Multifractal Description of Streamflow and Suspended Sediment Concentration Data from Indian River Basins

1 Multifractal Des cription of Stream flow and Suspend ed Sedi ment Concentration D ata from Indian R iver Basin s Adarsh S 1* , Drisya S Dharan 1 , Nandhu AR 1 , Anand Vishnu B 1 , Vysakh K Mohan 1 , M Wątorek 2 1 TKM College of Engineering Kollam, Kera la, India 1 *Corresponding author, Adarsh S, Ph.D., Associate Professor, TK M College of Engineering Kollam, Kerala, India adarsh_lce@yahoo.co.in ; adarsh1982@tkmce.ac.in Mob :+91-9446915388 2 Complex Systems Theory Department, Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Kraków , Poland 2 Multifractal Description of Streamflow and Suspended Sediment Conce ntration Data from Indian River Basins Adarsh S, Drisya S Dharan, Nandhu AR, Anand Vishnu B, Vysakh K Mohan, M Wątorek Abstract This study investiga tes the multifractality of streamflow data of 1 92 stations located in 13 river basins in India using the Multifractal Detr ended Fluctuation An alysis (MF-DFA). The streamflow datasets of d ifferent river basins disp layed multifractality and long term persistence with a mean ex ponent of 0.585. The streamflow records of Krishna basin displa ye d least persistence and that of Godavari basin displa y ed strongest multifractalit y and complex ity. Subsequently, the streamflow-sediment li nks of five major river basins are evaluated using th e novel Multifractal Cross Correlation Analy sis (MFCCA) method of cross correlation studies. The results showed that t he joint persistence of streamflow and total suspended sediments (TSS) is approximately the me an of the persistence of individual series. The streamflow displa yed higher persistence than TSS in 60 % of the stations while in majorit y of stations of Godavari basin the trend was opposite . The annual cross correlation is higher than seasonal cross correlation in majorit y of stations but at these time scales strength of their association differs with river basin. Keywords : streamflow, multifractal, sediment, persistence, correlation Introduction The estimation of local fluctuations and long term dependency of hydrologic time series is a long standing problem in h y drolog y. Hurst exponent (Hurst 1951) is perhaps one of the most debated properties of h ydro-meteorological datasets, which is mainly used to elucidate the persistence of the time series. Mandelbrot (1982) paved the wa y of existence of fractal geometry of geophysical fields. Over the years, a large number o f methods evolved for estimation of dependency stru cture and fractal behavior of hydrologic time seri es. It includes the rescaled range analysis, double trace moments (Tessier et al. 1996) , Fourier spectral anal ysis (Hurst et al. 1965; Pandey et al. 1998) , ex tended self similarity p rinciples (Dahlstedt and J ensen 2005) , W avelet Transform Modula Maxima (WTMM) (Muzy et al. 1991), arbitrary order Hilbert spectral analysis 3 (AOHSA) (Huang et al. 2009; Adarsh et al. 201 8a). Peng et al. (1994) proposed an efficient method namel y Detrended Fluctuation Analysis ( DFA) to perform the fractal analysis based on a detrending procedure. Kantelhardt et al. (2002) proposed the multifractal e xtension of DFA procedure now popula rly known as multifractal D FA (M F-DFA). Multifra ctal is the appropriate framework for scaling fields of time series and thus c an provide the natural framework for analysing and modelling various geoph ysical processes . For h ydrological time series multifractal description can be regarded as a ‘fingerprint’ and it serves as an efficient nontrivial test bed for the performance of state-of-the-art precipitation-runoff models K antelhardt et al. (2006). Therefore D FA or MF -DFA was successfull y applied for characterization of various h ydro- meteorological time series (Yuan et al. 2010; Yu et al. 2014; Baranowski et al. 2011; Krz y szczak et al. 2019; Adarsh et al. 2019). Kantelhardt et al. (2003) applied the MF-DFA procedure for runof f and prec ipitation from different p arts of globe and compared the results with W TMM method . Koscielny-Bunde et al. (2003) applied DFA, M F-DFA and wavelet anal y sis to discharge records from 41 hydrological stations around the globe for investigating their temporal correlations and multifractal properties. The study found that the daily runoff re cords were long-term correlated above some crossover time in the order of weeks, and the y were characterized by a correlation function that follow a power law behaviour w ith exponents varying between 0.1 to 0.9. Kantelhardt et al. (2006) studied the multifractal behaviour of 99 long ter m dail y p recipitation records and 42 long term daily runoff records from different parts of the world. They found that t he precipitation records generally show short te rm persistence while runoff records showed long t erm persistence with a mean exponent of 0.73. Zh ang et al. (2008 ) applied the MF-DFA proc edure to analyse the multifractal characteristics of streamflow f rom four gauging stations in Yangtze river in China. The stud y d etected the non -stationarity of different time series and analysed the differences in multifractality among th e records from stations at upper and lower Yangtze basin. Zhang et al. (2009) applied MF -DFA method to study the scaling behaviors of the long daily streamflow series of four h ydrologica l stations in the mainstream of East River in China. The results indicated that streamflow series of the East River basin were characterized b y anti -persistence and showed simil ar scaling behaviour at different shorter ti me scale. Further their stud y applied the technique to investigate the effect of water storage structures on streamflow records and found that the streamflow magnitude wa s mai nly influenced by the precipitation magnitude 4 while the fluctuations of the streamflow records were affected b y the human interventions like construction of control structures. La bat et al. (2004) applied DFA to investigate the multifractality of streamflow series of two karstic watersheds in the sout hern France, suggesting that the correlation properties exist in small scales and anti -correlated p roperties exist in large scales. Hirpa et al. (2010) anal y zed and compared the long-ran ge co rrelations of river flow fluctuations from 14 stations in the Flint River Basin in the state of Georgia in the southeastern United States. The study investigated the effect of basin area on the multifractal characteristics of streamflow time series at different locations and it was found that in general, higher the b asin area lower will be the degree of multifractalit y. Rego et al. (2013) applied the MF-DFA to analyse th e multifractality of water level records of 12 p rincipal Brazilian rivers, and th e results indicated that the p resence of mul