8/30/06 Colorado State University Department of Civil and Environmental Engineering CE 322
Precipitation by Jose D. Salas Table of Contents 1. Types of Precipitation 2. Formation of Precipitation 3. General Characteristics of Storm Types 4. Measurement of Precipitation 5. Temporal and Spatial Variability of Precipitation 6. Estimating Mean Precipitation Over an Area 6.1 Arithmetic mean 6.2 Thiessen polygons 6.3 Isohyetal method 6.4 Statistical spatial analysis techniques 7. Estimating Missing Precipitation Data 7.1 Causes of Missing Data 7.2 Methods for Estimating Missing Data 8. Consistency Analysis of Precipitation Data 8.1 Inconsistent Data 8.2 Methods of Consistency Analysis 9. Modification of Precipitation Problems References
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Colorado State University Department of Civil and Environmental Engineering CE 322
Precipitation by Jose D. Salas Precipitation is one of the most important components of the hydrologic cycle. It is the beginning of a whole chain of events that occurs in watersheds and river basins. Precipitation is the source of water that replenishes soil moisture, streamflows, lakes, glaciers, etc. In this chapter we study the typical forms of precipitation, how precipitation is formed, the general characteristics of storm types, how precipitation is measured, temporal and spatial variability of precipitation, the estimation of mean precipitation over an area, the estimation of missing precipitation data, consistency analysis of precipitation data, and modification of precipitation. 1. Types of Precipitation Precipitation is all forms of moisture emanating from the clouds and falling to earth. Typical examples are rainfall (rain), snowfall (snow), and hailstorm (hail). Rain is precipitation of liquid water while snow and hail contain water in its solid state. Raindrops generally have sizes bigger than 0.5 mm (0.02 in) in diameter. Rainfall is commonly reported as light, moderate, and heavy depending on its intensity, e.g. i ≤ 2.5 mm / hr is light rain while i > 7.6 mm / hr is heavy rain and moderate rain is in between. Snow is made up of ice crystals, which may reach several cm in diameter. The average density (specific gravity) of snow if about 0.1, e.g. 1000 mm of snow will produce 100 mm of liquid water on the average. Hail is precipitation in the form of ice. Hail may have smooth or irregular shapes, size ranging from about 5 to over 125 mm in diameter, and density of about 0.8. Other forms of precipitation are called drizzle, glaze, rime, and sleet (Chow, 1964; check also Linsley et al). Drizzle is a fine sprinkle of numerous but very small waterdrops, with diameter less than 0.02 in. (0.5 mm). The drops are so small that they appear to float in the air. Glaze (or freezing rain) occurs when rain falls into a cold layer of air and freezes when it strikes objects on the ground. Rime consists of white layers of ice crystals and forms when the droplets of under-cooled clouds and fogs strike obstacles. Sleet is melting snow or a mixture of snow and rain. 2. Formation of Precipitation
The presence of clouds in the atmosphere signals the conditions where precipitation may fall from the sky. However, we often see clouds just above the sky and no precipitation occurs. We will examine the physical mechanisms underlying the occurrence of precipitation. The necessary conditions for the occurrence of precipitation are: (a) creation of saturation conditions, (b) formation of water droplets or ice crystals, and (c) growth of water droplets or ice crystals. (a) Creation of saturation conditions. We know that as the air rises, the air temperature drops, and eventually the air can become saturated. There are certain physical mechanisms for the lifting of the air (and cooling of the air). Generally they are classified into three as shown below.
2
•
Cyclonic. It is the lifting of the air converging into a low pressure area. Frontal and nonfrontal cyclones are examples. Frontal cyclones are called extratropical because they generally occur beyond the tropics while non-frontal cyclones generally occur in tropical regions of the world. Examples of non-frontal cyclones are the hurricanes (typhoons). Other names associated with the formation of non-frontal cyclones are: -
Tropical depressions: Tropical storms: Hurricanes (typhoons):
v < 40 mph 40 ≤ v < 75 mph v ≥ 75 mph
A graphical sketch of frontal cyclones are shown below. The figure shows a warm front and a cold front. In both cases warm air is lifted as the air mass pushes through the converging boundary.
