Development of Intensity Duration Frequency Curves for Precipitation in North Lakhimpur (ASSAM)

Development of Intensity Duration Frequency Curves for Precipitation in North Lakhimpur (ASSAM)

Tina Moni Boruah*, Senchumbeni. M. Patton**, Shyamoly Phukan***, Neha Tiwari****, Pompy Chutia*****

Department of Civil Engineering

Royal School of Engineering & Technology, Ghy-35, Assam

Abstract:- An Intensity- Duration- Frequency (IDF) curves is the graphical representation of the probability that a given average rainfall intensity will occur in a particular region for a given duration of storm having desired frequency of occurrence. The intensity of rainfall is the rate of precipitation, i.e., depth of precipitation per unit time. This can be either instantaneous intensity or average intensity over the duration of rainfall. Here, in this project we have considered average intensity of rainfall. The Intensity Duration Frequency (IDF) relationship of rainfall amounts is considered as one of the most commonly used tools in water resource engineering for planning, design and operation of water resources project, or for various engineering projects against design floods. The objective of this research is to derive IDF relationship of rainfall for watershed of North Lakhimpur, Assam. These relationships are useful in design of urban drainage works, for example storm sewer, culverts and other hydraulic structure. In this study, rainfall depth for 11 years viz. 2006 to 2016 has been collected from the Regional Meteorological centre, Guwahati. Gumbles frequency analysis technique has been used to calculate the return periods for a period of 2yrs, 5yrs, 10yrs, 50yrs and 100yrs from the maximum intensity. Finally, regression analysis has been to develop the Intensity Duration Frequency (IDF) curve.

Keywords: IDF, Return Period, Regression Analysis, North Lakhimpur

1. INTRODUCTION

   In many parts of the world, flooding is probably the most severe hazard among the natural hazards occurring due to change in rainfall pattern. Development of rainfall Intensity-Duration-Frequency (IDF) relationship is a primary basic input for the design of the storm water drainage system for cities. Intensity-Duration-Frequency (IDF) relationship of rainfall amounts is one of the most commonly used tools in water resources engineering for planning, design, and operation of water resources projects. Rainfall Intensity-Duration-Frequency IDF curves are graphical representations of the amount of water that falls within a given period of time in catchment areas. IDF curves are used to aid the engineers while designing urban drainage works. A design flood is the flood magnitude selected for the use as a criterion in designing flood control works. The objective of the rainfall IDF curves is to estimate the maximum intensity of rainfall for any duration and return period. This frequency analysis uses annual or seasonal maximum intensity of rainfall for any duration and return period. This frequency analysis uses annual or seasonal maximum series, or independent values above a high threshold selected for different durations. IDF analysis takes into account the different durations in a single study, and prevents curves intersecting.

2. STUDY AREA

   Lakhimpur is an administrative district in the state of Assam in India. It is situated at 27°13'60 N and 94°7'0 E.The district headquarters are located at North Lakhimpur. The district is bounded on the north by Siang and Papump are District of Arunachal and on the east by Dhemaji District and Subansiri River. Majuli Sub Division of Jorhat District stands on the southern side and Gohpur sub division of Sonitpur District is on the West. The Brahmaputra is navigable for steamers in all seasons as far as Dibrugarh, in the rainy season as far as Sadiya; its navigable tributaries within the district are the Subansiri, Ranganadi and Dikrong. Lakhimpur's climate is classified as warm and temperate. The summers here have a good deal of rainfall, while the winters have very little. In Lakhimpur, the average annual temperature is 25.0 °C. The rainfall here averages 1200 mm.

   Fig 1.1: Study Area Map of North Lakhimpur district (Source: Ground water booklet of Lakhimpur district)

   1. Objectives of the study

      1. To collect the data of rainfall depth of North Lakhimpur for 11 years from Regional Meteorological centre, Guwahati. The data of 24 hours rainfall was obtained from the year 2006-2016.

      2. To calculate the return period for the period of 2yr, 5yr, 10yr, 50yr and 100yr from the maximum intensity using Gumbels distribution.

      3. To establishment of Intensity-Duration-Frequency curves for several purpose :

         • The estimation of extreme rainfall for design purposes.

         • The assessment of the rarity of observed rainfalls.

         • Comparison of methods to estimate design rainfall.