tifractality and long -range correlations for all the stations after eliminating the climatic periodicity. Li et al. (2015) applied the MF -DFA method to the streamflow time series of four stations of Yellow river in China. The y detected the crossover point at annual scale in all the ti me series. After removing the tr end by the s easonal trend decomposition, they found that a ll decomposed series we re characterized by the long term persistence. Also the stud y not ed that the multifractality of streamflow series was because of the correlation properties as well as the probabilit y densit y function. Tan and Gan (2017) used MF - DFA for determination of multifractal behaviour of 145 streamflow and 100 daily precipitation series of Canada. The y reported that all precipitation time series showed long term pe rsistence (LTP) at both small and large time scales, while streamflow time series generally showed LTP at large time scales. Recentl y, Adarsh et al. (2018b, 2019) performed the multifractal analysis of streamflow records of f our stations of Brahmani river basin and one station of Kallada river basin in India. Eventhough man y studie s performed multifractal chara cterization of streamflow emplo y ing the MF -DFA procedure worldwide, according to the author’s knowl edge, no comprehensive study has bee n reported considering streamflow data from I ndian rivers and such an anal ysis on sediment concentration data is really scarce in literature. The specific o bjectives of this paper include: (i) mul tifractal cha racterization of streamflow data of different rivers in India; (ii) investigate th e streamflow – suspended sediment link of five major basins in India using multifractal cross correlati on analysis (MFCCA). The nex t section presents the theoretical details on MF-DFA and MFCC A. The details of data u sed in the stud y are presented in the se ction 5 thereafter. Subsequently, results of MF -DFA analysis of streamflow and MFCCA on streamflow-total suspended sediment (TSS) links of five major b asins are presented along with relevant discussions. Then the major conclusions drawn from the study are prese nted. Materials and Meth ods This section presents the theoretical det ails on the M ultifractal Detrended Fluctuation Analysis (MF-DFA) and Multifractal Detrended Cross Correlation Analysis (MFCCA) used in this study. Multifractal Detrended Fluctuation Analysis (MF-DFA) The multifractal detrended fluctuation anal y sis ( MF-DFA) is a popular tool used for the s caling characterization of non-stationar y time series.The different steps involved in MF-DFA computational procedure can be described as follows: Consider a time series X ( x 1 , x 2 … x N ), where N is the length of the t ime series. The accum ulated deviation of the series (known as ‘profile’) is calculated a s :       i k k x x i X 1 ) ( (1) where i =1, 2,...., N , k =1,2..., N , x is the mean of the series x k Divide the profile X (i) in to int( / ) s N N s  non-overlapping segments of length, here s i s the segment sample size (so called scale) chosen for the anal ysis and int( N / s ) is the int eger p art of ( N / s ). As N need not be a mul tiple of s alwa y s, there is a chance of omission of small portion of the time series at the end, and to include su ch segments, the same procedure is repeated starting from the opposite end and a total of 2N s segments are considered in the analysis Calculate the local trend for each of the 2N s segments by a least squares fit of the series as:     2 1 2 ) ( ) 1 ( 1 ) , (       s i i x i s X s s F    for υ = 1,2,....., N s (2) And     2 1 2 ) ( ) ( 1 ) , (        s i s i x i s N N X s s F    for υ = N s +1,...., 2N s (3) Here x υ (i ) is the fitti ng polynomial in segment υ . Linear, quadratic, cubic etc., different t y pes of fitting can be made and accordingly DFA procedure is named a s DFA1, DFA2,.. ...DFAm etc. 6 Compute the q th order fluctuation func tion by averaging:   q N q s q s s F N s F / 1 2 1 2 / 2 ) , ( 2 1 ) (            (4) Here the index variable q can take any real value except zero and the zero th order fluctuation function is computed by following a logarithmic a veraging procedure :          Ns s s F N s F 2 1 2 0 )] , ( ln[ 4 1 exp ) (   (5) Analyse the scaling beh aviour of the fluctuation functions developing the log -log plot s of F q ( s ) versus s for each values of q . If the time se ries is long range pow er law correlated, F q ( s ) increases as: F q ( s ) ~ s h ( q ) and h ( q ), the slope of the plot i s referred as the g eneralized Hur st e xponent (GHE). For stationar y time series, 0 < h ( q = 2) < 1, is identical to the classical Hurst exponent (Hurst 1951) . For an unc orrelated series the value of Hurst ex ponent is 0.5. If the Hurst exponent falls between 0.5 and 1, it indicates the long ter m persistence (long me mory process) and if it falls between 0 and 0.5, it indicates a short term persistence (sho rt memory process). Long term persistence implies a positive autocorrelation in the time series (i.e., the effect of an observation on future observations remain significant fo r a l ong p eriod of time). For example an extreme event would have higher probability being follow ed b y another ex treme of same character (i.e., a flood followed b y another flood). The selection of scale ( s ) or segment sample siz e, the t y pe of polynomial chosen etc., are some of the key iss ues while applying t he MF -DFA method. Generally sufficient se gments are chosen between the bounds (minimum and maximum) scale range. Minimum scale can be chosen in such wa y that it is sufficiently larger than the polynomial order chosen to prevent error in computation of local fluctuations and maximum scale below 1/10 of the sample siz e. Also the pol ynomial order can be chosen 1 -3 probably sufficient to avoid overfitting problems within small segment sizes (Ihlen 2012; Oświęcimka et al. 2013 ). From the GHE, sev eral o ther t y pes of scalin g exponents can also be derived, which is helpful for the multifractal characterization of the time s eries. The q -ord er mass ex ponent ( )) ( ( q  and singularity exponent ( α ) are derived a s follows: 1 ) ( ) (   q qh q  (6) 7 dq q d ) (    (7) and ) ( ) ( q q f      (8) where f ( α ) provide s the sing ularity spectrum . The dependency of h ( q ) on q infe r m ultifractality of the tim e series and the s pread of GHE plot ∆ h ( q ) refer the strength of m ultifracta lity ( Grech 2016) . I f the variation of GHE plot is steeper the time series is more multifractal (higher degree of multifractality) and if i t is f latter the series is less multifractal (lower degree of multifractality) . T he base width of the singularity spectrum (spread of singularity exponent, ∆ α ) also reflects the streng th of the multifractality o f the tim e series. The shape and extent of