1. CEE 6410 Water Resources Systems Analysis
Data-Driven Modeling and Machine Learning Approach in Water Resource Systems
CEE 6410
Water Resources Systems Analysis

2. Illustrative Example

3. Sevier Valley/Piute irrigation canal in the Sevier River Basin, Utah.
Inputs:
Model 1:
Past daily information of the canal diversions at the head gauging station (Flow at Head)
Model 2:
Past daily information of the canal diversions (Flow at Head) and also information at four gauging stations along the canal: Willow Creek, Aurora, Clairon and End stations
Outputs:
Predictions of canal diversions 1 day ahead from the head gauging station (Flow at Head)

4. RVM Regression w is a weight matrix
Ф is a “design” matrix related with a kernel K(x1,xM)
For this example : Gaussian Kernel
Gaussian Kernel:
K(x,x(n)) = exp(-r-2||x- x(n)||2)
where r is the kernel width parameter.

5. Model Development
Daily data from the 2003 through 2006 irrigation seasons were used to train the RVM model. Daily data from the 2007 irrigation season were used to test the model.
Several RVM models were built with variation in kernel width and number of previous time steps.
The number of previous time steps was chosen from a range of 1-7 days previous to the prediction time.
The kernel width was chosen from a range of 1-5. The selected kernel width is the one with maximum E.
From the list of models with selected kernel width at different “number of previous days" values, we considered that the selected model is the one with the maximum E.

6. Results
nd= number of previous days
#RVs= number of relevance vector

7. Results – Model 1 Test phase from July to September 2007

8. Results – Model 2 Test phase from July to September 2007

9. Bootstrap Histogram - Model 2

10. Applications in
Water Resources Systems Research

11. Forecasting required releases from a multiple reservoir system
Problem: The operator has to release water from more than one reservoir taking into consideration different water requirements (irrigation, environmental issues, hydropower, recreation, etc.) in a timely manner.
“Multivariate Bayesian regression approach to forecast releases from a system of multiple reservoirs”, (2010) Andres M. Ticlavilca and Mac McKee, Water Resources Management.

12. Lower Sevier River Basin

13. x = [X1d-n, X2d-n, X3d-n, X4d-n, X5d-n] T
Inputs
x = [X1d-n, X2d-n, X3d-n, X4d-n, X5d-n] T
where,
d= day of prediction
n= number of days previous to the prediction time
X1d-n = diversions to the Central Utah canal, Vincent canal, Leamington canal, canal A, Abraham canal, Deseret High canal, and Deseret Low canal.
X2d-n = streamflow on the Sevier River near Lynndyl.
X3d-n = Sevier Bridge reservoir releases.
X4d-n = DMAD reservoir releases.
X5d-n = maximum and minimum daily Temperature from Oak city and Delta city.

14. Outputs t = [ R1d, R1d+1 , R2d, R2d+1]T where,
R1d = prediction of Sevier Bridge reservoir release one day ahead
R1d+1 = prediction of Sevier Bridge reservoir release two days ahead
R2d = prediction of DMAD reservoir releases one day ahead
R2d+1 = prediction of DMAD reservoir releases two day ahead

15. Results – Model Selection
Testing phase
Type of kernel function
Kernel width
Number of days
Statistics
R1d
R1d+1
R2d
R2d+1
Average
Gauss
2900 2 E
0.947
0.868
0.930
0.805
0.888 R
0.973
0.932
0.968
0.909
0.946
Laplace
3100 1
0.925
0.841
0.900
0.780
0.861
0.964
0.917
0.952
0.892
0.931
Cauchy
0.943
0.866
0.935
0.800
0.886
0.972
0.969
0.903
0.944
Cubic
1700
0.936
0.842
0.927
0.813
0.879
0.918
0.966
0.914
0.942

16. Results
The model only utilizes 39 RVs from the full data set (1248 observations) that was used for training (2001 through 2006 irrigation seasons).

17. Results - MVRVM
R1d
R1d+1
R2d
R2d+1

18. Results - ANN Testing phase Type of training function Size of layer
Number of days
Statistics
R1d
R1d+1
R2d
R2d+1
Average
Quasi-Newton 3 2 E
0.942
0.867
0.898
0.777
0.871 R
0.971
0.932
0.952
0.891
0.936
Conjugate gradient with Powell-Beale restarts 4
0.928
0.846
0.922
0.821
0.879
0.964
0.961
0.908
0.939
Levenberg-Marquardt 5
0.917
0.860
0.923
0.811
0.878
0.959
0.929
0.963
0.940
Scaled conjugate gradient
0.943
0.856
0.834
0.890
0.972
0.966
0.946

