Water Resour Manage (2007) 21:1829–1848 DOI 10.1007/s11269-006-9132-1 Another Look at Z-transform Technique for Deriving Unit Impulse Response Function R. K. Rai & M. K. Jain & S. K. Mishra & C. S. P. Ojha & V. P. Singh Received: 18 January 2006 / Accepted: 11 December 2006 / Published online: 20 January 2007 # Springer Science + Business Media B.V. 2007 Abstract This paper presents a technique to derive the unit impulse response functions (UIRF) used for determination of unit hydrograph by employing the Z-transform technique to the response function derived from the Auto Regressive Moving Average (ARMA) process of order (p, q). The proposed approach was applied to reproduce direct surface runoff for single storm event data registered over four watersheds of area ranging from 0.42 to 295 km2. It is observed that the UIRF based on ARMA (1, 2) and ARMA (2, 2) provides a better representation of the watershed response. Further, to test the superiority of the developed impulse response function form ARMA process, the direct runoff hydrographs were computed using the simple ARMA process and optimized Nash’s two parameter model and compared with the results obtained from UIRF’s of ARMA model. The performance of the models based on the graphical presentation as well as from the test statistics viz. RMSE and MAPE indicates that UIRF-ARMA (p, q) performs better than optimized Nash Model and mostly similar to simple ARMA (p,q) model. Further more, the ARMA process of order p≤2 and q≤2 is generally sufficient and less cumbersome than the Argand diagram based approach for UIRF derivation. R. K. Rai (*) Department of Hydrology, Indian Institute of Technology Roorkee, Roorkee 246 667 U.A., India e-mail: rai.rkhyd@gmail.com M. K. Jain Department of Hydrology, Indian Institute of Technology Roorkee, Roorkee 247 667 U.A., India e-mail: jain.mkj@gmail.com S. K. Mishra Department of WRDM, Indian Institute of Technology Roorkee, Roorkee 246 667 U.A., India e-mail: skm61fwt@iitr.ernet.in C. S. P. Ojha Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee 246 667 U.A., India e-mail: cojhafce@iitr.ernet.in V. P. Singh Department of Biological and Agricultural Engineering, Texas A & M University, College Station, TX 77843-2117, USA e-mail: vsingh@tamu.edu 1830 R.K. Rai, et al. Key words ARMA . transfer function . unit impulse response function . Z-transform Notations δ θ (B) φ (B) β0,β1......βq α1,α2, ......αp b Q(t) a1,a2,....ap ARMA B b0,b1,.....bq CE H(Z) H(t) H(t) H(t) I(t) I(Z) IRF n p q Q(t) Q(Z) R T UIRF Dirac delta function moving average operator autoregressive operator time-invariant parameters in continuous system time-invariant parameters in continuous system observed direct runoff ordinate at time t time-invariant parameters in discrete system Auto Regressive Moving Average backward shift operator Time-invariant parameters in discrete system coefficient of efficiency Z-transform of transfer function impulse response function at time t inverse transform of H(z) impulse response function at time t input (effective rainfall) at time t Z-transform of I (t) impulse response function time of last observation positive integer positive integer computed direct runoff ordinate at time t Z-transform of Q(t) radius of convergence sampling period Unit Impulse Response Function 1 Introduction Generally, the watershed’s rainfall-runoff process is nonlinear and complex