the singular it y spectrum cur ve contain significant information about the distribution characteristics and the singularit y content of the time se ries. A wider singularity spectrum indicates a highe r degree of multifractality and a narrow width indicates lesser degree of multifractality. For a multifractal ti me series the shape of sin gularity spectrum will be an inverted par abola whose ri ght and left hand wings c orrespond to negative and positive q respectively. Asymmetr y Index ( A α ) is a useful parameter for multifractal analysis derived from the properties of the spectrum. It is obtained by the following relation ( Drożdż et al. 2015):     R L R L A             (9) where min 0       L and 0 max       R are respectively, the width of le ft- and right- ha nd branches of the multi fractal spectrum curve; their values describe the distribution patterns of high and low fluctuations and 0  is the singularity exponent for q =0. The value of A α ranges from - 1 to 1. It quantifies the deviations of the multifractal spect rum curve. A α >0 suggests a left-hand deviation of the multifractal spectrum, likely to have resulted from some degree of loc al high fluctuations; A α <0 suggests a right hand deviation with local low fluctuations, and A α = 0 represents a symmetrical multifract al spectrum. The difference ∆ f ( α ) between maximum and minimum values of singularity provides an estimate of the spread in cha nges in fractal patterns. Since ∆ f ( α ) denotes the frequenc y ratio of the largest to the smallest fluctuations ∆ f ( α ) >0 means that the largest fluctuations are more frequent than smallest fluctuations. 8 Multifractal Cross Correlation Analysis (MFCCA) In order to determine th e int er-relationships between different hydro-meteorological variables, different statistical approaches h ave b een developed and sim plest of whic h is the estimation of Pearson correlation coefficient. However, thi s coefficient is not robust and can be misleading if outliers are present, as in real-world data charac terized by a high degree of non -linearity and non-stationarity. The Pearson correlation may display the spurious correlations in the presence of trend in non-stationar y time series. Podobnik and Stanley (2008 ) proposed a new method, detrended c ross-correlation analysis (DCCA), to investigate power -law cross correlations between two candidate non-stationarity time series in a multifractal framework. Some recent studies made detailed comparison on the person correlation and DCCA approach (Piao and Fu 2016) . DCCA was extended to multifractal case and named as Multifractal Detrended Cross Correlation Anal y sis (MF DC CA ) (Zhou 2008) and Mult ifractal Detrending Moving Ave rage Cross Correlation Analysis (MFXDMA) (Jiang and Zhou 2011) . Later on Oświcimk a et al. (2014 ) propounded a more generalized version of cross correlation anal ysis namely Multifractal Cross Correlation Anal ysis (MFCCA) which can also incorporate the sign of fluctuation function to their generalized moments. DCCA and its variants have successfully been applied to financial , biomedical and meteorologic al time se ries (Hajian and Movahed 2010; Shi 2014; Vassoler and Zebende 2012; Jiang et al. 2011; Wu et al. 2018; Dey and Mujumdar 2018). The different steps involved in MFCCA computational procedure can be described as follows: For two time series x i and y i ( i =1,2,…, N ); determine the profiles a s two new series:       j i i x x j X 1 ) ( (10) and       j i i y y j Y 1 ) ( (1 1) where, i = 1,2, ……., N ; x and y are the mean of the two series. Each se ries x i and y i are divided into N s non-overlapping segments bot h in progressive and retrograde directions, to avoid an y omi ssion of time series data at the beginning or end of the 9 series. For each 2 N s segments, loca l t rend o f both series x j and y j are computed by fitting polynomial of appropriate order ( m ). The subtraction of the fitted pol ynomial from the original segment gives the covariance:                   s i k m Y m X XY k p k Y k p k X s s f )) ( ) ) 1 (( ( )) ( ) ) 1 (( ( 1 ) , ( , , 2      (1 2) Calculate detrended covariance by summing over all overlapping a ll segments of length n:   2 / 2 1 2 0 2 ) , ( ) , ( 2 1 ) ( q XY N XY s XY q s f s f sig n N s F s        (1 3) F q XY ( s ) behaves as a power-law function of s (the scaling b ehavior), where s is the segmental sample size: F q XY ( s ) ~ s λ ( q ) (1 4) The cross-correlation exponent λ( q ) similar to the generalized Hurst ex ponent h ( q ) in MF-DFA and it c an be obtained by observin g the slope o f log -log plot of F ( s ) versus s b y ordinar y least squares. Determination of Cross Correlation coef ficient (ρ XY ) DCCA cross-correlation coefficient is defin ed as the ratio between the detrended covariance function XY F and the detrended variance functions X F and Y F (Zebende 2011; Kwapień et al. 2015) Y q X q XY q XY F F F   (1 5) Theoretically the value o f ρ XY ranges betw een −1 ≤ ρ XY ≤ 1. If the value r ange between ±0.666 to ±1 cross correlation it can be considered as strong positive (or negative); ±0.333 to ±0.666 it is medium and ±0 to ±0.33 3 it is weak (Brito et al. 2018). The MF CCA anal y sis facilitate the estimation of scale dependent correlation between two candidate ti me series, which can provid e better insight into the physical association betwee n the variables . It is to be noted that in this study MFCCA is retrieved for the moment order q =2. 10 Study area and D ata In this study long term dail y streamflow data of 192 stations falling in 1 3 river basins in India are collected from Water Resources Information S y stem (WRIS) India (w ww.india-wris.nrsc.gov.in ) operated b y the C entral Water Commission (CWC) I ndia, which one of the most reliable database p ertaining to India. The map showing d ifferent major river basins are presented in Fig. 1 . The data ran ging f rom 1969 to 2016 are c onsidered fo r the stud y . For brevity, the maximum and minimum data lengths of the basin along with the maxim um and minimum drainage area of stations of different basins, are provided in Table 1. As the total suspended sediment information is really s carce, the streamflow-sediment link is investigated in five major basins b y conside ring the longest common period for which both the streamflow and sediment data are ava ilable. [Insert Figure 1 Here] [Insert Table 1 Here] Results and Discus sions In this study, first dail y streamflow data of different stations are anal y sed using the MF -DFA method b y selecting mo ment order in the range -4 to +4 and minimum scale as 10, max imum as N /2, where N is the data length. Six different pr ominent multifractal properties such as Hurst exponent (H), spread of generalized Hurst exponent plot ) ( q h  , spread of singularit y p arameter ∆ α (called as spectral