Warm air
Warm air Cold air Warm front
Cold air
Cold front Fig.1
Likewise the sketch below shows a schematic of the air flow for non-frontal cyclones occurring in the northern hemisphere.
Fig.2 •
Convective. Lifting of the air is produced by uneven heating of the earth’s surface. Uneven heating may occur because of different cover of the earth (vegetation, soils, etc.). Convective cooling generally produces thunderstorms.
•
Orographic. This may occur when the air flow is deflected upward by orographic barriers such as mountains.
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(a) Convective cooling
(b) Orographic cooling Fig.3
(b) Formation of Water Droplets or Ice Crystals When the air is close to saturation it requires the presence of certain nuclei for water droplets or ice crystals to form. Nuclei are small particles of various substances of the order of 0.1 – 10 µ m (10-6 m). They are called condensation nuclei when they lead to the formation of water droplets and freezing nuclei when they lead to ice crystals. nuclei
water droplet or ice crystal
•
Condensation nuclei - salt particles - combustion products - oxides of nitrogen
•
freezing nuclei - clay minerals - CO2 Fig. 4
(c) Growth of water droplets or ice crystals. Upon nucleation the water droplet or ice crystal grows to visible size in a fraction of a second through diffusion of water vapor into it, but growth thereafter is slow. Generally the growth of water droplets and ice crystals occur due to: -
diffusion of water vapor collision of droplets or ice crystals coalescence (molecular attraction of particles that are close to each other).
3. General Characteristics of Storm Types
We have seen in the previous section that air may become saturated because of various types of cooling mechanisms. The type of cooling generally identifies the type of storm that may be generated from them. They include: (a) Frontal cyclonic storms, (b) non-frontal cyclonic storms, (c) convective storms, and (d) orographic storms. Often though a storm may be primarily generated by a certain mechanism, e.g. a tropical cyclone, but at the same time may have the effect of other cooling mechanisms that may further enhance the cooling process. 4
Some general characteristics of the four types of storms are summarized below. (a) Storms caused by frontal cyclones (movement of extratropical cyclones generally at latitudes of 35 to 50 degrees North) • They generally cover large areas • Low to moderate intensity • Highest intensity associated with cold fronts because the steeper frontal surface gives a stronger lift • It travels in the direction west to east • It travels with a speed of about 30 mph (b) Storms caused by non-frontal cyclones (tropical cyclones) • Cooling and lifting of the air caused by convergence in the central core. • It originates with a weak disturbance along the thermal Equator • The air movement towards disturbance is deflected by the Coriolis force which forms a vortex. • The water vapor from a warm ocean is condensed by rising air at the center • The storm system generally moves westward due to the westerly wind system • The storm system decays when reaching land because of the greater ground friction and the fact that the moisture (energy) supply is cutoff . • Also it dissipates when reaching colder oceans. • They can be design storms for 1,000 to 2,000 sq.mi areas in coastal regions such as southern and southeastern USA, Philippines, Taiwan, and Japan. (c) Storms Caused by Thermal Convection (convective storms) • Cooling and lifting caused by thermal heating at the ground • Generally occurs in the afternoon and early evening after the earth has been thoroughly heated by the sun. • Typically covers small areas of the order of 5-10 sq-mi. although they may about 300 sq.mi. • High intensity over a small area and a large spatial and temporal variation of intensity. Sometimes accompanied by hail. • They serve to construct design storms for small basins. (d) Storms Caused by Orographic Lifting • Cooling and condensation caused by lifting of the moist air mass over a topographic divide. • Precipitation is a function of elevation, rate of rise and direction or moist air movement. • Generally widespread in a transverse direction along the orographic barrier. • Moderate to low intensity. • Duration of the storm persists as long as the moist air flow continues. • Orographic patterns often superimposed on other types of storms (e.g. a hurricane reaching an mountain range). 4. Measurement of Precipitation
4.1 Non-Recording and Recording Rain and Snow Gages 5