3. METHODOLOGY

   The classical approach for building IDF curves has three steps. In the first step, a probability distribution function is fitted to each duration sample. In a second step, the quantities of several return periods T are calculated using the estimated distribution function from step1. Lastly, the parameters of the IDF equations and coefficient of different return periods are calculated by using non-linear multiple regression method. The results obtained showed that in all the cases the correlation coefficient is very high indicating the goodness of fit of the formulae to estimate IDF curves in the region of interest.

   3.1 Intensity-Duration-Analysis

   It is observed that the most intense storms last for every short durations. As the duration of storm increases, the maximum average intensity of storm decreases. If the observed maximum rainfall intensities at a place for various durations such as 10min, 15min, 1hr, 3hr, 5hr, 7hr, etc are plotted against respective durations, a graph known as intensity-duration graph is obtained.

   3.1.1 Empirical IDF Equations

   The IDF formulae are the empirical equations representing a relationship among maximum rainfall intensity (as dependent variable) and other parameters of interest such as rainfall duration and frequency (as independent variable). There are several commonly used functions found in the literature of hydrology applications, four basic forms of equations used to describe the rainfall intensity duration relationship are summarized as follows:

   Talbot equations:

   I=

   +

   Bernard equations:

   I=

   Kimijima equation:

   I=

   . (3.1)

   . (3.2)

   . (3.3)

   Sherman equation:

   I= (+)

   . (3.4)

   Where I is the rainfall intensity (mm/hr); d is the duration (minutes); a, b and c are the constant parameters related to the meteorological conditions. These empirical equations show rainfall intensity decreases with rainfall duration for a given return period. All functions have been widely used for hydrology practical applications.

   3.1.2 Calculation of Recurrence Period by Gumbels method

   The extreme value distribution was introduced by Gumbel and is commonly known as Gumbels distribution. The estimate of rainfall intensity of given duration for different return period is obtained by this method. It is one of the most widely used probability-distribution functions for extreme values in hydrologic and meteorological studies for protection of flood peaks, maximum rainfalls, maximum wind speeds, etc. It was confirmed that the Gumbel distribution well describes the variation of annual series of maximum rainfall intensity. According to this theory of extreme vents, the probability of occurrence of an event equal to or larger than a value of Xo is,

   o

   o

   P(Xx ) =1- .(3.5)

   In which y is a dimensionless variable given by

   y=(x-a)

   a=x-0.45005X

   =1.2825/X

   Thus, y=1.2825() + 0.577 . (3.6)

   Where, =mean and X=standard deviation of the variable X

   In practice it is the value of X for a given P that is required and as such equation 3.5 is transposed as, Yp=-ln[-ln(1-P)]

   . (3.7)

   Noting that the return period T=1/Pand designating y1= the value of y, commonly called the reduced variable, for a given T Y1 = -[ln.ln ] . (3.8) Or

   1

   Y1= -[0.834 + 2.303log log

   1

   . (3.8 a)

   Now by rearranging equation 3.6, the variable X with a return period T is, Xt=x+k

   Where, K=0.577

   1.2825

   Gumbels Equation For Practical Use are as follows

   (3.10)

   Equation 3.9 gives the value of the variable X with a recurrence interval T is used as XT = + Kn-1

   Where, n-1 =standard deviation of the sample of size N K = frequency factor expressed as K=

   In which y=2 reduced variable, a function of T and is given by, YT = -[. ] Or

   1

   YT = – [0.834 + 2.303 log ]

   1

   = reduced mean,a function of sample size N

   Sn = reduced standard deviation, a function of the sample size

   1. Intensity-Duration Frequency Analysis

      Every storm in a year is analyzed to find the maximum intensities for various duration. Thus each storm gives one value of maximum intensity for duration. The largest of all such values is taken to be the maximum intensity in that year for that duration. Likewise the annual maximum intensity is obtained for all the duration. Similar analysis yields the annual maximum intensities for various durations in different years. It will then be observed that the annual maximum intensity for any given duration is not the same every year but varies from year to year. In other words it behaves as a random variable.