width), Asymmetry index ( A α ), ∆ f ( α ), singularity parameter for zero moment order ( 0  ) etc. are evaluated. The spatial dist ribution of the different multifractal parameters is shown in Fig. 2. Further, the non-parametric Kernel density estimator (KDE) is used to develop the probabilit y densit y function and CDF of all the six mul tifractal parameters andthe results are presented in Fig. 3. [Insert Figure 2 and Figure 3 Here] From the results it is noted that m ost of the streamflow series displ ayed long term persistence (71.3 %) with a mean value of 0.585, which is less than the universal value of 0.73 report ed b y Kantelhardt et al. (2006 ). Similarly the hi gh multi fractal width and spread ( ) ( q h  ) a re noted in the database, which shows that there is a large variation in dist ribution of high and low 11 fluctuations, indicating irre gular and non-homogeneous distribution. This is quite obvious because of the high inter mittent character of river flows in th e basins cons idered in the stud y. It is to be noted that the database considered the s tations located in the southern/peninsular part of India, where in most of the rivers the streamflow is intermittent in nature and comprising of continuous zero or ver y low discharge values. I n the no rthern I ndia, abundant alluvial and perennial rivers are p resent, but most of them are trans -boundary in character for which the d ata sharing is not flexible. From Fig. 2 it is also noted that river basins Periyar, C auvery, Pennar, Vaippar, which are nea r to the southern coastal regions have high degree of multifractality. The Asymmetry index value is positive for most of stations (181 stations out of 192), which indicates left hand deviations of the spectra with local high fluctuations. From Fig. 3, it is noted that as expected the distribution of spectral width and spread (which convey the similar message on degree of multifractality) irre spective of their numerical values. The PDF of Hurst ex ponent shows a densit y concentration around 0. 5-0.7, where Hurst exponent lies in this range for most of the stations (49 %). A near s ymmetrical distribution is noted for the value of ) (  f  and the dominant densit y of 0  is in the range of 0.8-1.2. Now, for a comparison of multifractal properties of streamflow of dif ferent basins, five major basins, namely Godavari, Krishna, Mahanadi, W FR Tadri to Kanyakumari (WFR T -K) and C auvery are considered (for which datasets of minimum 10 stations are available). The P DFs and CDFs of different multifractal parameters are presented in Fig. 4. [Insert Figure 4 Here] From the PDFs and CDFs of streamflow data of river basins it is clear that the data of Krishna has least persistence (foll owed by M ahanadi) as compared with that of other basins. The hi ghest degree of multifractality is noted for the streamflows of Godavari basin which is having over 400 major and minor dams and other regulation struct ures which control the streamflows. From Fig. 4 , it is also noted that streamflows of Godavari basin has higher α 0 as compared with other basins, which infer the complexit y o f the series.From the plot of α0 it is noted that the streamflow of Krishna and WFR T -K has almost similar comple xit y which possess finer structure. In th e WFR T-K basin, no major flo w regulation structures are pr esent and the drainage areas of di fferent stations are similar in magnitude (varies between 238-5755 km 2 from Table 1 ). To get an insight 12 into the effect of drainage area and d ata len gth on the mul tifractality an d persistence, the plots between drainage area and H drainage area and spectral width, data len gth vs H, data length vs spectral width are prepared and pre sented in Fig. 5. [Insert Figure 5 Here] It is evident from the Fig. 5 it is noted that that most of the Hurst ex ponent values are centered around 0.55-0.65 and there is no major change in the value of the Hurst Exponent with drainage area. This evidently concludes that change in drainage area has no effect o n the persistence of the different se ries. No direct conclusions can be m ade from the other two plots ex cept that area and data length independently seem to hav e no significant effect on th e multifractality and persistence. MFCCA between Streamflow and Suspended Sedim ent Multifractal Cross Correlation Analysis (MFCCA) between streamflow and total suspended sediment (TSS) was performed for 5 major basins in I ndia - Cauvery, Krishna, Godavari , Mahanadi and W FR T- K b y cho osing the moment order -4 to +4, max imum scale as N/2 and minimum scale is selected as more than th e length of lon gest stretch of zero values . From the MFCCA, the individual persistence, joint persistence and cross correlation coefficient at annual and the overall correlation are determined for each case. For Cauvery basin 11 stations for which long and continuous streamflow and TSS data ar e available are considered for MFCCA analysis. The annual c ross correlation coefficient along with Hurst exponents obtained are given in Table 2. [Insert Table 2 Here] Results obtained by the MFCCA analysis for str eamflow and sedi ment data for Cauver y basin (Table 2 ) it is noted that the persistence of streamflow is more than that of TSS except for two stations. At all stations of Cauver y basin, the joint persistence is found to be nearl y the average of individual persistence of streamflow and TSS. The joint persistence is found to be stron g with a mean value of 0.733. The annual correlation is found to be more than 0.5 in five stations, but the overall correlation is found to be weak and it is less than 0.5 in all stations .The mean annual correlation is found to be 0.492 while the mean overall correlation is only 0.33. On examining 13 the correlations it was found that, 7 out of 11 stations weak seasonal correlation (at 90 day scal e) was also detected in this basin. Except for the data of Savandpur andThengumarahada stations, the annual correlation is found to be more than th at of seasonal co rrelation. Fig. 6 shows t ypical plots of multifractal analysis along with the variability of cross correlation with time scale of Kudige station. The annual and overall correlation along with Hurst exponent s of datasets of different stations of Godavari basin are given in Table 3. [Insert Table 3 Here] [Insert Figure 6 Here] From Table 3 it is noted that unlike for Cauvery basin, for majority o f the stations in Godavari basin (i.e., 14 out of 26), the persistence of TSS is more than that of streamflow. The persistence is strong and long term for both streamflow and TS S series with a mean of 0.803 and 0.789 respectively. There exists a strong annual correlation between streamflow and TSS in this basin (mean value of 0.702). The annual correlation is greater than 0.5 in 2 3 cases, out of which in 1 7 cases