Traditionally precipitation has been measured by the so-called non-recording and recording gages. Non-recording gages measure the total precipitation that falls at a given point during a specified time interval ∆t. Generally precipitation readings are made on a daily basis although 6hr and 12-hr intervals are used sometimes. On the other hand, recording gages measure precipitation in continuous time. This allows determining precipitation intensities from the paper chart, i.e. i = ∆P/∆t where ∆P = amount of precipitation (typically inches, mm, or cm) accumulated in the gage during the time interval ∆t (typically in minutes). Often precipitation gages are installed connected with data collection platforms (DCP), which enables transmitting data to satellites, which in turn relay the information to a receiver station at the site of interest. Furthermore, during the past years advances in remote sensing technology has made it possible to estimate precipitation by using radar signals (ref.). Typical measurement equipment utilized in the United States includes the National Weather Service (NWS) non-recording rain gage as shown in Fig.5 (b). It has a circular orifice with 8 inches in diameter and has 20 inches of capacity. It measures rain and snow with a resolution of 0.01 inches. Fig.5(b) also shows a rain and snow gage utilized by the US Forest Service, which has 8.25 inches of diameter and capacity of 7 inches. Raingage shelters and screens are utilized to protect the rain or snow catch from the wind. Fig. 5(c) shows an example of a rain gage wind screen.
(a)
(b)
(c)
Fig.5 (a) NWS non-recording rain and snow gage, (b) US Forest Service rain and snow gage, and (c) rain gage with Alter-type wind screen (???) (source: http:/www.novalynx.com) ???? check with C. Fu) 4.2 Radar Measurement of Precipitation (check also Linsley) Radar is the acronym utilized for Radio-Detection-And-Ranging. It was developed during the World War II for detecting ships and aircrafts. Radar based techniques have been used for atmospheric observations in the United States since 1954. The current operating weather radar system in the US is the WSR-88D (Weather Surveillance Radar-1988 Doppler) system that consists of 158 NEXRAD (Next Generation Radar) sites across the U.S. (National Research Council, 2002, CoCoHaHS, 2005). Radars can be utilized from airplanes and satellites. NEXRAD are ground-based radars. 6
Radars detect objects in the atmosphere by sending out radio waves and then receiving the returning signals reflected from the objects. The returning signal is known as the reflectivity factor (Z). The values of Z can reveal the forms of precipitation. For example, the Z value for fog is 0.001 while for hail is 50,000,000. The value of Z in the logarithmic domain, i.e. dBZ = 10 × log10 ( Z ) are utilized to display the precipitation signals on radar screens. To estimate the amount of precipitation from the radar signal, the so-called “Z-R” relationship given by Z = A× RB
is employed where Z is the reflectivity factor, R is the rainfall rate, and A and B are parameters that are determined by assuming the rain drop size distribution. (????) The U.S. National Weather Service uses
Z = 300 × R1.4 or ⎛ Z ⎞ R=⎜ ⎟ ⎝ 300 ⎠
(1 / 1.4 )
for all the NEXRAD radar data in the U.S. (CoCoHaHS, 2005). Fig. 7 shows a picture of the NWS NEXRAD.
(a) (b) Fig. 7 (a) Outside view of a NWS NEXRAD radar, and (b) inside the radome view of a NWS NEXRAD radar (source: http://www.letxa.com/nexradtour.php) 5. Temporal and Spatial Variability of Precipitation
Precipitation data are generally highly variable in space and time depending on several factors. First of all, the temporal variability of precipitation depends on the time scale. If we measure the temporal variability by the coefficient of variation (Cv=S/m, i.e. standard deviation over the mean) one would expect that Cv decreases as the time step increases. For example, Fig. 8(a) shows a plot of the daily precipitation for …… Likewise, Figs. 8(b) and (c) show the corresponding plots for the monthly and annual precipitation data. And the values of Cv are respectively xx, yy, and zz. An important aspect of the temporal variability of precipitation is the effect of the so-called seasonality. This term reflects the effect of the annual revolution of the earth around the sun on a yearly basis (annual cycle). This causes the precipitation to be
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smaller during certain months of the year than during other months resulting in what is known as the dry and the wet periods. Figure 9 shows clearly the seasonality of precipitation for various locations of the United States. In addition, another feature of the temporal variability of precipitation pertains to the daily cycle that is exhibited in short-term precipitation such as hourly especially in mid-latitudes (reference). Furthermore, the variability of precipitation at all time scales is affected by the so-called low frequency, which means that precipitation statistics may evolve and change throughout time spans of several decades (e.g. refer to Gray et al to West Africa rainfall, etc.).
