Autocorrelation of Rainfall and Streamflow Minimums
GEOLOGICAL SURVEY PROFESSIONAL PAPER 434-B Autocorrelation of Rainfall and Streamflow Minimums
By NICHOLAS C. MATALAS
STATISTICAL STUDIES IN HYDROLOGY
GEOLOGICAL SURVEY PROFESSIONAL PAPER 434-B
UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1963 UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary
GEOLOGICAL SURVEY William T. Pecora, Director
First printing 1963 Second printing 1967
For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price 20 cents (paper cover) CONTENTS
Page Page Abstract ______Bl Methods for investigating nonrandomness Continued Introduction ___-__.____-____-______.-___. 1 Correlogram analysis____-______-_____--_--___ B4 Characteristics of time series______1 Investigation of nonrandomness in hydrologic time series. 5 Serial correlation______5 Effect of nonrandomness in hydrologic studies- 2 Generating process______6 Methods for investigating nonrandomness--___ 2 Effect of serial correlation on the estimate of the Trend analysis______2 variance______6 Measure of nonrandomness ______3 Discussion and conclusions______-_-_-_---_-.__ 7 Generating processes______3 References.__-__-_--_-__-_--______--__--_.-_-_-___ 7
ILLUSTRATIONS
Page Page FIGTJBES 1-6. Correlograms for annual minimum rainfall. FIGURE 7. South Fork Holston River at Bluff 1. Boston, Mass_____-_-.______-_ B8 City, Tenn______B9 2. Providence, R.I___~-______~_ 8 8. James River at Buchanan, Va______9 3. Charleston, S.C______8 9. Correlogram for annual minimum 1-day 4. Washington, D.C_-.-______8 discharge for Middle Branch Westfield 5. Baltimore, Md______8 River near Goss Heights, Mass....__ 6. Philadelphia, Pa______8 10. Expected value of the variance for a first- 7, 8. Correlograms for annual minimum 7-day order Markov process__-_-_-_------_ discharge.
TABLES
Page Pag* TABLE 1. Rainfall stations used in analyzing annual mini TABLE 4. Discharge data___-___-_-----__----__- BIO mum monthly rainfall______B9 5,6. Serial-correlation coefficients for orders 1 2. Gaging stations used in analyzing annual mini through 19. mum discharge_.-__---_-____-__-_____. 9 5. RainfaU data______10 3,4. Significance of first-order serial-correlation co 6. Discharge data.______-----__----_-_ 10 efficients. 7. Chi-square goodness of fit of first-order Markov 3. Rainfall data--_---______10 process to discharge Correlograms ______10 ra STATISTICAL STUDIES IN HYDROLOGY
AUTOCORRELATION OF RAINFALL AND STREAMFLOW MINIMUMS
By NICHOLAS C. MATALAS
ABSTRACT CHARACTERISTICS OF TIME SERIES Hydrologic time series of annual minimum mean monthly Time series may be classified as either stationary or rainfall and annual minimum 1-day and 7-day discharge, con sidered as drought indices, were used to study the distribution nonstationary. Assume that a time series is divided of droughts with respect to time. The rainfall data were found into several segments and that the data within each to be nearly random. The discharge data, however, were segment are characterized by statistical parameters found to be nonrandomly distributed in time and generated by such as the mean and variance. If the expected values a first-order Markov process. The expected value of the variance of these parameters are the same for each segment, for a time series generated by a first-order Markov process was compared with the expected value of the variance for a random the time series is said to be stationary. If the expected time series. This comparison showed that the expected value values are not the same for all segments, the time of the variance for a nonrandom time series converged to the series is nonstationary. In stationary time series, population variance with an increase in sample size at a slower absolute time is not important, and the series maybe rate than for a random time series. assumed to have started somewhere in the infinite past. INTRODUCTION However, in nonstationary time series, absolute time must be considered since the series cannot be assumed The object of the investigation was to determine if to have begun prior to the time of the initial observation. droughts are randomly or nonrandomly distributed Time series may be considered as composed of the in time and to modify tests of reliability to account sum of a random element and a nonrandom element. for the effect of nonrandomness. For these purposes, The nonrandom element consists of a trend and an the hydrologic time series used