the correlation is more than 0.7. The ov erall correlation was found to be mor e than 0.5 in 18 cases out of which the association is strong (>0.4) in 4 cases.For th e datasets of Bishnur, Bhatpalii and Satrapur stations, both the annual and overall correlation are found to be ver y weak. It was also noted the seasonal correlation (at 3 month time scale) was also detectable at 16 out of 26 stations and annual correlation was fo und to be greater than seasonal correlation for data of all stations ex cept Satrapur. At all stations of this basin, the joint persistence is found to be the average of persistence of streamflow and TSS. Fig. 7 shows t ypical plots of multifractal analysis along with the variability of cross correlation with time scale of Polavaram station in the Godavari basin. [Insert Figure 7 here] The annual and overall correlation between streamflow and TSS along with Hurst exponents of datasets of Krishna basin are given in Table 4. [Insert Table 4 Here] 14 From Table 4 it is clear t hat for 14 out of 23 stations, the persistence of st reamflow is more than that of TSS. In this c ase, the joint persistence (with a mean of 0.614 ) is found to be the average of the individual persistence of streamflow and T SS. Strong annual correlation (>0.7) is noted in 7 cases while it is more than 0.5 in 18 cases. In 9 cases seasonal correlation was also noted and the annual correlation is greater than that o f se asonal cor relation in these stations. The overall correlation was found to be weak (with a mean o f 0.375) and in 5 cases th e correlation is found to be more than 0.5. Fig . 8 shows typica l plots of multifractal analysis of streamflow and sediment data along with the variability of Cholachguda station in Krishna basin. [Insert Figure 8 here] The seasonal and annual cross correl ation coeffic ient along with Hurst exponents of datasets of Mahanadi basin is given in Table 5. [Insert Table 5 Here] From Table 5 it is noticed that in 81% of stations (i.e., 13 out of 16) the persistence of streamflow is more tha n that of TSS. Except in two cases, the seasonal correlation was detected only at Basantpur and Tikarapara station. Th is cro ss correlation coefficient is more than 0.7 at all stations except Kesinga indicating ver y strong positive correlation between the parameters in the basin and reasonably good overall correlation (>0.4) is noted at 14 stations. The mean value of annual correlation is found to be 0.748 while it is 0.495 for overall data . T he correlation plot and multifractal plots of Basantpur station is presented in Fig. 9. [Insert Figure 9 here] The results of MFCCA of streamflow and TSS of WFR Tadri Kan y akumari (WF R T-K) are given in Table 6. From Table 6 it is clear that the persistence of streamflow is more that of sediment for 9 stations . The joi nt persistenceis nearly the mean o f the individual persistence of streamflow and TSS stations of different stations. There exists reasonably good correlation at annualscalewith a mean correlation of 0.75 and the overall correlation was also more than 0.5 in 14 cases. The seasonal association was detectable at 9 station a nd the annual scale correlation is greater than the seasonal correlation for all the stations except for the data of Kumbidi station . 15 The annual cross c orrelation is greater than 0.5 in 18 cases out of which in 14 cases the correlation is found to be >0.7. Fig. 10 shows the multifractal plots of Ramamangalam station. [Insert Figur e 10 here] [Insert Table 6 here] In general, in most of the stations ( 57 out of 95 stations) the persistence o f streamflow is greater than that of TSS . In Godavari basin, majority of the stations the persistence of TSS is more than that of streamflow. The human interventions and flow re gulations might have influenced the persistence and multifractalit y of streamflow in this basin to a great extent . The investi gation using MFCCA provides the ti me (scale) dependent information of the association between streamflow and TSS against the unique and traditional linear correlation between them. i.e., eventhough the overall correlation between the two are less, at sp ecific time scale the association could be of considerable magnitude. In 45 st ations, seasonal (intra-annual) association between streamflow and TSS are also noticed, among which highest number of stations (18 stations) are located in Godavari basin. This also infers the role of flow regulations in streamflow-TSS links of this basin. Eventhough streamflow -TSS association varies with temporal scales and there is no systematic pattern in this variation for the datasets of different basins . But it is noted that the strength of their associat ion could var y si gnificantly with time scale a nd t heir association could significantly depend on the basin and climatic (precipitation) characteristics. Conclusions This stud y first investigated the multifractality of streamflow of 1 92 stati ons falling in 13 river basins in India using the Multifractal Detrended Fluctuation Anal y sis (M F-DFA). Subsequentl y, the Multifractal Cross Correlation Analysis (MFCCA) is emplo yed for inv estigating the streamflow-sediment link in a multi fractal perspective. From the results it is noted that t he streamflow datasets of d ifferent river basins disp layed multifractality and long term persistence with a mean ex ponent of 0.583. The streamflow records of Krishna basin displayed least persistence and that of Godavari displ ayed st rongest multifractalit y and complexit y. The streamflow-sediment links of five m ajor river basins evaluated usin g MFCC A showed that the joint persiste nce is nea rly the mean of the persistence of individual series. The stre amflow displayed higher persistence than total suspended sediment in majorit y of t he stations except that 16 in Godavari basin. The annual cross correlation between streamflow and sediment is higher than seasonal and overall cross correlation but the stre ngth of their association differs with river basin. Funding The authors received NO funding for performing this research wor k Conflict of Interest On behalf of all authors, the corresponding author states that there is no conflict of interest. References Adarsh S, Drisya SD , Anuja PK , Aggie S (201 8a) Unravelling the scaling characteri stics of daily streamflows of Brahmani river basin, India using Arbitrary Order Hilbert Spectral and Detrended Fluctuation Anal y ses, SN Applied Sciences, 1(2018),58. DO I: 10.1007/s42452 -018- 0056-1 Adarsh S, Drisya SD , Anuja PK (2018b) Anal y zing the H ydrolog ic Variabilit y of Kallada River, India Using Continuous Wavelet Transform and Fractal Theor y, Wat. Cons. Sci. En g. https://doi.org/10.1007/s41101-018-0060-8 Adarsh S, Kumar DN , Dee pthi B, Gayathri G, Aswathy SS , Bhag y asree S (2019) Multifractal characterization of mete orological drought in I ndia using detrended fluctuation analysis, Int . J. Climatol. DOI: 10.1002/joc.6070 Baranowski P, Krz yszczak J, Slawinski C, Hoffmann H, Koz yra J , Nieróbca A, Siwek K, Gluza A (2015) Mult ifractal a nalysis of meteorological time series to assess climate impacts, Clim . Res. 65:39 – 52 Brito AA, Santos FR , A.P. N de Castro, A.T. da Cunha Lima, G.F. Zebende, I.C. da C unha Lima, Cross- correlation in a turbulent flow: Ana lysis of the velocity fiel d using the σ DCCA coefficient, Europhy. Lett. 123(2018), 20011 Dahlstedt K, Jensen H (2005) Fluctuation spectrum and size scaling of river flow and level, Physica A 348:596 – 610 17 Dey P, Mujumdar PP (2018) Multiscale evolution of persistence of rainfall and streamflow, Adv . Wat. Resour. 