Areal Estimation techniques

Two types of technique:

1. Direct weighted averages

2. Surface fitting methods

DIRECT WEIGHTED AVERAGE METHODS

use the equation:

vg are weights between 0 and 1

pg are values of precip measured at each gage

G is the total number of gages

g are individual gages

P v pg g

g

G^

1

Arithmetic average

all gages have same weight

PG

p gg

G^

1

1

Thiessen Polygons

-area divided into G subregions each approximately centered on the G rain gages

-regions are such that all points in a subregion are closer to their own gage than any others

- once these polygons defined and areas (ag) calculated, weights calculated as vg=ag/A and spatial average calulated as

because

PA

a pg g

g

G^

1

1

a Ag

g

G

1

SURFACE-FITTING METHODS

Purpose is to use the gage values to identify a surface that represents the precipitation at all points in the basin (or region of interest)

Various approaches differ in methods used to construct the surface

Isohyetal concepts

calculates precipitation at all points in the ith subregion and pi- and pi+ are the values of isohyets the bound the ith subregion

regional average is calculated as

where αi is the area between the two contours within the region

Contours can be sketched by hand, but surface fitting and contouring programs much better

P P Pi i i

^

( ) 1

2

PA

pii

I

i

^

11

- With sufficient points, surface fitting can be done using contouring programs (e.g. surfer) that use statistical techniques (e.g. kriging) to describe the increasing difference or decreasing correlation between sample values as separation between them increases. The ultimate goal is to determine the value of a point in a heterogeneous grid from known values nearby.

Hypsometric method

Differs from other surface-fitting methods in that spatial average precip is not calculated from isohyets, but as a function of elevation. Not a complicated method, but many steps, so won’t be covered in detail here. Read in Dingman.

Briefly:1. Select elevation interval of basin2. Construct3. Construct precip vs elevation curve and determine precip at the

median of each interval4. Compute estimated areal average precip as: ( )P p Z ah

h

H

h

1

Precip at each median interval from step 3

Fractional area of interval

Comparison studies have shown that the optimal-interpolation/kriging method is by far the most accurate, but are also very complex

Choice should be based on:1. Objective,2. 3. time and computing resources available

Temporal Characteristics of Precipitation

- Our ability to forecast long-term temporal variation in precipitation is limited. The best tool hydrologists have for prediction is frequency analysis which allows the calculation of probabilities based on historical events

For annual precipitation:

-given enough years of record we can build a histogram of events which will take a gauzian shape

- the longer the record the better the confidence in the curve

- using standard statistical relationships...1 sd, 2sd

variance

65% within 1 SD

95% within 2 SD

99.7% within 3 SD

For lesser durations:

Depth-Duration-Frequency (DDF) analysis or IDF is used

We can calculate the exceedence probability (probability that a unit volume of precip in a unit time will be exceeded)

e.g. there is a 37% chance that the annual precipitation in Seattle will exceed 1.0 m

Inverse of exceedence probability is Return Period

Manipulating the probability

e.g. the return period of a 1 m annual rain in Seattle is 3 years

Duration, tr (min)

0 15 30 45 60 75 90 105 120 135 150 165 180

Inte

nsi

ty, i (m

m/h

r)

0

50

100

150

200

250

300

F requency100-year50-year25-year10-year5-year2-year

District 1

i = A/(B+tr)m

Frequency A B m2-year 1581 12.98 0.855-year 1995 14.19 0.8410-year 1874 13.14 0.8025-year 2175 13.49 0.7950-year 2121 12.64 0.76100-year 2603 13.71