8
12 8 4 0 3
5 7 Month
9
11
6.0 4.5 3.0 1.5 0.0 1
3
5 7 Month
9
Precipitation, in inches
1
1.6
11
10 8 6 4 2 0 1
1.2
3
5 7 Month
9
11
0.8 0.4 0.0 3
5 7 Month
9
11
3.2
Precipitation, in inches
1
Precipitation, in inches
Precipitation, in inches
Precipitation, in inches
Precipitation, in inches
16
2.4 1.6 0.8 0.0 1
3
5 7 Month
9
11
8 6 4 2 0 1
3
5 7 Month
9
11
Fig. 9 Map of the mean monthly precipitation at some climatic divisions of the states of Florida, Vermont, Arkansas, Colorado, Arizona, and Washington, United States.
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Fig. 10. Mean annual precipitation (inches) based on normal period 1961-1990 (source: NCDC website at http://www5.ncdc.noaa.gov/cgi-bin/script/webcat.pl)
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6. Estimating Mean Precipitation Over an Area
• • • •
Arithmetic mean Thiessen polygons Isohyetal method Statistical spatial analysis techniques
7. Estimating Missing Precipitation Data
7.3 Causes of Missing Data It is common to find in actual practice that precipitation data may be missing from the set of records. There are several reasons why precipitation data may be missing from the database. Some of them are listed below. • • • • •
Malfunctioning of precipitation gage and/or related equipment Effect of natural hazards (floods, landslides, hurricanes, etc.) Sabotage (terrorism) Human related problems (temporary absence of people in charge of reading gages, mistakes in handling data, etc.) Others
7.4 Methods for Estimating Missing Data Several methods are available for estimating missing precipitation data. Some methods are quite simple but others are more involved requiring some expertise in statistical analysis, time series analysis, neural networks, spatial analysis, etc. The type of methods to apply fore a particular case depend whether the missing data are temporal or spatial data. The method to apply for temporal data depends on the temporal resolution (e.g. annual, monthly, daily, hourly, etc.) and the length (amount) of missing data. Also the applicability of certain methods (e.g. methods based on time series analysis) depends on the available record length. The methods that we are going to discuss in this section are rather simple. Also note that for certain type of problems involving missing data in space the methods that we have discussed preciously for estimating mean precipitation over an area may be applicable and useful. • • • • •
Arithmetic mean Normal ratio method Inverse distance method Correlation and regression Others
(a) Arithmetic mean This method may be applied when the normal annual precipitation at the site of interest (site with the missing record) is within 10% of the normal annual precipitation at other sites where concurrent data are available. (need further comments on: typical uses of this method, - for example for preliminary studies -, reliability of the method, etc.) 11
(b) Normal ratio method Assume that we have gaging stations A, B, C, and Y where the normal annual precipitation NP(A), NP(B), NP(C), and NP(Y) are available. Also assume that on a given rainy day t, the precipitation at gage Y is missing. We would like to estimate Pt(Y) based on the gage readings at the other sites. The formula used is: ⎫ 1 ⎧ NP (Y ) NP (Y ) NP (Y ) Pt (Y ) = ⎨ Pt ( A) + Pt ( B) + Pt (C )⎬ 3 ⎩ NP ( A) NP ( B) NP (C ) ⎭
(1)
where Pt(Z) = precipitation at gage Z during the day t. This is a better method than the arithmetic one and is usually applied when the normal annual precipitation at the site with the missing record differs by more than 10% of the normal annual precipitation at the other sites where the concurrent data are available. Note that the method has been described assuming daily data, however it is also applicable for any type of data such as weekly, monthly, etc. Also it is applicable for any type of hydrometeorological data such as temperature, pan evaporation, etc. (need further comments on: the cases where there may be fewer than 3 sites, say 2 sites, about normal values, etc.) Example 1: Assume that the following data are available: Pt(A) = 98 mm, NP(A) = 1,008 mm Pt(B) = 80 mm, NP(B) = 842 mm NP(Y) = 880 mm Pt(Y) = ? Applying the normal ratio method to the available data we have: Pt (Y ) =