19. Results - ANN
R1d
R1d+1
R2d
R2d+1

20. Multivariate Relevance Vector Machine Artificial Neural Network
Results
Multivariate Relevance Vector Machine
Training
Testing
Statistic
R1d
R1d+1
R2d
R2d+1
Coefficient of efficiency E
0.95
0.87
0.89
0.77
0.93
0.80
Root mean square error RMSE, cfs
64.42
99.24
34.84
49.66
59.45
93.82
32.55
54.49
Artificial Neural Network
Training
Testing
Statistics
R1d
R1d+1
R2d
R2d+1
Coefficient of efficiency E
0.96
0.90
0.89
0.79
0.94
0.86
0.93
0.83
Root mean square error RMSE, cfs
55.47
88.21
34.50
48.29
61.77
98.11
33.02
50.29

21. Robustness
MVRVM
ANN
MVRVM allows more robust parameter estimation

22. Real-time forecasting of short term diversion demands for irrigation canals
Problem:
The canal operator has to divert water from the river into a canal taking into consideration the water requirement for crop irrigation, physical behavior of the irrigated agricultural area, and the behavior of irrigators.
The multi-reservoir operator (upstream of the irrigation canal) has to release water from more than one reservoir taking into consideration water demands including irrigation canal demands.

23. Irrigation Canals
“Real-time forecasting of short-term irrigation canal demands using a robust multivariate Bayesian learning model”, Andres M. Ticlavilca, Mac McKee, and Wynn R. Walker (2011), Irrigation Science.

24. Hourly Model - Inputs x= [D1h-n, D2h-n, D3h-n] T where,
h = hour of prediction
n = number of hours previous to the prediction time
D1d-n = Central Utah canal demand
D2d-n = Vincent canal demand
D3d-n = Leamington canal demand

25. Hourly Model - Outputs where,
t = [ D1h, D1h+12 , D1h+24 , D2h, D2h+12 , D2h+24 , D3h, D3h+12 , D3h+24]T
where,
D1 = prediction of Central Utah canal demand
D2 = prediction of Vincent canal demand
D3 = prediction of Leamington canal demand

26. x = [D1d-n, D2d-n, D3d-n , Tmaxd-n, Tmind-n] T
Daily Model - Inputs
x = [D1d-n, D2d-n, D3d-n , Tmaxd-n, Tmind-n] T
where,
d = day of prediction
n = number of days previous to the prediction time
D1d-n = Central Utah canal demand
D2d-n = Vincent canal demand
D3d-n = Leamington canal demand.
Tmaxd-n = maximum daily Temperature from Oak city.
Tmind-n = minimum daily Temperature from Oak city.

27. t = [ D1d, D1d+1, D2d, D2d+1, D3d, D3d+1]T
Daily Model - Outputs
t = [ D1d, D1d+1, D2d, D2d+1, D3d, D3d+1]T
where,
D1= prediction of Central Utah canal demand
D2 = prediction of Vincent canal demand
D3 = prediction of Leamington canal demand

28. Model Selection Gaussian kernel:
K(x,x(n)) = exp(-r-2||x- x(n)||2) ,where r is the kernel width parameter.
This type of kernel has been used by several authors in water resources and hydrology applications (Khalil et al. 2005b; Tripathi and Govindaraju 2006).

29. Results
The hourly model utilizes 26 RVs from the full data set (out of a possible 3000 observations) that was used for training.
The daily model only utilizes 26 RVs from the full data set that was used for training (1194 observations).

30. MVRVM – Hourly Model D1h D1h+12 D1h+24 D2h D2h+12 D2h+24 D3h D3h+12

31. ANN – Hourly Model D1h D1h+12 D1h+24 D2h D2h+12 D2h+24 D3h D3h+12

32. MVRVM – Daily Model
D1d
D1d+1
D2d
D2d+1
D3d
D3d+1

33. ANN – Daily Model
D1d
D1d+1
D2d
D2d+1
D3d
D3d+1

34. Statistics – Hourly Model
Central Utah Canal
Vincent Canal
Leamington Canal
Model
Statistics
1-hour
12-hours
24-hours
MVRVM (selected kernel width = 14) E
0.993
0.898
0.785
0.996
0.942
0.881
0.986
0.848
0.736
RMSE
2.879
10.913
15.824
0.549
2.177
3.136
1.460
4.864
6.415
ANN (selected hidden layer size = 5)
0.994
0.899
0.779
0.943
0.989
0.853
0.748
2.682
10.821
16.040
0.562
2.171
3.131
1.336
4.780
6.266