121:285-303 Drożdż S , Oświęcimka P (2015) Detecting and interpreting distortions in hierarchical organization of complex time series. Phys. Rev. E 91, 030902(R) Grech D (2016) Alternati ve measure of multifractal content and its application in finance, Chaos, Solitons& Fractals, 88:183-195 Hajian S, Movahed MS (2010) Mult ifractal detrended cross-correlation analy sis of sunspot numbers and river flow fluctuations, Physica A 389, 4942-57 Hirpa FA , Gebremichael M, Over TM (2010 ) River flow fluctuation analysis: Ef fect of watershed area, Wat. Resour. Res. 46(12) https://doi.org/10.1029/2009WR009000 Huang YX, Schmitt FG, Hermand JP, Gagne Y, Lu ZM, Liu YM (2011) Arbitrary-order Hilbert spectral anal y sis for time series possessing scaling statistics: a comparison stud y with detrended fluctuation analysis and wavelet leaders, P hys. Rev. E 84, 016208, 2011, 10.1103/PhysRevE.84.016208 Hurst H.E. Long ‐ term storage capacity of reservoirs, Trans ASCE 116 (1951) 770 – 808 Hurst HE , Black RP , Simaika YM (1965) Long ‐ term storage: An experimental study, Constable, London. Ihlen EAF (2012) Introduction to multifractal detrende d fluctuation analysis in MAT LAB, Frontiers in Physiology 3:141, doi: 10.3389/fphy s.2012.00141 Jiang ZQ , Wen-Jie Xi., Zhou W X, Sornette D (2011) Multifractal anal ysis of financial markets, Quantitative Finance. Retrieved from http://arxiv.org/abs/1805.04750 Jiang ZQ , Zhou W X (2011) Multifractal detrending moving-average cross-correlation analysis, Phy s. Rev. E 84, 016106. Kantelhardt JW , Bunde EK , Rybski D, B arun P, Bunde A, Havlin S (2006) L ong-term persistence and multifractality of precipitation and river runoff records, J . Geophys. Res: Atmos. 28 :1-13 18 Kantelhardt JW , R y bski D, Zschiegner SA , P. Braun, E. Koscieln y-Bunde, V. Livina, S. Havlin, A. Bunde , Multifractality of river runo ff and pr ecipitation: comparison of fluctuation anal y sis and wavelet methods, Phy sica A 330 (2003) 240 – 245 Kantelhardt JW , Zschiegner SA , Koscieln y ‐ Bunde E, Halvin H, Bunde A, S tanley HE , Multifractal detrended fluctuation analysis of non ‐ stationary time series, Physica A 316 : 87 – 114 Koscielny-Bunde E, Kantelhardt JW , Braun P, Bunde A, Havlin S (2003) Long-term persistence and multifractality of river runoff records: detrended fluctuation studies, J. Hydrol. 322:120 – 137 Krzyszczak J, Baranowski P, Zubik M, Kazandjiev V, Geor gieva V , Sł awiński C, Siwek K , Kozyra J , Nieróbca A (2018) Multifractal characterization and comparison of meteorolog ical tim e series from two climatic zones. Theoret . Appl Climatol. DO I : https://doi.org/10.1007/s00704-018-2705-0 Kwapień J , Oświęcimka P, Drożdż S (2015) Detrended fluctuation anal y sis made flexible to detect range of cross-correlated fluctuations. Phys. Rev. E. 92(2015), 052815. Labat D, Masbou J , Beaulieu E, Mangin A (2011) Scaling behavior of the fluctuations in stream flow at the outlet of karstic watersheds, France, J. Hydrol. 410:162-168 Li E, Mu X, Zhao G, Gao P (2015) Multifractal detrended fluctuation analysis of streamflow in Yellow river basin, China, Water 7: 1670-1686 Mandelbrot B, The fractal geometry of nature. WH Freeman Publishers. New York, 1982 Muzy JF , Bacr y E, Arneodo A (1991) Wavelets and multifractal formalism for sin gular si gnals: application to turbulence data, Phys. Re v. Lett. 67 (1991) 3515 – 3518 Oświcimka P , Drożdż S, Forczek M, Jadach S , Kwapień J (2014) Detre nded cross-correlation analysis consistently extended to multifractality, Phys. Rev E. 89, 023305. Oświęcimka P , Drożdż S , Kwapień J (2013) A. Górski, Effect of detrending on multifractal characteristics, Acta Phys. Pol. A 123: 597 – 603 19 Pandey G, L ovejoy S, Schertzer D (1998) Multifractal analysis of dail y river flows in cluding extremes for basins five to two million square kilometers, one da y to 75 years. J . Hydrol. 208: 62 – 81 Peng CK , Buldyrev SV , Simons M , Stanle y HE , Goldberger AL (1994) Mosaic organization of DNA nucleotides. Phys. Rev. E. 49:1685 – 1689 Piao L, Fu Z (2016 ) Quantifying distinct associations on different temporal scales: comparison of DCCA and Pearson methods, Scientific Reports 6, 36759 Podobnik B, Stanley HE (2008) Detrended cross-correlation anal ysis: a new method for analyzing two non-stationary time series, Phys. Rev. Lett., 100(8), 084102 Rego CRC , Frota HO , Gusmão MS (2013) Multifractality of Brazilian rivers, J . Hy drol. 495: 208 – 215 Shi K (2014) Detrended cross-correlation anal ysis of temperature, rainfall, PM 10 and ambient dioxins in Hong Kong. Atmos. Environ. 97 (2014), 130-135 Tan X, Gan TW (2017) Multifractality of Canadian precipitation and streamflow, I nt. J . Climatol. 37 (S1) : 1221-1236 Tessier Y, Lovejoy S, Hubert P , Schertzer D, P ecknold S (1996) M ultifractal ana l ysis and modeling of rainfall and river flows and scaling, causal transfer functi ons, J . Geophys. Res: Atmos 101:26427 – 26440 Vassoler RT, Zebende GF (2012) DCCA cross-correlation coefficient apply in time series of air temperature and air relative humidity, Physica A 391, 2438-2443 Wątorek M , Drożdż S , Oświęcimka P, Stanuszek M (2019) Multifractal cross-correlations between the wo rld oil and other financial markets in 2012 – 2017, Energy Economics 81 : 874- 885. Wu Y, He Y, Wu M, Lu C, Gao S, Xu Y (2018) Multifrac tality and cross correlation analysis of streamflow and sediment fluctuation at the apex of the P earl R iver Delta, S cientific R eports, 8 16553 , DOI:10.1038/s41598-018-35032- z 20 Yu ZG , Leung Y, Chen YD , Zhan g Q, Anh V, Zhou Y (2014) Multifractal analyses of daily rainfall time series in Pearl River basin of China, Phy sica A 405 : 193 – 202 Yuan N, Fu Z, Mao J (2010) Different scaling behaviors in daily temperature records over China, Physica A 389(19): 4087 – 4095 Zebende GF (2011) DCCA cross-correlation coefficient: Quantifying lev el of cross-correlation, Physica A 390: 614-618 Zhang Q, Xu CY , Chen DYQ , Gemmer M, Yu ZG (2008), Multifractal detrended fluctuation analysis of streamflow series of the Yangtze river basin, China, Hydrol. Process. 