1 ⎧ 880 880 ⎫ 98 + 80⎬ = 84.6 mm ⎨ 2 ⎩1,008 842 ⎭
(c) Inverse Distance Method Referring to the sketch below let us define the following information: dZ = distance between gage Y and Z, where Z =A, B, etc. Pt(Z) = precipitation at gage Z during time t, where Z =A, B, etc. Pt(Y) = missing precipitation at gage Y
A
B dA
dB Y
D
dD
dC
C
Fig. 5 12
The estimated missing precipitation at site Y by the inverse distance method is: 2
2
2
2
⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎟⎟ Pt (C ) + ⎜⎜ ⎜⎜ ⎟⎟ Pt ( A) + ⎜⎜ ⎟⎟ Pt ( B ) + ⎜⎜ ⎟⎟ Pt ( D) d d d d ⎝ B⎠ ⎝ D⎠ ⎝ C⎠ Pt (Y ) = ⎝ A ⎠ 2 2 2 2 ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎟⎟ + ⎜⎜ ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ ⎝ dA ⎠ ⎝ dB ⎠ ⎝ dD ⎠ ⎝ dC ⎠
(2)
Note that Eq.(2) uses the squares of the inverse distances, however other power exponents have been suggested (refs.) Example 2: Assume that three gages A, B, and Y are available. The data for gage Y is missing. Given that the following data are available, Pt(A) = 98 mm, Pt(B) = 80 mm, Pt(Y) = ? dA = 2 Km dB = 3 Km We would like to estimate the missing data at Y by using the inverse distance method. Applying Eq. (2) for the available data we have: 2
⎛ 1 ⎞ ⎛ 1 ⎜⎜ ⎟⎟ Pt ( A) + ⎜⎜ d ⎝ dB Pt (Y ) = ⎝ A ⎠ 2 ⎛ 1 ⎞ ⎛ 1 ⎜⎜ ⎟⎟ + ⎜⎜ ⎝ dA ⎠ ⎝ dB
2
⎞ ⎟⎟ Pt ( B) ⎠ =
⎞ ⎟⎟ ⎠
2
2
2
⎛1⎞ ⎛1⎞ ⎜ ⎟ 98 + ⎜ ⎟ 80 ⎝ 3⎠ ⎝ 2⎠ = 92.5 mm 2 2 ⎛1⎞ ⎛1⎞ ⎜ ⎟ +⎜ ⎟ ⎝ 2⎠ ⎝ 3⎠
(d) Method based on regression analysis Assume that two precipitation gages Y and X have long records of annual precipitation, i.e. Y1 , Y2 , ..., YN and X 1 , X 2 , ..., X N . The precipitation Yt is missing. We will fill in the missing data based on a simple linear regression model. The model can be written as: Yt = a + b X t
(3)
in which the parameters a and b can be estimated by:
aˆ = Y − bˆ X
(4)
S Y* ˆ b = rXY * SX
(5)
where Y and X are the sample means, S Y* and S X* are the sample unbiased standard deviations of Y and X, respectively, and rXY is the cross-correlation coefficient between X and Y. The latter term can be estimated as: 1 N ∑ [( X t − X )(Yt − Y )] N i =1 rXY = (6) S X SY 13
where S Y and S X are the sample biased standard deviations (add comments regarding: the limitation of the method, not enough data, degree of correlation, sample size, etc.) Example 3. The annual precipitation data for two sites are shown in the table below. The record for year 1971 (i.e. Y11) is missing. We would like to complete the record by using the simple linear regression method. We will use Eq.(3) to estimate the missing observation based on the concurrent record Xt. Year t 1961 1 1962 2 1963 3 1964 4 1965 5 1966 6 1967 7 1968 8 1969 9 1970 10 1971 11 Means: Biased stand. dev. S Unbiased stand. dev. S*
Xt 10.2 8.5 2.8 5.4 15.2 10.1 9.2 14.8 11.1 6.6 3.3 9.39 3.675 3.874
Yt 6.1 9.5 1.8 6.0 9.5 5.3 2.3 11.6 7.7 3.0
Equation (6) is applied to calculate the cross-correlation coefficient using the concurrent data 1961-1970. It gives rxy= 0.751. Then, from Eqs.(5) and (4) we get respectively: bˆ = (0.75 × 3.313) / 3.874 = 0.642 aˆ = 6.28 − 0.642 × 9.39 = 0.249 The missing data is obtained from (3) which gives: Y11 = 0.249 + 0.642 × 3.3 = 2.37 The fitted regression line and the scatter plot are shown on the figure below.
6.28 3.143 3.313
Precipitation at station Y
14 12 10 8 6 4 2 0 0
5
10
15
20
Precipitation at station X
Fig.6 8. Consistency Analysis of Precipitation Data
8.1 Inconsistent Data The precipitation measured at a given gage Y is inconsistent with that measured at another gage X when the time series pattern of Y and related statistics (e.g. the mean) are different than 14