in nature and involves number of variables in the modeling. The input–output mathematical models based on the linear theory of hydrologic system attempt to establish a causal linkage between input and output without detailed description of physical process under investigation and are widely accepted theory in hydrologic modeling. The unit hydrograph (UH) concept is one such theory and perhaps the most used modeling tool for computation of flood hydrograph. The unit hydrograph or unit impulse response function (UIRF) is an important tool for system identification in hydrological analysis and can be used as mathematical description of a linear system (Sherman 1932; Dooge 1959, 1973; Chow et al. 1988; Singh 1988; etc.). A number of transform techniques, such as, Harmonic series (O’Donnel 1960; Singh et al. 1982), Fourier transform (Blank and Delleur 1968; Blank et al. 1971), and Laplace transform (Diskin 1967; Blank and Delleur 1968; Dooge 1973) are available and used to develop response functions of the rainfall-runoff system in continuoustime domain. Since the hydrologic data are available in sampled or discrete form only, it is useful to employ discrete-time models and, therefore, a number of investigators have suggested Another look at Z-transform technique 1831 approaches for expressing the continuous process in discrete form. For example, Spolia and Chander (1974) presented a discrete and coincident form of the equal-reservoir cascade model (Nash 1957). O’Connor (1976) developed a discrete linear cascade model using the cascade concept of the Auto Regressive Moving Average (ARMA) and derived UIRF for a family of discrete-parametric models. Later, O’Connor (1982) used the transfer-function approach to derive the discretely coincident forms of several linear, continuous, and timeinvariant models. Wang and Wu (1983) represented discrete input data using unit step functions and, for developing a rainfall-runoff model for small watersheds, the discrete excess rainfall-runoff model was successfully applied for computing the runoff hydrograph (Wang et al. 1991, 1992). Among others, the Z-transform method can also be applied to develop the above response functions in discrete time domain (Singh 1988). This technique assumes that the rainfall-runoff process, an extremely complex process, behaves as a linear system for which a Z-transform of the direct runoff equals the product of the Z-transform of the transfer function and the effective rainfall. Turner et al. (1989), Singh (1997), and Ojha et al. (1999) among others utilized this technique to obtain various combinations of unit hydrograph and effective rainfall hyetograph. Very high (up to 27) order polynomials were considered to analyze the isolated storm events for derivation of unit hydrograph ordinates employing the Argand diagram (Turner et al. 1989; Singh 1997; Ojha et al. 1999). For pragmatic reasons, it is in order to explore the possibility of the polynomial order reduction and provide analytical expressions for obviating the cumbersome use of Argand diagram, which does not appear to have been reported in literature. Thus, the objectives of this paper are to (a) derive analytically the UIRF from the transfer functions of ARMA (p, q) process, designated as UIRF-ARMA (p, q), using the Ztransform and inverse Z-transform techniques; (b) investigate the fitting of ARMA type effective rainfall-direct runoff process of minimum (up to 2) order to eliminate the use of Argand diagram; and test the proposed methodology with field data and compare with a few available existing methods. 