35. Statistics – Daily Model
Central Utah Canal
Vincent Canal
Leamintong Canal
Model
Statistics
1-day
2-days
MVRVM (selected kernel width = 25) E
0.851
0.659
0.927
0.803
0.846
0.671
RMSE
12.998
19.648
2.515
4.116
4.890
7.158
ANN (selected hidden layer size = 5)
0.840
0.658
0.809
0.831
13.447
19.684
2.514
4.056
5.122
7.293

36. Robustness – Hourly Model
MVRVM
ANN
MVRVM is more robust
Central Utah Canal: (a) 1-hour ahead, (b) 12-hours ahead, (c) 24-hours ahead
Vincent Canal: (d) 1-hour ahead, (e) 12-hours ahead, (f) 24-hours ahead
Leamington Canal: (g) 1-hour ahead, (h) 12-hours ahead, (i) 24-hours ahead

37. Robustness – Daily Model
MVRVM
ANN
MVRVM is more robust
Central Utah Canal: (a) 1-day ahead, (b) 2-days ahead
Vincent Canal: (c) 1-day ahead, (d) 2-days ahead
Leamington Canal: (e) 1-day ahead, (f) 2-days ahead

38. Forecasting Daily Potential Evapotranspiration Using Machine Learning And Limited Climatic Data1
1: Agricultural Water Management 98 (2011) 553–562, doi: /j.agwat 38

39. Area of Study 39

40. Agricultural Command Area “Canal B” & fed canal “Canal A”40

41. Statistics Main crops: alfalfa, corn and small grains,
Irrigated area: 10,521 ha.,
Fed by 9 km. canal (A) from upstream reservoir (DMAD),
Water management based on local water masters and managers,
SCADA system implemented from reservoirs to ACA inlets,
Limited climatic data (local weather station),
Soil Moisture Monitoring sites implemented lately,
Water conveyance time: 12 hr (DMAD) or 3 days (Sevier Bridge). 41

42. Current Situation Water managers “guess” future water demands,
crop evapotranspiration: ETo
Issues:
ET0 models: for estimation, no forecasting,
Locations w/ limited available climatic data.

43. Approach Forecasted ET0 for water management: Used data:
forecast interval > 1 day, (large irrigation systems)
real time operation (daily basis),
Used data:
Historical daily air temp,
1985 Hargreaves ET0:
Mapping algorithm:
MVRVM:
Tmax, & Tmin : daily air temp (oC),
TC : 0.5 (Tmax + Tmin),
TR: Tmax – Tmin,
Ra: extraterrestrial solar rad (mm/day).

44. Methodology Time series concept: Direct Approach: Indirect Approach:
ET0 forecast,
Indirect Approach:
Air temp. forecast to calculate future ET0.

45. Data Local weather station, Delta, UT,
Historical records of maximum and minimum air temperatures, 10 irrigation seasons, ,
2000 – 2006: training dataset,
2007 – 2009: testing dataset.
Learning machine parameter:
MVRVM = kernel width,
MLP = number of hidden neurons
Values to forecast: 7 days
Values in the past as inputs: ??

46. Results – Test data:

47. Results – Direct Approach, peak season 2009

48. Results – Indirect Approach, peak season 2009

49. Results – Goodness-of-fit
MVRVM
MLP

50. Results – Goodness-of-fit

51. Results – Bootstrap Analysis
Direct Approach
Indirect Approach

52. Crops in ACA Canal B Crop Identification : 2009 Crop Area (ha) %
Alfalfa
3369.2
32.0%
Corn
723.6
6.9%
Small
Grains
323.3
3.1%
Fallow
6105.2
58.0%
Total
100.0%

53. Results – Average ET for ACA Canal B
Indirect Method using MVRVM has better performance than MLP,
Then using:
remote sensing: crops and areas,
local crop coefficients,
Combined ET for ACA B:

54. Machine Learning Approach for Error Correction of Hydraulic Simulation Models2
2: Coauthored by Andres M. Ticlavilca, Wynn R. Walker and Mac McKee,
under review, Journal of Irrigation and Drainage (2010). 54

55. Current Situation Actual conditions: Issues:
Modern irrigation systems: SCADA and simulation models,
SCADA data no noise free,
model: phenomena approximation.
Issues:
Rarely error sources can be isolated,
Over time, aggregate error affect system reliability and operation.

56. Simulation Model + Error Sources
3 types of error sources:
Parameter uncertainty (εp),
Noise in variables (εo),
Structural error (εs).
Combined effect:
Aggregate error (εA),
Impact on results:
Simulation error (εsim).

57. Approach Model Error Correction: Used data: Mapping algorithm:
better quality of real-time model response,
adequate linkage to numerical approximation,
mapping of error & variables.
Used data:
Data-driven model: same as simulation model:
Mapping algorithm:
RVM: Relevance Vector Machine.