in the investigations oscillation about the trend. Trend is usually thought were annual minimum monthly rainfall at 18 rainfall of as a smooth motion of the time series over a long stations and annual minimum 1-day and 7-day dis period of time. For any given time series, the sequence charge at 9 gaging stations. of values will follow an oscillatory pattern. If this A time series is defined as a sequence formed by the pattern indicates a more or less steady rise or fall, the values of a variable at increasing points in time. A pattern is defined as a trend. No matter what the time series may be composed of the sum of two com length of a time series, one can never state with cer ponents: a random element and a nonrandom element. tainty that an apparent trend is not part of a slow If the values of the time series are not independent of oscillation, unless the series ends. each other, the nonrandom element exists, and the An oscillatory movement is often confused with a values are said to be serially dependent. cyclical movement. In a cyclical time series, the The nonrandom element in a time series may be due maximum and minimum values occur at equal intervals to a trend or an oscillation about the trend or both. of time with constant amplitude. The random com In a given time series, the nonrandom element need ponent, if present, tends to distort this pattern. In an not be due to both of these causes. In order to analyze oscillatory time series, the amplitude and the interval properly a time series, the random and nonrandom ele of time between maximum and minimum values are ments must be isolated and studied separately. Trend distributed about mean values. A cyclical time series must be eliminated from the nonrandom element is oscillatory, but an oscillatory time series is not before studying the oscillatory character of a time series. necessarily cyclical. The oscillatory movement in a trend-free time series In order to analyze properly a time series, it is neces may be due to 1 of 3 schemes: (a) moving averages, sary to separate the random and the nonrandom (b) sums of cyclic components, and (c) autoregression. components. Trend must be eliminated from the Bl B2 STATISTICAL STUDIES IN HYDROLOGY nonrandom component before studying the oscillatory of 690 sets of records and found the range to be of the character of the time series. form (4) EFFECT OF NONRANDOMNESS IN HYDROLOGIC STUDIES The value of k varied slightly with different phe The design of dams and water-storage reservoirs is nomena. For the 690 sets of records, the mean and based on the expected hydrologic conditions many standard deviation of k were 0.729 and 0.091, respec years in the future, depending upon the life expectancy tively. of the hydraulic structures. To obtain long-range By this analysis, Hurst concluded that estimates of estimates, the frequency with which hydrologic events the standard deviation based on N observations are occur must be known, if it is assumed that the future more variable for a nonrandom time series than for a will follow the pattern shown by the past. In a random time series. In a continuation of this study, nonrandom time series, there is a tendency for low Hurst (1956) pointed out that statistical averages values to follow low values and for high values to based on 30 or 40 observations from a nonrandom time follow high values. In the case of a water-supply series do not yield close approximations to the true reservoir, the reservoir is filled during a wet season to averages. meet the anticipated demand during a dry season. If The difference between the ranges defined by equa droughts are nonrandomly distributed in time, then to tions 1 and 4 may be due to the data not being randomly overcome the tendency for low values to follow low distributed in time. This, however, requires that the values, the reservoir capacity must be increased to nonrandomness be of a very special type. Moran (1959) provide sufficient water to meet the need for more than suggests that the experimental series used by Hurst are, one dry season. as a result of nonrandomness, of insufficient length for Hurst (1950), in his study of long-term storage the asymptotic formula (eq. 1) to become valid. Hurst capacities of reservoirs, concluded that annual total (1957) constructed a probability model which tends to discharge is nonrandomly distributed in time. This illustrate this point. study was based on an analysis