2 Mathematical Formulation 2.1 ARMA (p, q) Rainfall-runoff Process For a linear continuous system, the general relationship between the time-dependent input I (t) and output Q(t) is described by the following differential equation (Casti 1977). " 1þ p X i¼1 i # αi D QðtÞ ¼ " q X j¼0 βj D j # I ðtÞ ð1Þ where Di is the ith order differential operator with respect to time t; α and β are unknown time-invariant parameters; and Q(t) and I(t) are output (direct runoff in m3/s) and input (effective rainfall intensity in m3/s), respectively. For hydrologic applications, the values of p and q generally do not exceed four (Chow 1972). Box and Jenkins (1976) discretized Eq. 1 as:  1 a1 B a2 B2





   ap Bp QðtÞ ¼ b0 þ b1 B þ b2 B2





þ bq Bq I ðtÞ ð2Þ 1832 R.K. Rai, et al. in which B is the backward shift operator such that B½Qðt ފ ¼ Qðt 1Þ. Equation 2 represents an autoregressive and moving average process of order p and q or ARMA (p, q) and can also be written as (Box and Jenkins 1976; Wang and Yu 1986): QðtÞ ¼ ½θð BÞ=φð BފI ðtÞ ð3Þ where   θðBÞ ¼ b0 þ b1 B þ b2 B2





þ bq Bq and  φð BÞ ¼ 1 a1 B a 2 B2





ap Bp ð4Þ  ð5Þ The operator θð BÞ=φð BÞ in Eq. 3 is called the transfer function, and θ(B) and φ(B) are polynomials of degree q and p in B, respectively. The general relationship between the time dependent input I(t) and output Q(t) for the discrete linear model of order (p, q) or ARMA (p, q) process (Eq. 2) can be given as follows. Qðt Þ ¼ a1 Qðt 1Þ þ a2 Qðt þ





þ bq I ðt 2Þ þ





þ ap Qðt pÞ þ b0 I ðt Þ þ b1 I ðt qÞ 1Þ ð6Þ In above relationship as and bs are the model parameters. The Eq. 6 represents an ARMA (p, q) process and can be employed to develop unit impulse response functions (UIRF). The computational time step used in hydrograph synthesis is usually very small compared to the travel time in the channel reach which may be of order of several time steps. Consequently, the current outflow from the reach would be expected to depend on the inflows and outflows of few time units back. While using an ARMA type of rainfall-runoff relation, several investigators had reported that the ARMA process of order less than three generally produces satisfactory results (Chow 1972; Wang et al. 1991, 1992). Therefore, in the present investigation, p and q values up to 2 has been considered. 3 Parameter Estimation of ARMA (p, q) Process Methods of fitting mathematical models to numerical data have been presented in a number of references such as Box and Jenkins (1976), Wang and Yu (1986) and others. The leastsquares method was used to estimate the model parameters of the ARMA (p, q) process from input (effective rainfall) and output (direct runoff) data. This method involves estimating the parameters by minimizing the sum of squares of residuals or errors between the observed and the computed Q(t). If the residual be e(t), then: b ðt Þ eðtÞ ¼ Q QðtÞ for t ¼ 1; 2; . . . . . . :m ð7Þ b ðt Þ is the observed direct runoff (m3/s) and Q(t) is the computed direct runoff (m3/s) where Q from the model using the observed effective rainfall I(t) (m3/s). Equation 9 may be written in matrix form as: e¼Q Aβ ð8Þ Another look at Z-transform technique 1833 where, e ¼ ½eð1Þ; eð2Þ;





