Example 3.3: Rainfall reported at a group of five stations (see Fig. 3.7) is as follows. Kundla. Sabli

Size: px

Start display at page:

Download "Example 3.3: Rainfall reported at a group of five stations (see Fig. 3.7) is as follows. Kundla. Sabli"

James Ramsey
6 years ago
Views:

1 3.4.4 Spatial Cosistecy Check Raifall data exhibit some spatial cosistecy ad this forms the basis of ivestigatig the observed raifall values. A estimate of the iterpolated raifall value at a statio is obtaied o the basis of the weighted average of raifall observed at the surroudig statios. If the differece betwee the observed ad the estimated values exceed the expected limitig value, such values are cosidered as suspect ad are flagged for further ivestigatio ad ascertaiig the possible causes of departures. Spatial cosistecy checks for raifall data are carried out by relatig the observatios from surroudig statios for the same duratio with the raifall observed at the statio. This is achieved by iterpolatig the raifall at the statio uder questio with raifall data of eighborig statios. The statio beig cosidered is called the test statio. The iterpolated value is estimated by computig the weighted average of the raifall observed at eighborig statios. Ideally, the statios selected as eighbors should be physically represetative of the area i which the statio uder scrutiy is situated. The followig criteria are used to select the eighborig statios: (a) (b) (c) The distace betwee the test ad the eighborig statio must be less tha a specified maximum correlatio distace; too may eighborig statios should ot be cosidered for iterpolatio; ad to reduce the spatial bias i selectio, it is advisable to cosider a equal umber of statios i each quadrat. Example 3.3: Raifall reported at a group of five statios (see Fig. 3.7) is as follows. Chado Kudla Virpur Idar Sabli Fig. 3.7 Locatio of statios for spatial cosistecy check. Statio Kudla Idar Virpur Chadop Sabli Raifall Durig the quality cotrol process, the data at Idar is idetified as doubtful. Check this data for spatial cosistecy.

2 Solutio: The raifall at Idar is estimated usig the distace power method ad compared with the observed value. From the four quadrats aroud Idar (Fig. 3.6), the statio earest from each quadrat is selected for estimatio of raifall at Idar. Usig the referece coordiate system, the distace of each of the estimator statios from Idar is determied ad the raifall at Idar is estimated. S. N. Statio Distace from Idar D i 2 1/D i 2 R i /D i (km) 1. Kudla * Virpur * Sabli * Total 14.02* Raifall at Idar = [(R i /D i 2 )] / [(1/D i 2 )] = 0.17/14.02*10-4 = mm. Sice the observed value is very much differet from the estimated value, it is rejected ad replaced by the estimated value. Note that there is a possibility that the decimal poit was wrogly placed while recordig the data at Idar. 3.5 Spatial Averagig of Raifall Data Precipitatio observatios from gauges are poit measuremets. However, i the hydrological aalysis ad desig, we frequetly require mea areal precipitatio over a area. A characteristic of the precipitatio process is that it exhibits appreciable spatial variatio though the values at relatively short distaces may have good correlatio. Numerous methods of computig areal raifall from poit measuremets have bee developed. While usig precipitatio data, oe ofte comes across missig data situatios. Data for the period of missig raifall could be filled usig various techiques. Due to the spatial structure of precipitatio data, some type of iterpolatio makig use of the data of earby statios is commoly adopted. Let the precipitatio data be available at statios, spread over a area ad P i be the observed depth of precipitatio at the i th statio. Usig a liear iterpolatio techique, a estimate of precipitatio over the area ca be expressed by P * = PiW i where W i is the weight of the i th statio. The spatial averagig techiques differ i the method of evaluatio of these weights. Weights of a optimal iterpolatio techique are decided such that the variace of error i estimatio is the miimum. (3.7)

3 The most commoly used methods are for Spatial Averagig of Precipitatio Data: (a) (b) (c) (d) (e) Arithmetic average, Normal ratio method, Distace power method, Thiesse polygo method, ad Isohyetal method. The choice of the method is depeds o the quality ad ature of data, importace of use ad required precisio, availability of time ad computer. Some of the commoly used methods are described below Arithmetic Average The simplest techique to compute the average precipitatio depth over a catchmet area is to take a arithmetic average of the observed precipitatio depths at gauges withi the catchmet area for the time period of cocer. The average precipitatio is: P= PiW i (3.8) where P is the average catchmet precipitatio from the data of statios, P i is the precipitatio at statio i, ad W i is the weight of i th statio. If the gauges are relatively uiformly distributed over the catchmet ad the raifall values do ot have a wide variatio, this techique yields good results Thiesse Polygo The Thiesse Polygo method is based o the cocept of proximal mappig. Weights are assiged to each statio accordig to the catchmet area which is closer to that statio tha to ay other statio. This area is foud by drawig perpedicular bisectors of the lies joiig the earby statios so that the polygos are formed aroud each statio (Fig. 3.8). It is assumed that these polygos are the boudaries of the catchmet area which is represeted by the statio lyig iside the polygo. The area represeted by each statio is measured ad is expressed as a percetage of the total area. The weighted average precipitatio for the basi is computed by multiplyig the precipitatio received at each statio by its weight ad summig. The weighted average precipitatio is give by: P= PiW i i which W i = A i /A, where A i is the area represeted by the statio i ad A is the total catchmet area. Clearly, the weights will sum to uity. (3.9)