of the cumulative sum of departures of annual total discharges from the mean METHODS FOR INVESTIGATING annual total discharge. The absolute value of the NONRANDOMNESS difference between the maximum and the minimum of TREND ANALYSIS the cumulative sums was defined as the range. To test for nonrandomness, Hurst determined the expected Trend is thought of as a smooth motion of the time values of the range for various natural phenomena, and series over a long period of time. If the oscillatory compared these values with the expected value of the pattern of a time series indicates a more or less steady range for a random normally distributed time series. rise or fall, the pattern is defined as a trend. To study For a random normally distributed time series, Hurst the oscillatory characteristics of a time series, the trend, defined the expected value of the range, R, as if present, should be removed. Various methods of removing trend are available. All the methods, however, are not fully understood as to how they affect the time series. The most commonly used method is that of where a is the standard deviation and N is the sample moving averages. size. Feller (1951) showed that the variance of the Assume that the observations &i, $2, . . ., XN are range is taken at equal intervals of time. The method of moving V(R) =0.074 (3) The weights of the moving average, blf b2, and b8, are Hurst investigated 75 different phenomena consisting such that their sum equals 3. In general, for moving AUTOCORRELATION OF RAINFALL AND STREAMFLOW MINIMUMS B3 i N-k i /N-k \ /N-k JL x! ~\ * ' (6) averages of m, the sum of the weights equals m. The where weights may be positive or positive and negative. A 0=1 V '*=r-t f (7) simple moving average refers to the case where each of the weights equals one. Although a simple moving ing the time series, and rk is the Arth-order serial- average tends to smooth out the data, it does not correlation coefficient. preserve the main features of the time series as well as a Under the assumption that x t varies linearly with weighted moving average. %i+K, ft serves as a measure of linear dependence. Generally, even a smooth trend obtained by the If a time series is random, rK=Q for all values of k>l. method of moving averages cannot be represented con However, for a finite sample, computed values of rK veniently by a mathematical equation. The simplest may differ from zero because of sampling errors. For mathematical expression is a straight line. However, a hydrologic time series, N is small, so that the sampling time series is apt to be such that a single linear trend errors may be quite large. Thus the values of rK can not be used throughout the time of observation. must be tested to determine if they are significantly In such cases, it is possible to approximate the trend by different from zero. using linear trends for portions of the time series. Anderson (1942) developed a test of significance After the trend line has been established, the trend based on a circular definition of the serial-correlation can be removed from the data in one of several ways. coefficients which supposes that the last event is fol One way is to take as a new variable the deviations lowed by the first event so that about the trend line. These deviations must comprise a 1 N 1 N stationary time series. In some cases, the deviations may not comprise a stationary time series, but the rk=- (8) deviations divided by their corresponding trend values may comprise a stationary time series. If the method of moving averages is used to determine For N large and k small, the values of rk given by the trend in a time series which has an oscillatory move equation 6 are nearly equal to those given by equa ment about a trend, then a long-period oscillation tends tion 8. to be included as part of the trend. Oscillations which For a random normal time series, Anderson showed are comparable in period to the length of the moving that TI is approximately normally distributed with average, m, or shorter are damped out. The moving mean [-!