22: 4997 – 5003 Zhang Q, Y.X. Chong, Z.G. Yu, C.L. Liu, D.Y .Q. Chen, Mult ifractal analysis of streamflow records of the East River basin (Pearl River) , China, Ph ysica A 388 ( 2009) 927 – 934 Zhou WX (2008) Multifractal detrended cross -correlation analysis for two non -stationary signals. Phys. Rev. E. 77, 066211 21 Table 1 Details of streamflow data used for the study Sl No Basin Number of stations Drainage Area (km 2 ) Data length Minimum Maximum Minimum Maximum 1. Krishna 31 1850 251360 1095 18615 2. Brahmani-Baitarani 9 830 33955 4015 15330 3. Sabarmati 6 1421 19636 5840 9490 4. Mahi 7 1510 32510 3285 13805 5. Mahanadi 19 1100 124450 4015 15695 6. Subarnarekha 5 1330 12649 6205 14235 7. Tapi 5 8487 58400 3285 5110 8. Cauvery 31 258 66243 2555 16425 9. WFR Tadri-Kanyakumari 28 238 5755 1460 16425 10. EFR Pennar-Kanyakumari 13 850 16230 4015 16060 11. Godavari 23 2500 307800 1019 13111 12 Pennar 7 2486 37981 1245 10606 13 WFR-Kutch- Saurashtra-Luni 8 345 6960 6865 15111 22 Table 2 Hurst exponents of streamflow and TSS data of Ca uver y ba sin along with the cross correlation Station Hx (Streamflow) Hy (TSS ) Scaling Exponent (Hxy) ρ XY (Annual) ρ XY (Overall) Biligundulu 0.797 0.669 0.733 0.404 0.274 Kodumudi 0.904 0.742 0.823 0.627 0.414 Kollegal 0.736 0.672 0.704 0.428 0.172 Kudige 0.745 0.655 0.700 0.431 0.249 Musiri 0.656 0.641 0.649 0.664 0.504 Muthankera 0.561 0.779 0.670 0.716 0.494 Savandpur 0.752 0.658 0.705 0.450 0.386 T Narasipur 0.781 0.633 0.707 0.208 0.090 TK Halli 0.688 0.673 0.681 0.566 0.415 Tehngudi 0.823 0.883 0.853 0.725 0.360 Thengumarahada 0.839 0.829 0.834 0.241 0.275 23 Table 3 Hurst exponents of streamflow and TSS data of Goda vari basin along with the cross correlation Station Hx (Streamflow) Hy (TSS ) Scaling Exponent(Hxy) ρ XY (Annual) ρ XY (Overall) Ashti 0.844 0.920 0.882 0.823 0.602 Babli 0.982 0.959 0.970 0.657 0.442 Bamini(Balharsha) 0.727 0.778 0.752 0.691 0.566 Basar 1.00 0.965 0.994 0.588 0.521 Bhatpalli 0.587 0.644 0.615 0.357 0.267 Bishnur 0.739 0.457 0.598 0.231 0.188 Dhalegaon 0.653 0.669 0.661 0.631 0.460 G.R.Bridge 0.767 0.737 0.752 0.714 0.561 Hivra 0.633 0.570 0.601 0.619 0.498 Jagdalpur 0.838 0.882 0.860 0.713 0.530 Konta 0.786 0.845 0.815 0.811 0.548 Kumhari 0.910 0.893 0.901 0.871 0.599 Mancheri al 0.860 0.796 0.828 0.518 0.391 Nandgaon 0.724 0.751 0.737 0.768 0.614 Nowrangpur 0.850 0.877 0.864 0.754 0.662 P.G. (Penganga) Bridge 0.489 0.316 0.402 0.710 0.489 Pathagudem 0.822 0.894 0.858 0.886 0.722 Pauni 0.718 0.782 0.750 0.807 0.575 Perur 0.889 0.898 0.893 0.950 0.915 Polavaram 0.908 0.830 0.869 0.932 0.855 Purna 0.782 0.747 0.764 0.722 0.594 Rajegaon 0.998 1.00 1.000 0.908 0.549 Saigaon 0.597 0.562 0.580 0.769 0.584 Satrapur 0.957 0.877 0.917 0.117 0.277 Tekra 0.848 0.867 0.858 0.884 0.685 Yelli 0.942 0.965 0.953 0.809 0.755 24 Table 4 Hurst exponents of streamflow and TSS data of Kr ishna basin along with the cross correlation Station Hx (Streamflow) Hy (TSS ) Scaling Exponent(Hxy) ρ XY (Annual) ρ XY (Overall) Bagalkot 0.540 0.541 0.540 0.441 0.205 Bawapuram 0.577 0.505 0.541 0.644 0.434 Byaladahalli 0.912 0.870 0.891 0.813 0.607 Cholachguda 0.597 0.682 0.639 0.808 0.660 Haralahalli 0.751 0.683 0.717 0.383 0.296 Honnali 0.967 1.027 0.997 0.589 0.194 Huvanahedgi 0.721 0.650 0.685 0.241 0.174 K Agraharam 0.713 0.621 0.667 0.677 0.430 Karaad 0.480 0.449 0.465 0.659 0.346 Keesara 0.591 0.548 0.569 0.570 0.302 Kurundwad 0.420 0.487 0.453 0.938 0.795 Malkhed 0.655 0.639 0.647 0.721 0.210 Mantralayam 0.559 0.557 0.558 0.575 0.353 Marol 0.525 0.578 0.552 0.396 0.143 Pondugala 0.645 0.857 0.751 0.337 0.112 Yadgir 0.490 0.392 0.441 0.686 0.524 Warunji 0.655 0.654 0.654 0.731 0.493 Wadanapalli 0.675 0.750 0.713 0.572 0.336 Wadakb al 0.582 0.558 0.570 0.617 0.484 Vijayawada 0.656 0.590 0.623 0.702 0.330 Takli 0.468 0.365 0.416 0.504 0.232 Shimogs 0.557 0.628 0.592 0.917 0.686 Sarati 0.421 0.446 0.434 0.523 0.320 25 Table 5 Hurst exponents of streamflow and TSS data of Ma hanadi basin along with the cross correlation Station Hx (Streamflow) Hy (TSS ) Scaling Exponent(Hxy) ρ XY (Annual) ρ XY (Overall) Andhiyarkore 0.527 0.341 0.434 0.721 0.490 Bamnidhi 0.517 0.506 0.512 0.759 0.489 Baronda 0.498 0.416 0.457 0.757 0.423 Basantpur 0.691 0.701 0.696 0.816 0.552 Ghatora 1.000 0.991 1.00 0.782 0.629 Jondhra 0.537 0.505 0.521 0.801 0.513 Kantamal 0.538 0.415 0.477 0.726 0.489 Kesinga 0.99 1.000 1.00 0.316 0.364 Kurubhata 0.573 0.571 0.572 0.892 0.740 Manendragarh 0.665 0.777 0.721 0.779 0.528 Rajim 0.499 0.396 0.448 0.700 0.379 Rampur 0.483 0.378 0.430 0.835 0.484 Salebhata 0.462 0.386 0.424 0.763 0.453 Simga 0.487 0.400 0.444 0.720 0.403 Sundaragarh 0.465 0.387 0.426 0.833 0.574 Tikarapara 0.762 0.721 0.741 0.765 0.420 26 Table 6 Hurst exponents of streamflow and TSS data of WFR-Tadri to Kany akumari basin alon g with the cross correlation Station Hx (Streamflow) Hy (TSS ) Scaling Exponent (Hxy) ρ XY (Annual) ρ XY (Overall) Ambarampalayam 0.963 0.648 0.806 0.431 0.368 Arangaly 0.500 0.665 0.582 0.855 0.611 Ayilam 0.605 0.558 0.582 0.749 0.581 Bantwal 0.391 0.517 0.454 0.866 0.710 Erinjipuzha 0.722 0.686 0.704 0.897 0.669 Kalampur 0.585 0.636 0.611 0.786 0.564 Kallooppara 0.522 0.593 0.557 0.783 0.420 Karathodu 0.717 0.776 0.747 0.834 0.701 Kidangoor 0.594 0.803 0.699 0.657 0.348 Kumbidi 0.733 0.772 0.752 0.714 0.637 Kuniyil 0.584 0.626 0.605 0.711 0.589 Kuttyadi 1.000 0.997 1.00 0.595 0.415 Malakkara 0.560 0.645 0.603 0.678 0.533 Neeleswaram 1.00 0.916 0.956 0.858 0.615 Pattazhy 0.695 0.685 0.690 0.641 0.555 Perumannu 0.881 0.764 0.822 0.904 0.695 Pulamanthole 0.844 0.772 0.808 0.810 0.621 Ramamangalam 0.681 0.663 0.672 0.782 0.501 Thumpamon 0.595 0.674 0.634 0.730 0.483 27 Fig.1 Map showing river basins in India 28 60 70 80 90 100 5 10 15 20 25 30 35 40 Longitu de Latitude <0.5 0.5-0. 7 >0.7 60 70 80 90 100 5 10 15 20 25 30 35 40 Longitu de Latitude <1 1-2 >2 60 70 80 90 100 5 10 15 20 25 30 35 40 Longitu de Latitude <1 1-2 >2 60 70 80 90 100 5 10 15 20 25 30 35 40 Longitu de Latitude <0 0-0.4 >0.4 60 70 80 90 100 5 10 15 20 25 30 35 40 Longitu de Latitude <0 0-0.5 >0.5 60 70 80 90 100 5 10 15 20 25 30 35 40 Longitu de Latitude <0.5 0.5-1 >1 (a) H u rst Ex pon en t (b)  h ( q) (c) Spectr al Width (d) Assy met ry I n dex (e)  f(  ) (f )  0 Fig. 2 Spatial distribution of multifractal parameters of streamflow all over India (a) Hurst exponent; (b) ) ( q h  ;(c) spectral width;(d)Asymmetry index; (e) ∆ f ( α ); (f) 0  29 0.1 0.2 0. 3 0. 4 0. 