those of X. Conversely, analyzing the consistency of data at Y is meant in relation to another data set, e.g. X, i.e. consistency analysis is testing that the time series pattern and/or statistical characteristics of Y are similar to those of X. It is assumed that the time series Xt is reliable and its pattern and statistics are a true representation of the natural variability that is expected at the site. Consider the two cases shown in the figure below. In Fig.(a) precipitation series Yt appears to be inconsistent (relative to X) because its time series pattern is markedly different than that of Xt . On the other hand series Yt in Fig.(b) appears to be consistent with series Xt. Yt
Yt
t Xt
t Xt
t
t
Fig.7 (a)
Fig.7 (b)
One may like to test the consistency of the data Yt when there is some evidence that the data are not be reliable, the data may have been subjected to systematic (non-random) errors, the equipment (gage) in recent years may be significantly different than that used in previous years, or the gage may have been moved in recent years to another location. The time series plot of the suspected series compared to other series in the study area (as shown in Figs. a and b) may suggest a possible inconsistency. Generally though exhaustive graphical and statistical procedures must be followed. 8.2 Methods of Consistency Analysis One may like to test the hypothesis that a given data set is consistent. Rejection of this hypothesis will imply that the data are inconsistent and accordingly one must adjust the records. Conversely, non-rejection of the hypothesis will imply that the data set is consistent and no adjustment is necessary. There are a number of methods and procedures that can be utilized for testing the consistency hypothesis of a given data set. Some of them are simple graphical procedures while others are statistically based. Sometimes both graphical and statistical procedures can be combined. Among the graphical procedures the so-called double mass method is the traditional one and perhaps the most widely used in practice. (comment on the effect of outliers on the double mass) (a) Double mass method. This is essentially a simple graphical method but statistical concepts and tests can be also utilized. Let us assume that we wish to check whether the data y1 , y 2 , ..., y N (N= sample size) are consistent data or not. For this purpose we will use another data set x1 , x 2 , ..., x N , which is known to be reliable. The latter data set could be data measured at another gage or more generally the average of the data records available at 15
several sites located in the same region as the suspected gage y. cumulative partial sums as
We will define the
S t ( y ) = S t −1 ( y ) + y t , t = 1,2,..., N
(7a)
S t ( x) = S t −1 ( x) + xt , t = 1,2,..., N
(7b)
in which S 0 ( x) = S 0 ( y ) = 0 . Thus we have two sequences of partial sums, namely: S1 ( y ), S 2 ( y ),..., S N ( y ) , and S1 ( x), S 2 ( x),..., S N ( x) . The double mass plot is constructed as shown in the sketch below. St(y)
s2
St(y)
s1 St(x)
St(x)
Fig. 8(a)
Fig.8 (b)
Figure 8(a) reflects the inconsistency shown in Fig.7(a) while Fig.8(b) reflects that series yt in Fig.7(b) has a pattern consistent with that of series xt. Referring to Fig.8(a) the point where the slope changes from s1 to s2 marks the time where the inconsistency occurs. Therefore part of the record of y must be adjusted. Which part of the record must be adjusted (before the break point or after) depends on the particular case. For example, if the inconsistency in y has occurred because the gage was moved from the original location to another location (the current location), then it is logical to adjust the record before the break point. It may be shown that the first part of the record may