/(# !)] and variance (A/-2)/(JV-l)2. average also introduces an oscillatory movement into Thus a computed value of TI may be tested for sig the random component of the time series. These con nificance by sequences of the moving-average method are referred ri=- to as the Slutzky-Yule effect (Slutzky, 1937; Yule, N-l (9) 1921). Because of the Slutzky-Yule effect, care must be taken hi discussing the oscillatory character of a time where ta is the standard normal variate corresponding series if its trend has been removed by the moving- to a probability level a. At a given probability level, average method. if r^>r\, then rt is considered to be significantly different from zero. MEASURE OF NONRANDOMNESS GENERATING PROCESSES A time series is said to be randomly distributed if A generating process refers to the manner by which each event is independent of all preceding and following the causal forces act to produce a time series. Some events. In analyzing sunspot data for periodicities, processes can be expressed mathematically, in which Yule (1927) found that the scatter diagrams for events case it is possible to determine directly the various k time units apart were linear. As a measure of linear characteristics of the time series. Often a time series dependence, Yule proposed equations 6 and 7. is approximated by a certain process. The choice of In equation 6, x t and xi+Jc are the events at times i this process is based upon how well the mathematical and i-\-k, respectively, N is the number of events form- structure of the process conforms to the physical B4 STATISTICAL STUDIES IN HYDROLOGY characteristics underlying the time series. The proc a physical model in hydrology, only a finite period of esses which have been studied extensively and which antecedency was considered. If the antecedent period have found wide application in practice are the moving is considered to be infinite, then a value of x at time average, the sum of harmonics, and the autoregression. i is a function of all previous values of x. This condi The moving average process may be expressed as tion may be expressed as an autoregression process by defining x t in terms of previous values of x and a Xi=b0 +biy i+b2yi-l + ... +6m7/<_ (m_1) , (10) random component. An example of such a process is where y is a random variable and m is the extent of the moving average. Equation 10 may be taken as a xi+2=axi+i -f b$i (14) model representing the relation between annual runoff, where a and b are constants and e is the random x, and annual effective precipitation (Folse, 1929). component. The effective precipitation prior to a given time interval Kendall (1951) shows that by letting p V and is referred to as antecedent effective precipitation. cos 0 a/(2V&), the serial-correlation coefficients for Since the contribution of effective precipitation to the the autoregression process given by equation 13 are runoff converges to zero very rapidly with an increase defined as in antecedent time, the effective precipitation for only a .(15) finite period of antecedency affects the runoff. This '* ' sin^ ' finite period of antecedency, which is defined as the carryover period, is a function of the water-retardation where tan \f/= g tan B. (16) characteristics of the river basin and the distribution of the effective precipitation with respect to time. It is assumed that -\fb is taken with a positive sign Wold (1933) shows that the serial-correlation coeffi and that 46>o2. Moreover, it is assumed that ifb is cients for a series generated by a moving average not greater than unity. process are given by A special type of autoregression is given by (ID where r\ is the first-order serial-correlation coefficient. 1=0 This process is often referred to as the first-order Markov process. The serial-correlation coefficients are given by rk Q;k>m. rk=r\. (18) Many hydrologic studies involve the use of time series which exhibit a cyclic movement; that is, se CORRELOGRAM ANALYSIS quences of daily or monthly discharges. The generat A graph with the serial-correlation coefficients plotted ing process for a cyclic time series may be represented by as ordinates and their respective orders as abscissas is Xi=A sin Oi+yt, (12) called a correlogram. To reveal the features of the where A and B are the amplitude and period, re correlogram, the plotted points are joined each to the spectively, of the cyclic movement, and y is a random next by a straight line. component. The random component, y, tends to For a moving-average process, the values of rt are distort the amplitude and the period. If the random given by equation 11. Since rk 0 for k>m, the cor component is large, A and B may be badly distorted. relogram may oscillate, depending upon the values of For a cyclic time series, the serial-correlation coeffi b, but the