; eðmފ ð8aÞ h i b ð1Þ; Q b ð2Þ; . . . . . . :; Q b ðmÞ QT ¼ Q ð8bÞ   β ¼ a1 ; a2 ; . . . . . . ; ap ; b0 ; b1 ; . . . . . . ; bq and 0 0 0 0 B Qð1Þ B B Qð2Þ Qð1Þ B .. .. A¼B B . . B . . @ .. .. Qðm 1Þ Qðm 2Þ ð8cÞ I ð1Þ 0 I ð2Þ I ð1Þ I ð3Þ I ð2Þ .. .. . . .. .. . ...... . I ðmÞ I ðm 1Þ ...... ...... ...... 1 . . . :: C C . . . :: C C C . . . :: C C . . . :: A ð8dÞ The least squares estimates are obtained with respect to b by solving the following matrix equation (Wang and Yu 1986):   1 T  β ¼ AT A A Q ð9Þ 4 Derivation of UIRF Using Z-transform The Z-transform is one of the transform methods used to solve linear difference equations (Muth 1977; Bree 1978; Dooge 1973), which are functional equations defining sequences of their discrete counterparts. The details are available elsewhere (Muth 1977). It is used for derivation of UIRF for various orders of the ARMA process. The UIRF of the ARMA(2, 2) process is given in the following section, however the UIRF of the ARMA (p, q) processes for p≤2 and q≤2 are given in Table 1 and the full derivation of UIRF-ARMA(1,1) UIRFARMA(1,2) and UIRF-ARMA(2,1) are explained in Appendix. 5 UIRF-ARMA (2, 2) Using Eq. 2, the transfer function of the ARMA(2, 2) process, by considering the order p=2 and q=2 of ARMA (p, q) process, is expressed as:    1 a1 B a2 B2 I ðtÞ ð10Þ QðtÞ ¼ b0 þ b1 B þ b2 B2 Applying the Z-transform technique Eq. 10 takes the form as:   1 a1 Z 1 H ðZ Þ ¼ QðZ Þ=I ðZ Þ ¼ b0 þ b1 Z 1 þ b2 Z 2 a2 Z Simplifying Eq. 11a yields:   2 H ðZ Þ ¼ b0 Z 2 þ b1 Z þ b2 Z a1 Z a2  2  ð11aÞ ð11bÞ 1834 Table 1 UIRFs of ARMA(p, q) processes ARMA (p, q) Parameters of ARMA (p, q) UIRF ARMA (1, 1) a1, b0, b1 hðtÞ ¼ Aδt 1 ARMA (1, 2) a1, b0, b1, b2 hðtÞ ¼ Aδt 1 þ Bδt ARMA (2, 1) a1, a2, b0, b1 When r1 and r2 are real and unequal roots: hðtÞ ¼ Ar1t 1 þ Br2t 1 Parameters of UIRF þ Bat1 1 A¼ ðb1 =a1 Þ; B ¼ ðb0 a1 þ b1 Þ=a1 ; δt 1 ¼  1; for t ¼ 1 0; otherwise A¼ t 2 þ Ca1 ½b1 þ ðb2 =a1 ފ=a1 ; B ¼ ðb2 =a1 Þ; C ¼ b0 þ ½b1 þ ðb2 =a1 ފ=a1 ;  1; for t ¼ 2 : δt 2 ¼ 0; otherwise   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ffi 2; A ¼ ðb0 Z þ b1 Þ=ðZ 2; r1 ¼ a1 a21 þ 4a2 r1 ¼ a1 þ a1 þ 4a2 1 B ¼ ðb0 Z þ b1 Þ=ðZ a¼ When r1 and r2 are complex conjugate roots: t 1 hðtÞ ¼ Mac Sinfðt 1Þθ þ φg ARMA (2, 2) a1, a2, b0, b1, b2 When r1 and r2 are real and unequal roots: hðtÞ ¼ Aδt 1 þ Br1t 1 þ Cr2t  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi  2  ð a2 Þ; b ¼ a1 =2; c ¼ a1 þ 4a2 2; θ ¼ tan 1 ðc=bÞ; M ¼ jW j qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi  2  1 a1 þ 4a2 ; θ ¼ tan ð a2 Þ; b ¼ a1 =2; c ¼ 2    M ¼ jW j; φ ¼ Arg ðW Þ; W ¼ b0 Z 2 þ b1 Z þ b2 Z Z¼bþic A¼ 1Þθ þ φg ðb2 =a2 Þ; a ¼ 1  r1 ފ Z¼r2 ðc=bÞ R.K. Rai, et al. When r1 and r2 are complex conjugate roots: t 1 Ma hðt Þ ¼ Aδt 1 þ  Sinfðt C r1 ÞjZ¼r2 φ ¼ ArgðW Þ; W ¼ ðb0 Z þ b1 ÞjZ¼bþic qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2; A ¼ ðb2 =a2 Þ 2; r1 ¼ a1 a21 þ 4a2 r1 ¼ a1 þ a21 þ 4a2      B ¼ b0 Z 2 þ b1 Z þ b2 ½Z ðZ r2 ފ Z¼r1 C ¼ b0 Z 2 þ b1 Z þ b2 ½Z ðZ 1 r2 ÞjZ¼r1 Another look at Z-transform technique 1835 The inverse Z-transform can be performed for real and unequal, and complex conjugate roots of the denominator, as follows: 5.1 Real and Unequal Roots For real and unequal roots, Eq. 11b yields H ðZ Þ b0 Z 2 þ b1 Z þ b2 ¼ Z ðZ r1 ÞðZ r2 Þ Z ð12Þ where r1 and r2 are computed from Eqs.13 and 14, respectively.  