HYDROLOGY
FOR
ENGINEERS
(SOLVED PROBLEMS)
, PRECIPITATION
1. Assuming a rain falling vertically, express the catch of a gage inclined 20° from
the vertical as a percentage of the catch for the same gage installed vertically.
GIVEN: 𝑖𝑛𝑐𝑙𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑔𝑎𝑔𝑒 = 20°
REQUIRED: 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑜𝑓 𝑐𝑎𝑡𝑐ℎ
SOLUTION: 𝑈𝑠𝑒 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎:
𝑃𝑟𝑜𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑟𝑒𝑎
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐𝑎𝑡𝑐ℎ = 𝑥 100%
𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑟𝑒𝑎
𝐴 𝑐𝑜𝑠20°
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐𝑎𝑡𝑐ℎ = 𝑥 100%
𝐴
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐𝑎𝑡𝑐ℎ = 𝟗𝟑. 𝟗𝟕%
2. Shielded and unshielded gages at a station indicate a storm rainfall of 100 and
90 mm, respectively. Estimate the actual rainfall. Assume b = 1.7.
𝑮𝑰𝑽𝑬𝑵: 𝑠ℎ𝑖𝑒𝑙𝑑𝑒𝑑 𝑔𝑎𝑔𝑒 (𝑃𝑆𝐺) = 𝟏𝟎𝟎 𝒎𝒎.
𝑢𝑛𝑠ℎ𝑖𝑒𝑙𝑑𝑒𝑑 𝑔𝑎𝑔𝑒 (𝑃𝑈𝐺) = 𝟗𝟎 𝒎𝒎.
𝑐𝑎𝑙𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 (𝑏) = 1.7
REQUIRED: 𝑎𝑐𝑡𝑢𝑎𝑙 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 (𝑃𝐴)
SOLUTION: 𝑈𝑠𝑒 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (3 − 1) 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑏𝑜𝑜𝑘:
𝑃𝑈𝐺 𝑃𝑈𝐺
ln ( ) = 𝑏 ln ( )
𝑃𝐴 𝑃𝑆𝐺
90 90
ln ( ) = 1.7 ln ( )
𝑃𝐴 100
𝑃𝐴 = 𝟏𝟎𝟕. 𝟔𝟓𝟒 𝒎𝒎.
, 3. With a Z value of 300,000 mm6/m3, what are the rainfall rates, in mm/hr,
indicated by the Z-R relationships with a and b values of (a) 200 and 1.6 and (b)
300 and 1.4, respectively?
𝑮𝑰𝑽𝑬𝑵: 𝑍 = 300,000 𝑚𝑚6/𝑚3
(𝑎) 𝑎 = 200 𝑎𝑛𝑑 𝑏 = 1.6(𝑏) 𝑎 = 300 𝑎𝑛𝑑 𝑏 = 1.4
𝑹𝑬𝑸𝑼𝑰𝑹𝑬𝑫: 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 𝑟𝑎𝑡𝑒 (𝑅)
𝑺𝑶𝑳𝑼𝑻𝑰𝑶𝑵: 𝑈𝑠𝑒 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 :
𝑅 = 𝑎𝑍 𝑏
For: a = 200 and b = 1.6
𝑅 = (200) (300 000)1.6
𝑅 = 𝟏. 𝟏𝟔𝟎 𝒙 𝟏𝟎𝟏𝟏 𝒎𝒎/𝒉𝒓
For: a = 300 and b = 1.4
𝑅 = (300) (300 000)1.4
𝑅 = 𝟏. 𝟑𝟗𝟕 𝒙 𝟏𝟎𝟏𝟎 𝒎𝒎/𝒉𝒓
4. Comparison of rain-gage measurements with rainfall rates estimated by radar
using the standard Z-R relationship shows that the rates measured by the gage
are 2.5 times those indicated by radar. If the b exponent is assumed to be correct,
what is the value of a?
5. Precipitation station X was inoperative for part of a month during which a storm
occurred. The respective storm totals at three surrounding stations, A, B, and C,
were 98, 80, 110 mm. The normal annual precipitation amounts at stations X, A,
B, and C, are, respectively, 880, 1008, 842, and 1080 mm. Estimate the storm
precipitation for station X.
GIVEN: storm precipitation at A = 98 mm.
storm precipitation at B = 80 mm.
storm precipitation at C = 110 mm.
normal annual precipitation at X = 880 mm.
normal annual precipitation at A = 1008 mm.
normal annual precipitation at B = 842 mm.
normal annual precipitation at C = 1080 mm.