correlogram will vanish for k>m. The val cients are given by ues of rk for the harmonic process are given by equa A2 tion 13. For this process, the correlogram will oscillate r*=2^2 cos 0*» (13) but never vanish. The oscillation of the correlogram is strictly cyclic with amplitude A^feal and period B. where al is the variance of the x's. Note that the period of the correlogram is identical Equation 12 is a special case where only one har to the period of the time series. Equation 15 indi monic, namely B, is involved. Some hydrologists cates that the correlogram of the autoregression proc argue that there are hidden harmonics, or periodicities, ess given by equation 14 will oscillate, be damped, but in hydrologic data. These periodicities are called will not vanish. If the autoregression process is given hidden because, if there are a large number of differ by equation 17, then it is seen by equation 18 that ent periods, the series is very erratic seemingly random. the form of the correlogram will be "exponential." In the discussion of the moving-average process as Thus for infinitely long sequences, the correlogram pro- AUTOCORRELATION OF RAINFALL AND STREAMFLOW MINIMUMS B5 vides a theoretical basis for distinguishing among the termined and are given in tables 3 and 4, respectively. three types of oscillatory time series. The first-order serial-correlation coefficient given in In practice, N is small so that observed correlograms tables 3 and 4 range from 0.296 to 0.300 for rainfall show less damping than theoretical correlograms be data and from 0.085 to 0.385 for discharge data. cause the observed serial-correlation coefficients are Although the discharge data exhibit higher first-order inflated by sampling errors. Thus one cannot deter serial-correlation coefficients than the rainfall data, mine the generating process by simply observing the the variability of the first-order serial-correlation co correlogram. The goodness of fit of a given generat efficients is greater for the rainfall data than for the ing process fitted to an observed correlogram must be discharge data. determined. Tests of goodness of fit which have been These values of r^ were tested for significance at the developed are based on N being very large. 90-percent and 95-percent levels by equation 9. An in In this study only the autoregression process is con spection of tables 3 and 4 shows that at the 90-percent sidered. For this generating process, a chi-square test level of significance, the data for 6 rainfall and 4 gag developed by Quenouille (1949)' can be used to test ing stations exhibit significant values of r\. At the the goodness of fit. If the autoregression model is 95-percent level, the data for 4 rainfall and 3 gaging defined by equation 17, the value of chi-square, xL stations yield significant values of TV Thus at the associated with the &th-order serial-correlation coeffi usual levels of significance employed in hydrologic cient is given by studies, approximately K to K of the stations for both phenomena possessed data yielding significant first- 2 ~_ (19) order serial-correlation coefficients. where (20) The data which exhibited nonsignificant values of rx are not necessarily random in time. Serial-correlation The sum of the values of chi-square forms the basis coefficients of order higher than one, if significant, for testing the goodness of fit. If would indicate a lack of randomness. To determine if the data were random in time, the serial-correlation (21) coefficients for orders 1 through 19 were considered for k=l series having both significant and nonsignificant values of TI. The data for 6 rainfall and 3 gaging stations where xM«) is the value of chi-square at the probability- were used. The values of rk for k=l, 2, . . ., 19 for level a with m degrees of freedom (m is the highest the rainfall and discharge data are given in tables 5 order for which a serial-correlation coefficient is deter and 6. Equation 9 was used as an approximate test mined), then the first-order Markov process is rejected. of significance. The letter a denotes significance at INVESTIGATION OF NONRANDOMNESS IN HYDROLOGIC the 90-percent level. TIME SERIES If serial-correlation coefficients were determined for a random time series, an occasional significant value SERIAL CORRELATION would be expected to occur by chance. With respect If a time series is randomly distributed, the popula to the 6 rainfall records, 114 