ERS 482/682 (Fall 2002) Lecture 3 - 1

Information sources

ERS 482/682Small Watershed Hydrology

ERS 482/682 (Fall 2002) Lecture 3 - 2

Issues

• Model errors– Assumptions– Function

QPET

Water balance

ERS 482/682 (Fall 2002) Lecture 3 - 3

Issues

• Model errors• Measurement errors

Figure 2 (Sullivan et al. 1996)

ERS 482/682 (Fall 2002) Lecture 3 - 4

Issues

• Model errors• Measurement errors

• Spatial variability• Temporal variability

Figure 2 (Sullivan et al. 1996)

ERS 482/682 (Fall 2002) Lecture 3 - 5

Model errors

• Check and be aware of assumptions• Calibration• Validation• Verification

ERS 482/682 (Fall 2002) Lecture 3 - 6

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

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

0

500

1000

1500

2000

2500

3000

1915 1935 1955 1975 1995

year

dis

char

ge (

cfs)

0

5

10

15

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

Mor

e

discharge (cfs)

f(x)

frequency

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

RELATIVE frequency

0

0.05

0.1

0.15

0.2

0

200

400

600

800

1000

1200

1400

1600

1800

2000

discharge (cfs)

f(x)

n ns,observatio # total

frequency

0

5

10

15

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

Mor

e

discharge (cfs)

f(x)

ERS 482/682 (Fall 2002) Lecture 3 - 9

Normal distribution

0

0.05

0.1

0.15

0.2

0.25

- 20 - 10 0 10 20

x

f(x)

2

2

1

2

1

x

exfKurtosis: flat vs. peaked

standard deviation

mean

particular error

ERS 482/682 (Fall 2002) Lecture 3 - 10

0

0.05

0.1

0.15

0.2

0.25

- 20 - 10 0 10 20

x

f(x)

Descriptive statistics

Mean,x

n

xx

n

ii

1

For errors, we hope this is 0!For errors, we hope this is 0!

ERS 482/682 (Fall 2002) Lecture 3 - 11

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

ERS 482/682 (Fall 2002) Lecture 3 - 12

0

0.05

0.1

0.15

0.2

0.25

- 20 - 10 0 10 20

x

f(x)

Descriptive statistics

Mean,x

2xx

n

ii xx

nss

1

22

1

1

WHY?

ERS 482/682 (Fall 2002) Lecture 3 - 13

0

0.05

0.1

0.15

0.2

0.25

- 20 - 10 0 10 20

x

f(x)

Descriptive statistics

Mean,x

s s

sx 96.1 sx 96.1p = .95

ERS 482/682 (Fall 2002) Lecture 3 - 14

Measurement errors

• Estimating missing data (Sec. 4.2.3)– Station-average method– Normal-ratio method– Inverse-distance weighting– Regression

ERS 482/682 (Fall 2002) Lecture 3 - 15

Station-average method

G

ggp

Gp

10

1ˆ

0p̂

p1

p2

p3

p4 p5

p6

p7G = # of gages with data

Use when gage values are similar

ERS 482/682 (Fall 2002) Lecture 3 - 16

Normal-ratio method

G

gg

g

pP

P

Gp

1

00

1ˆ

0p̂

p1

p2

p3

p4 p5

p6

p7

G = # of gages with dataP0 = average annual precip at gage 0Pg = average annual precip at gage g

Use when gage values are not similar

ERS 482/682 (Fall 2002) Lecture 3 - 17

Inverse-distance weighting

G

gg

bg pd

Dp

10

1ˆ

0p̂

p1

p2

p3

p4 p5

p6

p7

G = # of gages with datadg = distance of gage g from gage 0b = 1 or 2

G

g

bgdD

1

ERS 482/682 (Fall 2002) Lecture 3 - 18

Regression

GG pbpbpbbp 221100ˆ

0p̂

p1

p2

p3

p4 p5

p6

p7

G = # of gages with databg = regression coefficient for gage g

Caution: Series and data mustbe independent

ERS 482/682 (Fall 2002) Lecture 3 - 19

Spatial variability

A

dxdyyxpA

P ,1

P = total precipitation on the watershedA = area of watershed

ERS 482/682 (Fall 2002) Lecture 3 - 20

Weighted averages

G

ggg pwP

1

ˆ

wg = weight of gage g

G

ggw

11 10 gw

ERS 482/682 (Fall 2002) Lecture 3 - 21

Weighted averages

• Thiessen polygons

G

ggg pa

AP

1

1ˆ

A

aw g

g ag = area of subregion for gage g

ERS 482/682 (Fall 2002) Lecture 3 - 22

Weighted averages

• Isohyetal methods– isohyet: contour of equal precipitation

I

iii pa

AP

1ˆ

1ˆ

2

ˆ ii

ipp

p

ai = area of subregion between pi- and pi+ isohyets p0.5

p1.0

p1.5

p2.0

a2

ERS 482/682 (Fall 2002) Lecture 3 - 23

Temporal variability

• Exceedence probability or return period

Stochastic hydrology (GEOL 702J)

– PDF = probability distribution function• f(x) = p (X = x)

– 1 – CDF• 1-F(x) = p(X > x) exceedence probabilityexceedence probability

probabilityprobability

non-exceedence probabilitynon-exceedence probability

– CDF = cumulative distribution function• F(x) = p (X x)

ERS 482/682 (Fall 2002) Lecture 3 - 24

Displaying cumulative frequency

f(x)

Discharge (cfs)

1 2 3 4 5 6 More

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

+

ERS 482/682 (Fall 2002) Lecture 3 - 25

Displaying cumulative frequency

f(x)

Discharge (cfs)

1 2 3 4 5 6 More

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

+

ERS 482/682 (Fall 2002) Lecture 3 - 26

<12

34

56

More

0

0.2

0.4

0.6

0.8

1f(

x)

discharge (cfs)

Displaying cumulative frequency

F(x

)

0

1

ERS 482/682 (Fall 2002) Lecture 3 - 27

Normal distribution

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

x

f(x)

and F

(x)

CDF: Given x Find P(Xx)

CDFCDF

PDPDFF

ERS 482/682 (Fall 2002) Lecture 3 - 28

Normal distribution

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

x

F(x

)

Non-exceedenceprobability: P (Xx)

Exceedenceprobability: P (X>x)

ERS 482/682 (Fall 2002) Lecture 3 - 29

Return period

xXP

y,probabilit Exceedence

1 periodReturn

10-year design = 10.0

1

xXP or 1 – P(Xx)

50-year design = 02.0

1

xXP

01.0

1

xXP100-year design =

Does the 1-yearstorm occur

every year???

ERS 482/682 (Fall 2002) Lecture 3 - 30

Normal distribution

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

x

F(x

)

10-year design = 10.0

1

xXP

Given F(x)=P(Xx)What is x?

ERS 482/682 (Fall 2002) Lecture 3 - 31

Using the normal distribution as a model

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

x

f(x)

and F

(x)

70% non-exceedence probability

ERS 482/682 (Fall 2002) Lecture 3 - 32

Using the normal distribution as a model

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

x

f(x)

and F

(x)

+ +++

+

++

++

++++

30% exceedence probability

ERS 482/682 (Fall 2002) Lecture 3 - 33

Figures 2-7, 2-8, 2-9 (Dunne & Leopold 1978)