4 A advatage of this method is that the data of statios outside the catchmet may also be used if these are believed to help i capturig the variatio of raifall i the catchmet. The method works well with o-uiform spacig of statios. A major drawback of this method is the assumptio that precipitatio betwee two statios varies liearly ad the method does ot make allowace for variatio due to orography. I this method, the precipitatio depth chages abruptly at the boudary of polygos. Also, wheever a set of statios are added to or removed from the etwork, a ew set of polygos have to be draw. The method fails to give ay idea as to the accuracy of the results. If a few observatios are missig, it may be more coveiet to estimate the missig data tha to costruct the ew set of polygos. Example 3.4: For a catchmet, the raifall data at six statios for July moth alog with their weights are as give i Table 3.4. Fid the weighted average raifall for the catchmet by usig the Thiesse polygo method. Solutio: Usig the observed raifall ad statio weight, weighted raifall at each statio is computed. Summatio of these values gives the weighted average raifall for the catchmet. The computatios are show i Table 3.1. Table 3.1 Estimatio of the mea areal raifall by the Thiesse polygo method. S. N. Statio Name Statio weight Raifall Weighted raifall 1. Sohela Bijepur Padampur Paikmal Bika Bolagir Weighted catchmet raifall 275.8

5 Thiesse Polygo plot for Betwa up to Basoda (SEP-1995) Siroj Sl. Statio Name Weight 1 Basoda Berasia Bhopal Gairatgaj Raise Basoda 6 Sehore Siroj Vidisha Berasia Vidisha Sehore Bhopal Raise Gairatgaj Fig. 3.8 The Thiesse polygo method for computig the mea areal raifall Isohyetal Method The isohyetal method employs the area ecompassed betwee isohyetal lies. Raifall values are plotted at their respective statios o a suitable base map ad cotours of equal raifall, called isohyets, are draw. I regios of little or o physiographic ifluece, drawig of isohyetal cotours is relatively simple matter of iterpolatio. The isohyetal cotours may be draw take ito accout the spacig of statios, the quality, ad variability of the data. I regios of proouced orography where precipitatio is iflueced by topography, the aalyst should take ito cosideratio the orographic effects, storm orietatio etc. to adjust or iterpolate betwee statio values. Computers are beig used to draw isohyetal maps these days, by usig special software. As a example, the isohyetal map for a area is show i Fig The total depth of precipitatio is computed by measurig the area betwee successive isohyets, multiplyig this area by the average raifall of the two ishohyets, ad totalig. The average depth of precipitatio is obtaied by dividig this sum by the total area. The average depth of precipitatio (P i ) over this area is obtaied by: P = Pi Ai Ai Avg. raifall: (estimated usig a combiatio of 8 raigauges) (3.10) where A i is the area betwee successive isohyets ad P i is the average raifall betwee the two isohyets.

Fig. 3.8 The isohyetal method for computig the mea areal raifall.. Table. 3. 2 Estimatio of mea areal catchmet raifall byy Isohyetal Method Isohyetal rage Average value Areaa (km 2 ) Volume (10 5 m 3 ) 521.

8 13044.2 Example 3.5: Usig the poit raifall data for a catchmet, isohyetal lies weree draw as show i Fig. 3..8. The area eclosed by each isohyet was calculated as give i Table 3.3. Compute the average catchmet raifall.

6 Fig. 3.8 The isohyetal method for computig the mea areal raifall.. Table Estimatio of mea areal catchmet raifall byy Isohyetal Method Isohyetal rage Average value Areaa (km 2 ) Volume (10 5 m 3 ) Average catchmet raifall = 232.9mm Example 3.5: Usig the poit raifall data for a catchmet, isohyetal lies weree draw as show i Fig The area eclosed by each isohyet was calculated as give i Table 3.3. Compute the average catchmet raifall. Solutio: For each isohyet, the average value is worked out (the maximum observed raifall was 108 cm ad the miimum 38 cm). This, multiplied by the area eclosed by that isohyet gives the volume of raifall for that isohyet. Now the volumes for differett isohyetals are summed ad

7 divided by the area of the catchmet to get average catchmet raifall. The computatios are show i Table 3.3. Table 3.3 Estimatio of mea areal raifall by the isohyetal method. Isohyet value (cm) Average value (cm) Area eclosed (km 2 ) Net area (km 2 ) Raifall volume (km 2 - cm) < Total Average catchmet raifall = /5.41= cm

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the