serial-correlation coeffi tion values of the serial-correlation coefficients, called cients were determined. If the data were randomly autocorrelation coefficients, between events for all orders distributed and if the data for different stations were are zero. For such a time series, the serial-correlation uncorrelated, 12 of these values would be expected to coefficients are not significantly different from zero at be significant at the 90-percent level. However, at a probability-level a. the 90-percent level, 22 values were found to be sig To investigate the distribution of droughts with nificant. For the 3 discharge records, 57 serial- respect to time, the annual minimum mean monthly correlation coefficients were determined. If the data rainfall and the annual minimum 1-day and 7-day dis were random, 6 values would be expected to be sig charge were considered as indices of droughts. The nificant by chance. Actually 10 values were found to rainfall and discharge stations are listed in tables 1 be significant. and 2, respectively. A preliminary investigation for These results seem to indicate that the data are non- trend was made. Only the rainfall data for Charleston, random. However, this investigation is handicapped S.C., strongly indicated a trend. However, the ap by many factors. The number of stations used is parent trend was not removed. very small, and the data for different stations, particu By using equation 6, the first-order serial-correlation larly the rainfall stations, are probably correlated. coefficients for the rainfall and discharge data were de Also, most of the significant serial correlations for the B6 STATISTICAL STUDIES IN HYDROLOGY rainfall data belonged to the two records having the where x is the estimate of the mean defined by x=^ xltN' largest values of r\. One of these records was the Charleston, S.C., rainfall which apparently is not It follows that equation 22 can be written as trend free. If these factors are taken into considera N' N' tion, the rainfall data can be regarded as random _/ i j_i i in time. For the rainfall data to be nonrandom? there 02_i=l_____»'==!____N' (N')2 (23) would need to be some form of atmospheric storage. The discharge data may be nonrandomly distributed The expectation of xt is E(xt)=0 and the expectation in time, with the nonrandomness being due to storage of x* is E(xl) = -as FIGUBE 1. Correlogram for annual minimum monthly rainfall at Boston, Mass. FIGURE 2. Correlogram for annual minimum'monthly rainfall at Providence, R.I. 1.0 0.5- -0.5 FIGUBE 3. Correlogram for annual minimum monthly rainfall at Charleston, 8.C. FIQOEE 4. Correlogram for~annual minimum monthly rainfall at Washington, B.C. 1.01 0.5 /..o ./.\./N/: Y X / V 5 & v -0.5 FIGUBE 5. Correlogram for annual minimum monthly rainfall at Baltimore, Md. FICTOBE 6. Correlogram for annual minimum monthly rainfall at Philadelphia, Pa. AUTOCORRELATION OF RAINFALL AND STREAMFLOW MINIMUMS B9 1.0 f i.O 0.5 -0.5 FIGURE 8. Correlogram for annual minimum 7-day discharge for South Fork Hols- 7. Correlogram for annual minimum 7-day discharge for James River at ton River at Bluff Citv. Terra. Buchanan, Va. 1.0 O.5 v . r\. -0.5 FIGURE 9. Correlogram for annual minimum 1-day discharge for Middle Branch FIQTTRE 10. Expected value of the variance for a first-order Markov process. Westfield River near Ooss Heights, Mass. TABLE 2. Gaging stations used in analyzing annual minimum discharge TABLE 1. Rainfall stations used in analyzing annual minimum monthly rainfall Length Length Gaging station Pertod of of Gaging station Period of of Length Length record record record record Rainfall station Period of of Rainfall station Period of of (years) (years) record record record record (years) (years) James River at Souhegan River Buchanan, Va.... 1898-1954 57 at Merrimack, 1836-1960 115 Albany, N.Y...... 1826-1928 103 N.H... ..- 1909-55 46 New Bedford, Providence, R.I- ... 1832-1928 97 Holston River at. Busquehanna 1814-1060 137 New York, N.Y.... 1826-1943 118 Bluff City, Tenn. 1901-49 49 River at Lowell Locks and Charleston, B.C.... 1832-1953 122 French Broad Conklin, N.Y.... 1906-60 46 Canals, Lowell, St. Louis, Mo. ... 1837-1953 117 River at Little Tennessee 1865-1950 96 Washington, D.C-. 1852-1953 103 Asheville, N.C 1903-63 50 River at Judson, 1826-1925 100 1854-1953 100 N.C...... 1896-1941 46 Waltham, Mass .... 