   2. Regression Analysis

   Regression analysis tries find out the average relationship between the variables. It refers to the methods by which estimates are made of the values of one variable from the knowledge of the values of one or more other variables. In regression analysis however one variable is taken as the dependent variable and the other taken as the independent variables, thus making it possible to study the cause and effect relationship. However, the maximum intensity varies inversely with the duration and

   generally an equation of the for, I=

   (+)

   4.1 Calculation of Return Period:

   .(3.14)

4. RESULTS AND CALCULATIONS

   The calculation of return period is done according to Gumbels distribution method. The return period for 2 years, 5 years, 10 years, 50 years, 100 years are found out.

   Table 4.1 : For 1 day, the return period of 2yr, 5yr, 10yr, 50yr, and 100yr :

   Order No.	INTENSITY	SUM	AVERAGE	X-Xavg	(X-Xavg)2	S=(X- Xavg)2	S/(N-1)	n-1=S/(n-1)
   	(X)		(Xavg)
   	mm/day
   1	103.8	436.7	39.7	64.1	4108.81	8354.87	835.48	28.9
   2	67.6		39.7	27.9	778.41
   3	66		39.7	26.8	718.24
   4	40		39.7	0.3	0.09
   5	38.9		39.7	-0.8	0.64
   6	34.1		39.7	-5.6	31.36
   7	28.6		39.7	-11.9	141.61
   8	17.7		39.7	-22	484
   9	14.8		39.7	-24.9	620.01
   10	13.8		39.7	-25.9	670.81
   11	11.4		39.7	-28.3	800.89

   For N= 11, Yn= 0.4996, Sn= 0.9676

   T	T-1	T/T-1	ln(T/T-1)	ln(ln(T/T-1))	Y= -ln(ln(T/T-1))
   2	1	2	0.6931	-0.3666	0.3666
   5	4	1.25	0.2231	-1.5001	1.5001
   10	9	1.1111	0.1053	-2.2509	2.2509
   50	49	1.0204	0.0202	-3.9021	3.9021
   100	99	1.0101	0.01	-4.6052	4.6052

   Y	Yn	Y-Yn	Sn	K=(Y-Yn)/Sn
   0.3666	0.4996	-0.1333	0.9676	-0.137
   1.5001	0.4996	1.0005	0.9676	1.034
   2.2509	0.4996	1.7513	0.9676	1.809
   3.9021	0.4996	3.4025	0.9676	3.516
   4.6052	0.4996	4.1056	0.9676	4.243

   Return Period	Xavg	K	n-1	K*n-1	Xt=Xavg+K*n-1
   2	39.7	-0.137	28.9	-3.959	35.741
   5	39.7	1.034	28.9	29.882	69.582
   10	39.7	1.809	28.9	52.28	91.98
   50	39.7	3.516	28.9	101.612	141.312
   100	39.7	4.243	28.9	122.622	162.322

   Table 4.2 : For 3 day, the return period of 2yr, 5yr, 10yr, 50yr, and 100yr :

   	Order No.	INTENSITY (X) mm/day	SUM	AVERAGE (Xavg)	X-Xavg	(X-Xavg)2	S=(X- Xavg)2	S/(N-1)	n-1=S/(n-1)
   	1	164.4	1003.9 6	91.27	73.13	5347.99	21753.1	2175.3	14.75
   	2	138.7		91.27	47.43	2249.6
   	3	136.1		91.27	44.83	p>2009.72
   	4	112.6		91.27	21.33	454.96
   	5	109.8		91.27	18.53	343.36
   	6	85.1		91.27	-6.17	33.06
   	7	74.8		91.27	-16.47	271.26
   	8	74.8		91.27	-16.47	271.26
   	9	68.2		91.27	-23.07	532.22
   	10	21.1		91.27	-70.17	4923.83
   	11	18.36		91.27	-72.91	5315.86
   For N= 11 ,Yn= 0.4996, Sn=0.9676
   T			T-1	T/T-1	ln(T/T-1)		ln(ln(T/T-1))	Y= – ln(ln(T/T-1))
   2			1	2	0.6931		-0.3666	0.3666
   5			4	1.25	0.2231		-1.5001	1.5001
   10			9	1.1111	0.1053		-2.2509	2.2509
   50			49	1.0204	0.0202		-3.9021	3.9021
   100			99	1.0101	0.01		-4.6052	4.6052