5 0.6 0.7 0 .8 0.9 1 1.1 0 0.5 1 1.5 2 2.5 Hu rst Ex ponen t PDF 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.2 0.4 0.6 0.8  h( q) PDF 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 Spectral Width PDF -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 Asym metry I n dex PDF -0.6 -0.4 -0.2 0 0.2 0.4 0 .6 0.8 1 1.2 0 0.5 1 1.5 2  f(  ) PDF 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1 .8 2 0 0.5 1 1.5  0 PDF (a) Hu rst Ex pon ent (b)  h (q) (c) Spectral Width (d) Assy metry I n dex (e)  f(  ) (f )  0 Fig. 3. PDF of different multifractal parameters of streamflow data 30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 1 2 3 4 Hu rst Ex ponen t PDF Godav ari WFR T -K Cauv ery Mah anadi Krishn a 0 0.2 0.4 0.6 0. 8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 Hu rst Ex ponen t CDF Godav ari WFR T -K Cauv ery Mah anadi Krishn a 0 0.5 1 1.5 2 2.5 3 3 .5 0 0.5 1 1.5 2  h (q) PDF Godav ari WFR T -K Cauv ery Mah anadi Krishn a 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1  h (q) CDF Godav ari WFR T -K Cauv ery Mah anadi Krishn a 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Spectral width PDF Godav ari WFR T -K Cauv ery Mah anadi Krishn a 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 Spectral width CDF Godav ari WFR T -K Cauv ery Mah anadi Krishn a -0.4 -0.2 0 0.2 0.4 0. 6 0.8 1 0 1 2 3 4 5 R PDF Godav ari WFR T -K Cauv ery Mah anadi Krishn a -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 R CDF Godav ari WFR T -K Cauv ery Mah anadi Krishn a -0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5  f(  ) PDF Godav ari WFR T -K Cauv ery Mah anadi Krishn a -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1  f(  ) CDF Godav ari WFR T -K Cauv ery Mah anadi Krishn a 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.5 1 1.5 2 2.5  0 PDF Godav ari WFR T -K Cauv ery Mah anadi Krishn a 0.2 0.4 0.6 0 .8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1  0 CDF Godav ari WFR T -K Cauv ery Mah anadi Krishn a (a) Hu rst Ex pon ent (b)  h (q) (c) Spectral Width (d) Assy metry I n dex (e)  f(  ) (f )  0 Fig. 4 PDFs and CDFs of different multifractal parameters for basin wise analysis of streamflow datasets 31 2.8 3 3.2 3.4 3 .6 3.8 4 4.2 4.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 log(DataLen gth ) Hurst Exponent 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 log(DataLen gth ) Spectral Width 2 2 .5 3 3.5 4 4 .5 5 5.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 log(Drain age Area) Hurst Exponent 2 2. 5 3 3.5 4 4.5 5 5. 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 log(Drain age Area) Spectral Width Fig. 5 Influe nce of drainage area and data length on persistence and multifractality (a) Hurst Exponent vs log(Drainage Area); (b) spectral width vs log(Draina ge Area); (c) Hurst Exponent vs log (Data length); (d) spectra l width vs log(Data length) 32 1.5 2 2.5 3 3 .5 -1 0 1 2 3 4 log (Scale) log (F(q,s)) SF TSS SFvsTS S 1.5 2 2.5 3 3 .5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 log(Scale)  DCCA -4 -2 0 2 4 0 0.5 1 1.5 2 2.5 q order H(q)of Fxy SF TSS SFvsTS S -4 -2 0 2 4 -10 -8 -6 -4 -2 0 2 q order  (q) SF TSS SFvsTS S 0 0.5 1 1.5 2 2 .5 -0.2 0 0.2 0.4 0.6 0.8 1 1.2  f(  ) SF TSS SFvsTS S (a) Scaling ex pon en t pl ot (b) Mass ex pon ent pl ot (c) Mu lt ifractal spectr um (d) Plot of Flu ct ul ation fu ct ion f or q= 2 (e) Tempor al v ariability of cross correl ation Fig. 6 Plots of mul tifractal analysis of data o f Kudige station along with the variability of cross correlation (a) Sc aling exponent plot; (b) mass ex ponent plot; (c) multifractal spectrum; (d) lo g - log plot of fluctuation function vs scale for q =2; (e) temporal variabilit y of cross correlation coefficient 33 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 log (Scale) log (F(q,s)) SF TSS SFvsTS S 1.5 2 2.5 3 3.5 4 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 log(Scale)  DCCA -4 -2 0 2 4 0 0.5 1 1.5 2 2.5 q order H(q)of Fxy SF TSS SFvsTS S -4 -2 0 2 4 -12 -10 -8 -6 -4 -2 0 2 q order  (q) SF TSS SFvsTS S 0 0.5 1 1.5 2 2 .5 3 -0.2 0 0.2 0.4 0.6 0.8 1 1.2  f(  ) SF TSS SFvsTS S (a) Scaling ex pon en t pl ot (b) Mass ex pon ent pl ot (c) Mu lt ifractal spectr um (d) Plot of Flu ct ul ation fu ct ion f or q= 2 (e) Tempor al v ariability of cross correl ation Fig. 7 Plot s of multifractal analysis of data of P olavaram station along with the variability of cross correlation (a) Scaling exponent plot; (b) mass exponent plot; (c) multi fractal spectrum; (d) log-lo g plot of fluctuation function v s scale q =2; (e) temporal variabilit y of cross correlation coefficient 34 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4 log (Scale) log (F(q,s)) SF TSS SFvsTS S 1.5 2 2.5 3 3.5 4 0.4 0.5 0.6 0.7 0.8 0.9 log(Scale)  DCCA -4 -2 0 2 4 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 q order H(q)of Fxy SF TSS SFvsTS S -4 -2 0 2 4 -10 -8 -6 -4 -2 0 2 q order  (q) SF TSS SFvsTS S 0 0.5 1 1.5 2 2 .5 -0.2 0 0.2 0.4 0.6 0.8 1 1.2  f(  ) SF TSS SFvsTS S (a) Scaling ex pon en t pl ot (b) Mass ex pon ent pl ot (c) Mu lt ifractal spectr um (d) Plot of Flu ct ul ation fu ct ion f or q= 2 (e) Tempor al v ariability of cross correl ation Fig. 8 Plots of mul tifractal anal y sis of data of Cholachguda station along with the variability of cross correlation (a) Scaling ex ponent plot; (b) mass exponent plot; (c) multi fractal spectrum (d) log-log plot of fluctuation function vs scale for q =2; (e) temporal variability of cross co rrelation coefficient 35 1.5 2 2.5 3 3.5 4 -1 0 1 2 3 4 5 log (Scale) log (F(q,s)) SF TSS SFvsTS S 1.5 2 2.5 3 3.5 4 0.5 0.6 0.7 0.8 0.9 log(Scale)  DCCA -4 -2 0 2 4 0 0.5 1 1.5 2 2.5 q order H(q)of Fxy SF TSS SFvsTS S -4 -2 0 2 4 -10 -8 -6 -4 -2 0 2 q order  (q) SF TSS SFvsTS S 0 0.5 1 1.5 2 2 .5 -0.2 0 0.2 0.4 0.6 0.8 1 1.2  f(  ) SF TSS SFvsTS S (a) Scaling ex pon en t pl ot (b) Mass ex pon ent pl ot (c) Mu lt ifractal spectr um (d) Plot of Flu ct ul ation fu ct ion f or q= 2 (e) Tempor al v ariability of cross correl ation Fig. 9 Plots of multifr actal analysis o f dat a of Basantpur station along with the variabilit y of cross correlation (a) Scaling exponent plot; (b) mass exponent plot; (c) mul tifractal spectrum; (d) lo g-log plot of fluctuation function vs scale for q =2; (e) temporal variability of cross correlation coefficient 36 1.5 2 2 .5 3 3.5 4 -2 -1 0 1 2 3 4 log (Scale) log (F(q,s)) SF TSS SFvsTS S 1.5 2 2 .5 3 3.5 4 0.4 0.5 0.6 0.7 0.8 0.9 1 log(Scale)  DCCA -4 -2 0 2 4 0.4 0.6 0.8 1 1.2 1.4 q order H(q)of Fxy SF TSS SFvsTS S -4 -2 0 2 4 -8 -6 -4 -2 0 2 q order  (q) SF TSS SFvsTS S 0 0.5 1 1 .5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4  f(  ) SF TSS SFvsTS S (e) Tempor al v ar iabi lity of cross correl ation (d) Plot of Flu ct ul ation fu ct ion f or q= 2 (a) Scaling ex pon en t pl ot (b) Mass ex pon ent pl ot (c) Mu lt ifractal spectr um Fig. 10 Plots of multifractal cross-correlation analysis of data o f Ramamangalam station along with the variability o f cross correlation (a ) Scaling exponent plot; (b) mass ex ponent plot; (c) multifractal spectrum; (d) log-log plot of fluctu ation function vs scale for q =2; (e) temporal variability of cross correlation coefficient