1828-1930 102 New Burnswick, Westfield River Schroon River at 1841-1921 81 N.J...... 1854-1953 100 at Ooss Heights, River Bank, Boston, Mass...... 1818-1913 96 Baltimore. Md ..... 1818-1953 136 1910-55 45 N.Y ...... - 1908-63 46 1838-1928 91 Philadelphia, Pa... 1820-1953 134 Pemiguewasset Burlington, Vt River at Plymouth, N.H- 1903-55 62 BIO STATISTICAL STUDIES IN HYDROLOGY TABLE 3. Significance of first-order serial-correlation coefficients TABLE 4. Significance of first-order serial-correlation coefficients for rainfall for discharge Confidence level Confidence level Rainfall station n Gaging station ri 90-percent 95-percent 90-percent 95-percent 0.030 0.145 -0. 152 0.174 -0. 191 a 0.345 0.200 -0.236 0.241 -0.277 .070 .133 -.148 .160 -.175 South Fork Holston River at Bluff City, Lowell Locks and Canals, .178 .214 -.256 .258 -.300 .060 .157 -.178 .190 -.211 French Broad River at Asheville, N.C.. .154 .212 -.253 .259 -.300 1.166 .154 -.175 .186 -.206 Middle Branch Westfleld River near a -.213 .153 -.173 .184 -.204 .297 .222 -.268 .268 -.314 .059 .171 -.196 .205 -.230 Pemiguewasset River at Plymouth, .071 .159 -.178 .190 -.211 N.H--. - -.082 .208 -.248 .253 -.292 .089 .161 -.184 .195 -.217 Souhegan River at Merrimack, N.H .... *.385 .223 -.267 .267 -.311 Albany, N.Y _____ ...... -.070 .152 -.172 .183 -.203 Susquehanna River at Conklin, N.Y.... ».224 .223 -.267 .267 -.311 .070 .157 -.177 .189 -.210 Little Tennessee River at Judson, N.C.. .096 .223 -.267 .267 -.311 New York, N.Y._.___ ... -.139 .143 -.160 .171 -.188 .059 .223 -.267 .267 -.311 a. 300 .141 -.157 .168 -.185 a -.222 .143 -.161 .172 -.189 2 OOfi .152 -.172 .183 -.203 i Significant at 90-percent level. .119 .154 -.175 .186 -.206 * Significant at 95-percent level. -.046 .154 -.175 .186 -.206 Baltimore. Md _ ...... i-. 171 .134 -.149 .160 -.175 Philadelphia, Pa...... 000 .135 -.150 .162 -.177 i Significant at 90-percent level. TABLE 6. Serial-correlation coefficients for orders 1 through 19 * Significant at 95-percent level. for discharge South Fork Middle Branch James River at Holston River Westfield TABLE 5. Serial-correlation coefficients for orders 1 through 19 Buchanan, Va. at Bluff City, River near for rainfall Tenn. (Joss Heights, Mass. Phila Boston, Providence, Charles Washing Baltimore, 10.345 a 178 10.297 Mass. R.I. ton, B.C. ton, D.C. Md. delphia, Pa. .084 -.200 .061 -.131 i -.254 1.242 i -0.296 1-.220 .037 -.027 0.071 0.070 10.300 i -0. in 0.000 -.098 1.239 1-.221 n ...... 1.164 -.060 1.238 1.157 1.180 .024 J-.269 .081 n ...... 142 -.138 1.282 .150 -.067 .024 1.234 i -.249 -.114 -.070 .098 .130 -.072 1.317 .074 .049 -.030 .110 -.200 n ...... 078 -.004 1.196 -.013 .000 .087 .122 .148 -.093 r« .. .075 .140 1.163 -.100 .086 -.003 no . . . . 1.238 -.033 -.185 -.031 -.031 .132 -.066 .004 .113 .174 -.041 -.271 n ...... -.099 -.210 .120 -.034 .130 -.014 -.008 .078 -.063 -.129 -.008 1.244 -.060 r»...... 007 .064 rtt . .177 .070 -.184 Tit ...... 117 -.004 -.048 1.175 -.011 -.059 -.026 .123 -.246 ni ------1.303 .077 .009 i -. 217 -.043 -.069 -.130 -.038 -.140 ru .000 -.018 -.135 .147 .101 .028 .030 -.021 -.201 .051 ru -....- -.117 .150 i -. 175 .032 -.089 TIT - - « -.348 .164 .166 H4 -- . -.095 -.096 1-.163 -.066 1.143 -.096 -.122 -.077 .093 -.124 .007 -.037 -.037 -.039 -.023 -.203 -.061 ru .... -.023 .123 -.112 .105 .096 .044 -.166 .028 -.028 -.109 .062 .018 1-.162 ru ...... -.107 .140 I -.164 -.213 -.106 -.127 rtj .... .080 -.014 1-.169 1.278 .036 -.003 > Significant at 90-percent level. i Significant at 90-percent level. TABLE 7. Chi-square goodness of fit of first-order Markov process to discharge correlograms James River at Bu- South Fork Holston Middle Branch West chanan, Va. River at Bluff City, fleld River near Ooss k Tenn. Heights, Mass. Rk xl Rk 5d Rk xi 1 -0.304 6.664 -0.172 1.521 -0.271 3.879 2 -.035 .085 -.232 2.692 -.027 .036 3 -.148 1.524 -.177 1.540 .232 2.718 4 -.120 .976 .121 .706 -.166 1.346 6 .038 .100 .218 2.225 -.184 1.621 6 -.218 3.107 .150 1.032 .210 2.069 7 .053 .180 -.146 .949 .030 .041 8 .018 .019 .142 .883 -.251 2.809 9 .129 1.033 .107 .486 .034 .051 10 .150 1.368 -.082 .283 -.147 .914 11 .024 .036 -.025 .025 -.169 1.174 12 .152 1.345 -.046 .086 .187 .742 13 .029 .062 .090 .311 -.203 1.686 14 -.119 .787 .096 .348 -.136 .694 16 -.111 .666 -.080 .228 -.011 .004 16 -.101 .533 .068 .165 .092 .293 17 -.229 2.705 .146 .714 .126 .635 18 .094 .447 -.134 .592 -.047 .071 19 -.122 .229 -.170 .928 -.011 .004 3&J 22.356 16.714 20.686 . GOVERNMENT PRINTING OFFICE: 1967 O 248-124