1. Lecture 3 - 1 ERS 482/682 (Fall 2002) Information sources ERS 482/682 Small Watershed Hydrology

2. Lecture 3 - 2 ERS 482/682 (Fall 2002) Issues Model errors –Assumptions –Function Water balance

3. Lecture 3 - 3 ERS 482/682 (Fall 2002) Issues Model errors Measurement errors Figure 2 (Sullivan et al. 1996)

4. Lecture 3 - 4 ERS 482/682 (Fall 2002) Issues Model errors Measurement errors Spatial variability Temporal variability Figure 2 (Sullivan et al. 1996)

5. Lecture 3 - 5 ERS 482/682 (Fall 2002) Model errors Check and be aware of assumptions Calibration Validation Verification

6. Lecture 3 - 6 ERS 482/682 (Fall 2002) Measurement errors Estimate the error –Instrument error –Data error Standard deviation of normal distribution  95% probability that an error will be between ±1.96 SD of the true value

7. Lecture 3 - 7 ERS 482/682 (Fall 2002) frequency

8. Lecture 3 - 8 ERS 482/682 (Fall 2002) RELATIVE frequency

9. Lecture 3 - 9 ERS 482/682 (Fall 2002) Normal distribution Kurtosis: flat vs. peaked standard deviation mean particular error

10. Lecture 3 - 10 ERS 482/682 (Fall 2002) Descriptive statistics Mean, For errors, we hope this is 0!

11. Lecture 3 - 11 ERS 482/682 (Fall 2002) Measures of central tendency Mean Center of gravity Median Half the x-values are smaller and half are larger Mode Value of x with the largest frequency

12. Lecture 3 - 12 ERS 482/682 (Fall 2002) Descriptive statistics Mean, WHY?

13. Lecture 3 - 13 ERS 482/682 (Fall 2002) Descriptive statistics Mean, ss p =.95

14. Lecture 3 - 14 ERS 482/682 (Fall 2002) Measurement errors Estimating missing data (Sec. 4.2.3) –Station-average method –Normal-ratio method –Inverse-distance weighting –Regression

15. Lecture 3 - 15 ERS 482/682 (Fall 2002) Station-average method p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 G = # of gages with data Use when gage values are similar

16. Lecture 3 - 16 ERS 482/682 (Fall 2002) Normal-ratio method p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 G = # of gages with data P 0 = average annual precip at gage 0 P g = average annual precip at gage g Use when gage values are not similar

17. Lecture 3 - 17 ERS 482/682 (Fall 2002) Inverse-distance weighting p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 G = # of gages with data d g = distance of gage g from gage 0 b = 1 or 2

18. Lecture 3 - 18 ERS 482/682 (Fall 2002) Regression p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 G = # of gages with data b g = regression coefficient for gage g Caution: Series and data must be independent

19. Lecture 3 - 19 ERS 482/682 (Fall 2002) Spatial variability P = total precipitation on the watershed A = area of watershed

20. Lecture 3 - 20 ERS 482/682 (Fall 2002) Weighted averages w g = weight of gage g

21. Lecture 3 - 21 ERS 482/682 (Fall 2002) Weighted averages Thiessen polygons a g = area of subregion for gage g

22. Lecture 3 - 22 ERS 482/682 (Fall 2002) Weighted averages Isohyetal methods –isohyet: contour of equal precipitation a i = area of subregion between p i- and p i+ isohyets p 0.5 p 1.0 p 1.5 p 2.0 a2a2

23. Lecture 3 - 23 ERS 482/682 (Fall 2002) Temporal variability Exceedence probability or return period Stochastic hydrology (GEOL 702J) –PDF = probability distribution function f(x) = p (X = x) –1 – CDF exceedence probability1-F(x) = p(X > x) exceedence probability probability non-exceedence probability –CDF = cumulative distribution function F(x) = p (X  x)

24. Lecture 3 - 24 ERS 482/682 (Fall 2002) Displaying cumulative frequency f(x) Discharge (cfs) 123456More 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 +

25. Lecture 3 - 25 ERS 482/682 (Fall 2002) Displaying cumulative frequency f(x) Discharge (cfs) 123456More 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 +

26. Lecture 3 - 26 ERS 482/682 (Fall 2002) Displaying cumulative frequency F(x) 0 1

27. Lecture 3 - 27 ERS 482/682 (Fall 2002) Normal distribution CDF: Given x Find P(X  x)CDF PDF

28. Lecture 3 - 28 ERS 482/682 (Fall 2002) Normal distribution Non-exceedence probability: P (X  x) Exceedence probability: P (X>x)

29. Lecture 3 - 29 ERS 482/682 (Fall 2002) Return period 10-year design = or 1 – P(X  x) 50-year design = 100-year design = Does the 1-year storm occur every year???

30. Lecture 3 - 30 ERS 482/682 (Fall 2002) Normal distribution 10-year design = Given F(x)=P(X  x) What is x?

31. Lecture 3 - 31 ERS 482/682 (Fall 2002) Using the normal distribution as a model 70% non-exceedence probability

32. Lecture 3 - 32 ERS 482/682 (Fall 2002) Using the normal distribution as a model + + + + + + + + + + + + + 30% exceedence probability

33. Lecture 3 - 33 ERS 482/682 (Fall 2002) Figures 2-7, 2-8, 2-9 (Dunne & Leopold 1978)