   	Y	Yn	Y-Yn	Sn		K=(Y-Yn)/Sn
   	0.3666	0.4996	-0.1333	0.9676		-0.137
   	1.5001	0.4996	1.0005	0.9676		1.034
   	2.2509	0.4996	1.7513	0.9676		1.809
   	3.9021	0.4996	3.4025	0.9676		3.516
   	4.6052	0.4996	4.1056	0.9676		4.243
   	Return Period	Xavg	K	n-1	K*n-1	Xt=Xavg+K*n-1
   	2	91.27	-0.137	14.75	-2.02	89.25
   	5	91.27	1.034	14.75	15.25	106.52
   	10	91.27	1.809	14.75	26.68	117.95
   	50	91.27	3.516	14.75	51.86	143.13

   Table 4.3 : For 5 day, the return period of 2yr, 5yr, 10yr, 50yr, and 100yr :

   Order No.	INTENSITY (X) mm/day	SUM	AVERAGE (Xavg)	X-Xavg	(X-Xavg)2	S=(X- Xavg)2	S/(N-1)	n-1=S/(n- 1)
   1	341.7	1541.7	140.15	201.55	40622.4	85608.13	8560.81	92.52
   2	219.4		140.15	79.25	6280.56
   3	194.4		140.15	54.25	2943.06
   4	185.3		140.15	45.15	2038.52
   5	161.9		140.15	21.75	473.06
   6	120.8		140.15	-19.35	374.42
   7	102.1		140.15	-20.05	402
   8	96.6		140.15	-43.55	1896.61
   9	55.9		140.15	-84.25	7098.06
   10	31.8		140.15	-108.35	11739.72
   11	31.8		140.15	-108.35	11739.72

   For N= 11 ,Yn= 0.4996,

   Sn= 0.9676

   T	T-1	T/T-1	ln(T/T-1)	ln(ln(T/T-1))	Y= – ln(ln(T/T-1))
   2	1	2	0.6931	-0.3666	0.3666
   5	4	1.25	0.2231	-1.5001	1.5001
   10	9	1.1111	0.1053	-2.2509	2.2509
   50	49	1.0204	0.0202	-3.9021	3.9021
   100	99	1.0101	0.01	-4.6052	4.6052

   Y	Yn	Y-Yn	Sn	K=(Y-Yn)/Sn
   0.3666	0.4996	-0.1333	0.9676	-0.137
   1.5001	0.4996	1.0005	0.9676	1.034
   2.2509	0.4996	1.7513	0.9676	1.809
   3.9021	0.4996	3.4025	0.9676	3.516
   4.6052	0.4996	4.1056	0.9676	4.243

   Return Period	Xavg	K	n-1	K*n-1	Xt=Xavg+K*n-1
   2	140.15	-0.137	92.52	-12.67	127.48
   5	140.15	1.034	92.52	95.66	235.81
   10	140.15	1.809	92.52	167.37	307.52
   50	140.15	3.516	92.52	325.3	465.45
   100	140.15	4.243	92.52	392.56	532.71

   Table 4.4 : For 6 day, the return period of 2yr, 5yr, 10yr, 50yr, and 100yr :

   Order No.	INTENSITY (X) mm/day	SUM	AVERAGE	X-Xavg	(X-Xavg)2		S=(X- Xavg)2	S/(N-1)	n- 1=S/(n- 1)
   1	294.3	1976.5	179.68	114.62	13137.74	89135.91	8913.59	94.41
   2	256.8		179.68	77.12	5947.49
   3	256.8		179.68	77.12	5947.49
   4	241.7		179.68	62.02	3846.48
   5	225.7		179.68	46.02	2117.84
   6	199		179.68	19.32	373.26
   7	167.9		179.68	-11.78	138.76
   8	158.8		179.68	-20.88	435.97
   9	151.1		179.68	-28.18	794.11

   	10	24.4		179.68	-155.28	24111.87
   	11	0		179.68	-179.68	32284.9
   For N= 11 ,Yn= 0.4996, Sn= 0.9676

   T	T-1	T/T-1	ln(T/T-1)	ln(ln(T/T-1))	Y= -ln(ln(T/T-1))
   2	1	2	0.6931	-0.3666	0.3666
   5	4	1.25	0.2231	-1.5001	1.5001
   10	9	1.1111	0.1053	-2.2509	2.2509
   50	49	1.0204	0.0202	-3.9021	3.9021
   100	99	1.0101	0.01	-4.6052	4.6052