58. Methodology Coupled simulation-error correction model:
Under working conditions (SCADA, gates and model),
Minimization of aggregate error in real-time,
Error correction within the code of the simulation model

59. Hydraulic Simulation Model

60. Saint Venant Equations: Continuity & Momentum
Hydraulic Simulation Model
Saint Venant Equations: Continuity & Momentum
Q: flow rate (m3/s),
A: flow cross-section area (m2),
X: longitudinal distance in the direction of flow (m),
t: elapsed time (hr)
q: lumped expression of seepage, evaporation, tributary inflows and model error.
P: net hydrostatic pressure per unit weight of water (m2)
D: drag force, or the product of friction slope and area (m2).
So: canal slope,
g: acceleration of gravity (9.807 m/s2).

61. Hydraulic Simulation Model
Deformable Control Volume:
Eulerian space-time solution grid:

62. Hydraulic Simulation Model
minimizing εsim to obtain εA values along the irrigation season
εA estimation is done by:
modifying Continuity Eq:

63. Simulation Model Performance
Aggregate error estimated by:
fine-tuning simulation model,
minimizing simulation results (model results vs. SCADA water levels),
Considering irrigation season in Canal A (2008 & 2009).
Statistical characteristics of aggregate error

64. Data From hydraulic simulation model: Training data: Testing data:
Inflow rate at canal head (Qin),
On –demand variation (ODv),
Water level at canal head (hin),
Previous aggregate error (εA(t-1),…, εA(t-5))
Training data:
hourly values 2008 irrigation season,
Testing data:
hourly values 2009 irrigation season,
Learning machine parameter:
RVM = kernel width,
MLP = number of hidden neurons.

65. Results – Goodness-of-fit Test Data 2009
MLP
RVM

66. Results – Error Correction –RVM – 2009

67. Results – Error Correction – MLP - 2009

68. Results – Bootstrap Analysis
Both error correction models perform similar under similar conditions,
But in bootstrap analysis RVM is more stable to unseen data than MLP.

69. Discussion & Conclusions

70. Discussions
A critical factor for the use of ML models is data availability.
Data-driven tools calibration process is based on patterns that the algorithm can “learn” from the data.
Data should be enough to divide it in training and testing subsets.
For time series or historical data, it is recommendable at least three complete cycles of the phenomena (e.g. irrigation seasons or runoff years) and their respective inputs. Two cycles would be separated for training and the most recent for testing.
There are two aspects to considerate when calibrate a ML model:
The calibration of the data-driven tool. Each ML model has its unique parameters. For example ANN models require the selection of the number of hidden neurons and the learning function. RVM models require the selection of the type of kernel and a kernel width value.
The selection of the most adequate inputs. There is not a unique methodology. This is related to the synergy among variables. There are general techniques: forward and backward variable selection, automatic relevance determination (for Bayesian-based algorithms) or combination of these techniques.

71. Discussions How to determine the accuracy of the calibrated ML model:
The selection of the relevance goodness-of-fit parameters is important.
For regression problems the coefficient of efficiency along with the RMSE.
Also the error bar for Bayesian LM is an indicative of the error in the data and algorithm.
For classification-type problems the Confusion Matrix is along the Kappa Index, which measures the accuracy of the model predicting the different classes vs. the probability of random occurrence.
For time series models a couple of important issues occur along the time dimension.
First, a time lag between the simulated and the true signal could occurs. In most cases this is an indication of missing inputs into the model.
Second, the characteristics of the residuals or the difference between the simulated and the true signal. The residuals should comply with white noise characteristics (normal, independent, identically distributed).
The absence of these characteristics indicates that the pattern in the data is not fully captured by the ML model because of : inadequate ML model calibration, limited amount of historical data, missing inputs, etc).

72. Discussions
Finally, validation of the quality of data and its sources is a critical step for the application of ML models.
some extreme cases or outliers in the training data might get ignored by the data-driven algorithm,
the quality of the information should be verified by the user before its use.
Therefore QA/QC (quality assurance and control) techniques are of importance for data validation.

73. Conclusions ML models or data-driven techniques are:
additional tools for use in water resources engineering that
can complement, improve (or replace in some scenarios) physical-based models.
can capture complex nonlinear patterns and trends in the available data.
Physical-based models (e.g. rainfall-runoff) components allows for the analysis of their internal components,
Data-driven tools do not allow for a direct interpretation yet (black-box algorithms).

74. Conclusions
ML models in most cases have a better performance than a physical-based model;
Nevertheless they are limited by the information quality and availability.
The use of ML models is recommended under the following scenarios:
incomplete data to develop physical-based models,
extensive records of the phenomena and related causes or inputs,
data forecasting, and classification-type problems.

75. Thank you! 75