and level of decomposition by understanding the underlying feature of the process. Data with short, transient feature would be better represented by wavelets with smaller vanishing moment and data with long memory would be represented with a wavelet with higher vanishing moment. Level of decomposition of a data depends on the scale of information the model targets to incorporates. This is represented by dyadic scheme as given below: 2 units- Scale 1 4 units- Scale 2 8 units- Scale 3 etc. A’trous wavelet transform is more suitable for forecasting applications due to use of causal filters.
26 Thanks
27 Appendix Data DescriptionGCM CanCM4 (AR5, IPCC), grid size 2.8° X 2.8 Canadian Centre for Climate Modelling and Analysis Historical( ) RCP 4.5 scenario ( ) NCEP/NCAR National Centre for environmental Prediction, grid size 2.5° X 2.5° Reanalysis data ( ) IMD Indian Meteorological Department grid size 0.5° X 0.5° ( )
28 Appendix 2: GCM future scenarioIn AR5 four Representative Concentration Pathways (RCPs) were selected and defined by their total radiative forcing (cumulative measure of human emissions of GHGs from all sources expressed in Watts per square meter) pathway and level by 2100. RCP4.5 represents stabilization without overshoot pathway to 4.5 W/m2 at stabilization after 2100
29 Probable Atmospheric VariablesAppendix Probable Atmospheric Variables S.no Predictor Pressure levels (mb) 1 Atmospheric Temperature (TA) 1000, 925, 850, 700, 600, 500, 400, 300, 250, 200, 150, 100, 70, 50, 30, 20, 10 2 Eastward Wind (UA) 3 Northward Wind (VA) 4 Geo-potential Height (ZG) 5 Sea Level Pressure (PSL) 6 Surface Temperature (TS)
30 Variable Averaging- TAAppendix Variable Averaging- TA
31 Appendix 3 ACF ACF ACF PACF PACF PACF PRECIPITATION TEMPERATUREDISCHARGE
32 Appendix:Precipitation (Point B)
33 Appendix:Future Precipitation (Point B)
34 Appendix:Temperature (Point B)
35 Appendix:Future Precipitation (Point B)
36 Outcome of downscaling tool developed by our team
37 References “Wavelet Bootstrap Multiple Linear Regression based Hybrid Modeling for Daily River Discharge Forecasting”- V. Sehgal, M. K. Tiwari and C. Chatterjee (Water Resources Management, Springer, 28 (10), ) “Effect of Utilization of Discrete Wavelet Components on Flood Forecasting Performance of Wavelet Based ANFIS Models” –V. Sehgal, R. R. Sahay, C. Chatterjee (Water Resources Management, Springer, 28 (6), ) “Wavelet- ANFIS Models for Forecasting Monsoon Flows: Case Study for the Gandak River (India)” R. R. Sahay and V. Sehgal (Water Resources, Springer, 41 (5), ) “Wavelet Autoregressive models for flood stage forecasting in Rivers: A case study in Eastern India”- R. R. Sahay and V. Sehgal. (Journal of Flood Risk Management, Willey, 6 (2), ) “Auto Updating Wavelet Based MLR models for Monsoonal river discharge forecasting”- V. Sehgal, C. Chatterjee (International Journal of Civil Engineering Research ,5 (4), ) “Monsoon Flood Forecasting in Gandak River Using Discreet Wavelet Transform” - V. Sehgal ,R. R. Sahay (Recent Trends in Civil Engineering & Technology 2 (1), 11-19)
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