   Y	Yn	Y-Yn	Sn	K=(Y-Yn)/Sn
   0.3666	0.4996	-0.1333	0.9676	-0.137
   1.5001	0.4996	1.0005	0.9676	1.034
   2.2509	0.4996	1.7513	0.9676	1.809
   3.9021	0.4996	3.4025	0.9676	3.516
   4.6052	0.4996	4.1056	0.9676	4.243

   Return Period	Xavg	K	n-1	K*n-1	Xt=Xavg+K*n-1
   2	179.68	-0.137	94.41	-12.93	166.75
   5	179.68	1.034	94.41	97.61	277.29
   10	179.68	1.809	94.41	170.78	350.46
   50	179.68	3.516	94.41	331.94	511.62
   100	179.68	4.243	94.41	400.58	580.26

   B. Construction of Intensity-Duration-Frequency Curves:

   1. Intensity frequency curve

      The frequency analysis and the maximum intensity of rainfall for various return period can be obtained. Then from the result of these analysis graphsof maximum rainfall intensity against the return period for various durations are plotted.

      Maximum rainfall intensity

      (mm/hr)

      Maximum rainfall intensity

      (mm/hr)

      Table 45 : For 1 day Maximum Intensity

      Return Period	Xavg	K	n-1	K*n-1	Xt=Xavg+K*n-1
      2	39.7	-0.137	28.9	-3.959	35.741
      5	39.7	1.034	28.9	29.882	69.582
      10	39.7	1.809	28.9	52.28	91.98
      50	39.7	3.516	28.9	101.612	141.312
      100	39.7	4.243	28.9	122.622	162.322

      250

      200

      150

      100

      50

      Series1

      Expon. (Series1)

      250

      200

      150

      100

      50

      Series1

      Expon. (Series1)

      0

      20

      60

      Return Period(yr)

      100

      120

      0

      20

      60

      Return Period(yr)

      100

      120

      0

      0

      40

      40

      80

      80

      Fig 4.1 Intensity frequency curve for 1 day

      Table 4.6: For 3 day Maximum Intensity

      	Return Period	Xavg	K	n-1	K*n-1	Xt=Xavg+K*n-1
      	2	91.27	-0.137	14.75	-2.02	89.25
      	5	91.27	1.034	14.75	15.25	106.52
      	10	91.27	1.809	14.75	26.68	117.95
      	50	91.27	3.516	14.75	51.86	143.13
      	100	91.27	4.243	14.75	62.58	153.85

      180

      160

      140

      120

      100

      80

      60

      40

      20

      0

      Series1

      Expon. (Series1)

      180

      160

      140

      120

      100

      80

      60

      40

      20

      0

      Series1

      Expon. (Series1)

      0 20 40 60 80 100 120

      Return Period (yr)

      0 20 40 60 80 100 120

      Return Period (yr)

      Maximum rainfall intensity

      (mm/hr)

      Maximum rainfall intensity

      (mm/hr)

      Fig 4.2 Intensity frequency curve for 3 day

      Table 4.7 : For 5 day Maximum Intensity

      Return Period Xavg K n-1 K*n-1 Xt=Xavg+K*n-1

      532.71

      2	140.15	-0.137	92.52	-12.67	127.48
      5	140.15	1.034	92.52	95.66	235.81
      10	140.15	1.809	92.52	167.37	307.52
      50	140.15	3.516	92.52	325.3	465.45
      100	140.15	4.243	92.52	392.56

      700

      600

      500

      400

      300

      200

      100

      0

      Series1

      Expon. (Series1)

      700

      600

      500

      400

      300

      200

      100

      0

      Series1

      Expon. (Series1)

      0 20 40 60 80 100 120

      Return period(yr)

      0 20 40 60 80 100 120

      Return period(yr)

      Maximum rainfall intensity

      (mm/hr)

      Maximum rainfall intensity

      (mm/hr)

      Fig 4.3 Intensity frequency curve for 5 day

      Table 4.8: For 6 day Maixmum Intensity

      Return Period	Xavg	K	n-1	K*n-1	Xt=Xavg+K*n-1
      2	179.68	-0.137	94.41	-12.93	166.75

      	5	179.68	1.034	94.41	97.61	277.29
      	10	179.68	1.809	94.41	170.78	350.46
      	50	179.68	3.516	94.41	331.94	511.62
      	100	179.68	4.243	94.41	400.58	580.26