FOR
ENGINEERS
(SOLVED PROBLEMS)
, PRECIPITATION
1. Assuming a rain falling vertically, express the catch of a gage inclined 20° from
the vertical as a percentage of the catch for the same gage installed vertically.
GIVEN: 𝑖𝑛𝑐𝑙𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑔𝑎𝑔𝑒 = 20°
REQUIRED: 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑜𝑓 𝑐𝑎𝑡𝑐ℎ
SOLUTION: 𝑈𝑠𝑒 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎:
𝑃𝑟𝑜𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑟𝑒𝑎
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐𝑎𝑡𝑐ℎ = 𝑥 100%
𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑟𝑒𝑎
𝐴 𝑐𝑜𝑠20°
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐𝑎𝑡𝑐ℎ = 𝑥 100%
𝐴
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐𝑎𝑡𝑐ℎ = 𝟗𝟑. 𝟗𝟕%
2. Shielded and unshielded gages at a station indicate a storm rainfall of 100 and
90 mm, respectively. Estimate the actual rainfall. Assume b = 1.7.
𝑮𝑰𝑽𝑬𝑵: 𝑠ℎ𝑖𝑒𝑙𝑑𝑒𝑑 𝑔𝑎𝑔𝑒 (𝑃𝑆𝐺) = 𝟏𝟎𝟎 𝒎𝒎.
𝑢𝑛𝑠ℎ𝑖𝑒𝑙𝑑𝑒𝑑 𝑔𝑎𝑔𝑒 (𝑃𝑈𝐺) = 𝟗𝟎 𝒎𝒎.
𝑐𝑎𝑙𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 (𝑏) = 1.7
REQUIRED: 𝑎𝑐𝑡𝑢𝑎𝑙 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 (𝑃𝐴)
SOLUTION: 𝑈𝑠𝑒 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (3 − 1) 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑏𝑜𝑜𝑘:
𝑃𝑈𝐺 𝑃𝑈𝐺
ln ( ) = 𝑏 ln ( )
𝑃𝐴 𝑃𝑆𝐺
90 90
ln ( ) = 1.7 ln ( )
𝑃𝐴 100
𝑃𝐴 = 𝟏𝟎𝟕. 𝟔𝟓𝟒 𝒎𝒎.
, 3. With a Z value of 300,000 mm6/m3, what are the rainfall rates, in mm/hr,
indicated by the Z-R relationships with a and b values of (a) 200 and 1.6 and (b)
300 and 1.4, respectively?
𝑮𝑰𝑽𝑬𝑵: 𝑍 = 300,000 𝑚𝑚6/𝑚3
(𝑎) 𝑎 = 200 𝑎𝑛𝑑 𝑏 = 1.6(𝑏) 𝑎 = 300 𝑎𝑛𝑑 𝑏 = 1.4
𝑹𝑬𝑸𝑼𝑰𝑹𝑬𝑫: 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 𝑟𝑎𝑡𝑒 (𝑅)
𝑺𝑶𝑳𝑼𝑻𝑰𝑶𝑵: 𝑈𝑠𝑒 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 :
𝑅 = 𝑎𝑍 𝑏
For: a = 200 and b = 1.6
𝑅 = (200) (300 000)1.6
𝑅 = 𝟏. 𝟏𝟔𝟎 𝒙 𝟏𝟎𝟏𝟏 𝒎𝒎/𝒉𝒓
For: a = 300 and b = 1.4
𝑅 = (300) (300 000)1.4
𝑅 = 𝟏. 𝟑𝟗𝟕 𝒙 𝟏𝟎𝟏𝟎 𝒎𝒎/𝒉𝒓
4. Comparison of rain-gage measurements with rainfall rates estimated by radar
using the standard Z-R relationship shows that the rates measured by the gage
are 2.5 times those indicated by radar. If the b exponent is assumed to be correct,
what is the value of a?
5. Precipitation station X was inoperative for part of a month during which a storm
occurred. The respective storm totals at three surrounding stations, A, B, and C,
were 98, 80, 110 mm. The normal annual precipitation amounts at stations X, A,
B, and C, are, respectively, 880, 1008, 842, and 1080 mm. Estimate the storm
precipitation for station X.
GIVEN: storm precipitation at A = 98 mm.
storm precipitation at B = 80 mm.
storm precipitation at C = 110 mm.
normal annual precipitation at X = 880 mm.
normal annual precipitation at A = 1008 mm.
normal annual precipitation at B = 842 mm.
normal annual precipitation at C = 1080 mm.