be adjusted by y t′ = ( s 2 / s1 ) y t . On the other hand, if the second part of the record is needed to be adjusted use instead y t′ = ( s1 / s 2 ) y t . (show in Fig.8 a schematic about the effect of outliers). Example 4. The precipitation for a certain basin has been recorded for several years at 5 gages. The data at one of the gages (site Y) is suspected to be inconsistent relative to that at the other four sites. The table below shows data for Y and the average data for the other four sites (called X). (a) Analyze the consistency of the data at Y by the double mass method. (b) Adjust the data of the earlier years at Y (so that they are consistent with those of recent years). Equations (7a) and (7b) are applied to obtain the cumulative sums St(y) and St(x). The values are shown in columns 4 and 5 of the table (shown below), respectively. They are plotted as shown in Fig. 9. The figure shows that there is a break in the slope at year 1983. Therefore we will need to adjust the precipitation values for the gage for the year 1979-1982. For example, for year 1979 the adjusted precipitation becomes: ′ = (1.035 / 0.763) × 668 = 906. The rest of the adjusted values are shown with an asterisk y1979 in the table below. 16
Year
Gage yt
1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
Average of 4 gages xt
668 570 681 601 483 1186 940 616 950 773 646
780 762 878 754 761 1056 887 656 791 840 706
St(y)
St(x)
0 668 1238 1919 2520 3003 4189 5129 5745 6695 7468 8114
0 780 1542 2420 3174 3935 4991 5878 6534 7325 8165 8871
Adjusted gage y t′ 906* 773* 923* 815* 483 1186 940 616 950 773 646
Cumulative sum of y
10000 8000 6000 4000 2000 0 0
2000
4000
6000
8000
10000
Cumulative sum of x
Fig.9 (b) Statistical tests. A number of statistical tests can be applied for consistency analysis of precipitation data. In fact, the double mass method as described above can be used in conjunction with a statistical method. For example one could test whether the slope s2 is different than the slope s1. Other tests that can be applied include the t-test, F-test, and a number of non-parametric tests (refs.) 9. Modification of Precipitation
17
Problems
1. Annual precipitation data for two sites y and x for the period 1979-1989 are shown in the table below. You may observe that the data of 1983 for y is missing. Use the simple linear regression method to fill in such missing value. Year
Gage yt
1979 1980 1981 1982 1983 1984
668 570 681 601 1186
Average of 4 gages xt 780 762 878 754 761 1056
Year
Gage yt
1985 1986 1987 1988 1989
940 616 950 773 646
Average of 4 gages xt 887 656 791 840 706
2. Put a similar example as # 4 but adjustment on after the break (see Ex. #1) 3. Same as Ex. #1 but use arithmetic method and compare w/normal ratio. For normal ratio add another station. 4. Same as Ex. #2 but add another point. 5. Provide a modification of Ex.#4 to include an outlier. 6. The precipitation data of a certain basin have been recorded for several years at 5 precipitation gages. The data at one of the gages (site Y) is suspected to be inconsistent relative to that at the other four sites. The table below shows data for Y and the average of the data for the other four sites (called X). (a) Analyze the consistency of the data at Y by the double mass method. (b) Adjust the data of the earlier years at Y (so that they are consistent with those of recent years). (c) You may observe that the data of 1989 for Y is missing. Use the simple linear regression method to fill in such missing value. Year 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
Gage Y 668 570 681 601 483 1186 940 616 950 773 missing
Average of 4 gages X 780 762 878 754 761 1056 887 656 791 840 706 18
SY
SX
7. Explain how precipitation occurs, i.e. how it is formed and what are the key factors involved. 8. For the precipitation networks shown below draw the corresponding Thiessen polygons.