      700

      600

      500

      400

      300

      200

      100

      0

      Series1

      Expon. (Series1)

      700

      600

      500

      400

      300

      200

      100

      0

      Series1

      Expon. (Series1)

      0 20 40

      60

      Return period(yr)

      80

      100

      120

      0 20 40

      60

      Return period(yr)

      80

      100

      120

      Maximum rainfall intensity

      (mm/hr)

      Maximum rainfall intensity

      (mm/hr)

      Fig 4.4 Intensity frequency curve for 6 day

   2. Intensity frequency duration curve

      Table 4.9: Return period 2, 5, 10, 50,100 years maximum Intensity duration

      Maximum rainfall intensity

      Maximum rainfall intensity

      Return Period Intensity for duration (mm/hr)

      years	1day	3day	5day	6day
      2	35.741	89.25	127.48	166.75
      5	69.582	106.52	235.81	277.29
      10	91.98	117.95	117.95	350.46
      50	141.312	143.13	143.13	511.62
      100	162.332	153.85	153.85	580.26

      700

      600

      500

      400

      300

      200

      100

      0

      700

      600

      500

      400

      300

      200

      100

      0

      0

      50

      100

      Return Period

      150

      200

      0

      50

      100

      Return Period

      150

      200

      Fig 4.5 Return period 2, 5, 10, 50,100 years Intensity

5. CONCLUSION

This study had been conducted for the formulation and construction of IDF curves using rainfall data for the year 2006 to 2016 for North Lakhimpur area, Assam. The rainfall intensity is found to be non-uniform throughout the area. The actual method to construct IDF curves involves three main steps. The first step is to obtain the annual maximum intensity for each interval length. Then for each time interval, a statistical analysis has to be done to compute the quantiles for different return periods. In the third

step, the IDF curves are usually determined by fitting a specified parametric equation for each return period to the quantiles estimates, using regression techniques. Using this method the same can be found out for other cities as well as by collecting the rainfall data for the respective day. From this study, the following conclusions are made:

• The gradual exponential decrease of IDF curves for different return periods reveal that the conclusion of maximum intensity for all the years is satisfactory.

• The value of a for which the sum of the squared deviation is minimum is found out and when the corresponding value of c and b for the maximum squared deviation of a is put in the equation (3.14) and is back calculated, the values of intensity for the corresponding time interval is found out to be approximately same.

• The values of a, b and c are found to be 34, 0.97, and 12.133 respectively.

ACKNOWLEDGEMENT

At the very outset, we are pleased and highly honored to express our sincere and heartfelt gratitude to Mr. Priyanjit Purkaystha, Assistant Professor, Department of Civil Engineering, Royal School of Engineering & Technology, Betkuchi, Guwahati-35, for his constant guidance, support, inspiration and full co-operation throughout the project work.

REFERENCES

1. Parthasarathy, K. & Singh G., (1961) – Rainfall intensity-duration-frequency for India forlocal drainage design – Indian Journal of Meteorology & Geophysics, Vol.-12(2), 231-242.

2. Raiford, J. P., Aziz, N. M., Khanand, A. A., Powell, D. N. (2007). Rainfall Depth-Duration-Frequency Relationships for South Carolina, North Carolina, and Georgia, AmericanJournal of Environmental Sciences, 3 (2), 78-84.

3. Nhat, Le M., Tachikawa,Y. , Takara, K. (2006). Establishment of Intensity-Duration-Frequency Curves for Precipitation in the Monsoon Area of Vietnam. Annuals of Disas.Prev. Res. Inst., Kyoto Univ., No. 49 B, 93-103.

4. Ram Babu, Tejwani, K. K., Agrawal, M. C. & Bhusan, L. S. (1979) – Rainfall intensity duration-return period equations & nomographs of India, CSWCRTI, ICAR, Dehradun,India.

5. Kothyari, U.C., and Garde, R. J. (1992), – Rainfall intensity – duration-frequency formula forIndia, Journal of Hydraulics Engineering, ASCE, 118(2).

6. Trevor M. Daniell and Guillermo Q. Tabios III (2008), Rainfall Intensity-Duration-Frequency (IDF) Analysis for the Asia Pacific Region, Technical documents in Hydrology, No.2. International Hydrological Programme.