## ABSTRACT

Assume that the z value for a p and b p [i.e., z(a p , b p )] needs to be found with the known values at the four points as z(a 1, b 1), z(a 1, b 2), z(a 2, b 1), and z(a 2, b 2) as shown in Figure 9.1. a and b are coordinates and z(a, b) is the corresponding precipitation value for the coordinates as an example. At first, the z values of two points at (a p , b 1) and (a p , b 2) must be defined as z ( a p , b 1 ) = a 2 − a p a 2 − a 1 z ( a 1 , b 1 ) + a p − a 1 a 2 − a 1 z ( a 2 , b 1 ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429459580/1ca04197-23e8-41ae-92cf-91f6e0fd55ca/content/equ9_0001.tif"/> z ( a p , b 2 ) = a 2 − a p a 2 − a 1 z ( a 1 , b 2 ) + a p − a 1 a 2 − a 1 z ( a 2 , b 2 ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429459580/1ca04197-23e8-41ae-92cf-91f6e0fd55ca/content/equ9_0002.tif"/> z ( a p , b p ) ≈ b 2 − b p b 2 − b 1 z ( a p , b 1 ) + b p − b 1 b 2 − b 1 z ( a p , b 2 ) = 1 ( a 2 − a 1 ) ( b 2 − b 1 ) [ a 2 − a p    a p − a 1 ]     [ z ( a 1 , b 1 ) z ( a 1 , b 2 ) z ( a 2 , b 1 ) z ( a 2 , b 2 ) ]   [ b 2 − b p   b p − b 1 ] https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429459580/1ca04197-23e8-41ae-92cf-91f6e0fd55ca/content/equ9_0003.tif"/> Example 9.1

Define the daily precipitation z value at a p = 4, b p = 4 [i.e., z(4, 4)] while a 1 = 1, b 1 = 1 and a 2 = 5, b 2 = 5. Assume the z values of z(a 1, b 1) = 5 mm, z(a 1, b 2) = 10 mm, z(a 2, b 1) = 20 mm, and z(a 2, b 2) = 30 mm are known. Then,

Solution:

z ( 4 , 4 ) = 1 ( 5 − 1 ) ( 5 − 1 ) [ 5 − 4 4 − 1 ] [ 5 10 20 30 ] [ 5 − 4 4 − 1 ] = 22.81   mm https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429459580/1ca04197-23e8-41ae-92cf-91f6e0fd55ca/content/equ9_0004.tif"/>