Lecture ERS 482/682 (Fall 2002) Information sources ERS 482/682 Small Watershed Hydrology

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

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

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

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

Lecture 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

Lecture ERS 482/682 (Fall 2002) frequency

Lecture ERS 482/682 (Fall 2002) RELATIVE frequency

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

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

Lecture 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

Lecture ERS 482/682 (Fall 2002) Descriptive statistics Mean, WHY?

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

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

Lecture 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

Lecture 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

Lecture 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

Lecture 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

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

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

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

Lecture 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

Lecture 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)

Lecture ERS 482/682 (Fall 2002) Displaying cumulative frequency f(x) Discharge (cfs) More

Lecture ERS 482/682 (Fall 2002) Displaying cumulative frequency f(x) Discharge (cfs) More

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

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

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

Lecture 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???

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

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

Lecture ERS 482/682 (Fall 2002) Using the normal distribution as a model % exceedence probability

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