•
• xx
•
•
•
•
•
9. The precipitation at gages A and B are 4 and 6 inches, respectively. For the case shown in the graph below estimate the missing precipitation at site C. The distances between sites are given. Note that the method to apply in this case is the quadrant method (inverse distance), which requires estimating weights as a function of distance. The weights can be determined by: 1 d2 ωi = p i , p = number of points utilized ⎛ 1⎞ ∑j =1 ⎜⎝ d 2 ⎟⎠ j
A
10 mi
6 mi
B
8 mi C
10. A parcel of air at 800 mb pressure has a temperature of 9 EC and 80% of relative humidity. Calculate the specific humidity of the pocket of air. 11. Compute the mean annual precipitation for the watershed shown below by using the arithmetic, Thiessen polygons and isohyetal methods. (V. Singh) 19
• 80
60 •
• 65
•100 • 106
• 108 • 105
64 •
• 90 90 •
• 70
75 •
• 38
12. Do Problem 2.6 of the textbook (p.38). 13. 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, and 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. 14. The annual precipitation at station X and the average annual precipitation at 15 surrounding stations are as shown in the table below. Data for Station X is suspected to be inconsistent while data for the other fifteen stations appear to be reliable. (a) Check the consistency of the data at X by using the data at the other fifteen stations. (b) In what year a change in regime occurs? (c) Adjust the records of site X assuming that the change in regime occurred because the gaging station was moved to a new location. (d) Compare the mean and the standard deviation of the original record versus those for the record after adjustment. Is there any significant difference? Year
Station X
1950
47
Average for 15 Stations 29
1951
24
21
1952
42
36
20
1953
27
26
1954
25
23
1955
35
30
1956
29
26
1957
36
26
1958
37
26
1959
35
28
1960
58
40
1961
41
26
1962
34
24
1963
20
22
1964
26
25
1965
36
34
1966
35
28
1967
28
23
1968
29
33
1969
32
33
1970
39
35
1971
25
26
1972
30
29
1973
23
28
1974
37
34
1975
34
33
1976
30
35
1977
28
26
1978
27
25
1979
34
35
15. For the precipitation networks shown below draw the corresponding Thiessen polygons.
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16. The precipitation for a certain basin has been recorded for several years at 5 gages. The data at one of the gages (site Y) is suspected to be inconsistent relative to that at the other four sites. The table below shows data for Y and the average data for the other four sites (called X). (a) Analyze the consistency of the data at Y by the double mass method. (b) If the data Y is inconsistent adjust the data of the later years at Y (so that they are consistent with those of earlier years). The adjustment can be done by y t′ = r y t where r = ratio of the two slopes
Year
Gage yt
Average of 4 gages xt
1979 1980 1981 1982 1983 1984 1985 1986 1987
670 570 680 600 480 1200 980 680 990
780 760 880 750 760 1100 890 650 790
References
Chow, Ven Te, 1964, Handbook of Applied Hydrology: A Compendium of Water-resources Technology, McGraw – Hill Book Company. CoCoHaHS, 2005, Weather Radar Basics, http://www.cocorahs.org/media/docs/radar_basics.pdf 22
Collier, Christopher G., 1996, Applications of Weather Radar System: A Guide to Uses of Radar Data in Meteorology and Hydrology, second edition, John Wiley & Sons. Linsley et al, 1986 National Research Council, 2002, Weather Radar Technology-Beyond NEXRAD, National Academy Press, Washington D.C. Tabios, G. and Salas, J.D., 1985.
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