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a1 þ r1 ¼ a21 þ 4a2 2 r2 ¼  a1 1 2 The partial fraction expansion of Eq. 12 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a21 þ 4a2 ð13Þ ð14Þ H ðZ Þ A B C ¼ þ þ Z Z Z r1 Z r2 ð15Þ The inverse Z-transform of Eq. 15 yields H ðtÞ ¼ Aδt þ Br1t þ Cr2t ð16Þ Here, UIRF is zero at t=0. Thus, substituting (t−1) in place of t for t=1, 2, ....., n+1 in Eq. 16, UIRF can therefore be expressed as: hðtÞ ¼ Aδt 1 þ Br1t 1 þ Cr2t 1 ð17Þ where A¼  b0 Z 2 þ b1 Z þ b2  ¼ Z 2 a1 Z a2 Z¼0  b0 Z 2 þ b1 Z þ b2  B¼ Z ðZ r2 Þ Z¼r1 C¼ and δt  b0 Z 2 þ b1 Z þ b2  Z ðZ r1 Þ Z¼r2 1 ¼  1 for t ¼ 1 0 otherwise b2 a2 ð18Þ ð19Þ ð20Þ ð21Þ 1836 R.K. Rai, et al. 5.2 Complex Conjugate Roots If the roots are complex conjugate, as above, the Z-transform of Eq. 11b can be written as follows. H ðZ Þ b0 Z 2 þ b1 Z þ b2 ¼ 2 Z ðZ a1 Z a2 ÞZ ð22Þ Equation 22 can be written in its partial fraction as: H ðZ Þ A BZ þ D ¼ þ 2 Z Z Z a1 Z a2 ð23Þ The Eq. 23 will takes the form after simplification as follows. BZ 2 þ DZ ð24Þ a1 Z a2 where, B and D are the dummy variables of the partial fraction expansion. The inverse Ztransform of Eq. 24 gives (Muth 1977): H ðZ Þ ¼ A þ H ðt Þ ¼ Aδt þ Therefore, H ðtÞ ¼ Aδt 1 where þ Z2 Mat Sinðt θ þ φÞ c Mat Sinfðt c 1Þθ þ φg ðb2 =a2 Þ ð27Þ a¼ pffiffiffiffiffiffiffiffiffiffiffiffi ð a2 Þ ð28Þ 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ffi a1 þ 4a2 θ ¼ tan W¼ ð26Þ A¼ b ¼ a1 =2 c¼ ð25Þ 1 ðc=bÞ ð29Þ ð30Þ ð31Þ M ¼ jW j ð32Þ φ ¼ ArgðW Þ ð33Þ  b0 Z 2 þ b1 Z þ b2   Z Z¼bþic ð34Þ Another look at Z-transform technique 1837 6 Results and Discussion To test the proposed methodology, two cases are considered: Case 1 UIRFs derived from other approaches using Argand diagram (Ojha et al. 1999) for the single storm event of Nenagh, an Irish catchment (Area=295 km2) are tested for their reproducibility by the present approach. Case 2 Direct runoff hydrographs are reproduced from the UIRF generated using the proposed approach. Here, the data of (a) Arki watershed (Area=24.60 km2) of Satluj river catchment (India), (b) Catsop watershed (Area=0.4156 km2) of Netherlands, and (c) Banha watershed (17.17 km2) of India (Jain and Singh 2005) are considered. 6.1 Case 1 To show the applicability of the present approach, rainfall-runoff data of a single storm runoff event (Ojha et al. 1999) are considered. The performance is evaluated using the Coefficient of efficiency (CE) (Nash and Sutcliffe 1970): i2 n h P b ðtÞ Qðt Þ Q ð35Þ CE ¼ 1 t¼1 i2 n h P b ðtÞ Q Q t¼1 b ðtÞ and Q(t) are the observed and computed DRH, respectively, and Q is the mean where Q DRH. The values of CE for ARMA(1, 2) and ARMA(2, 2) (Table 2) indicate that the ARMA (1, 2) has best fitted. Using this data set, the A and Q matrix for the estimation of model parameters of ARMA(1, 2) used in least-squares method (Eq. 9) are obtained as follows: 3 2 0:00000 24:1736 0:00000 0:00000 6 2:4712 0:00000 24:1736 0:00000 7 7 6 6 1:3407 81:6167 0:00000 24:1736 7 7 6 6 9:1251 0:00000 81:6167 0:00000 7 7 6 6 12:2212 0:00000 0:00000 81:6167 7 7 6 6 13:9808 0:00000 0:00000 0:00000 7 7 6 .. .. .. .. 7 A¼6 7 6 . . . . 7 6 . . . . 7 6 .. .. .. .. 7 6 7 6 .. .. .. .. 7 6 . . . . 7 6 6 0:45570 0:00000 0:00000 0:00000 7 7 6 4 0:33490 0:00000 0:00000 0:00000 5 0:23770 0:00000 0:00000 0:00000 Table 2 Coefficient of efficiency of UIRF-ARMA (p, q) for Cases 1 to 4 Cases Case Case Case Case 1 2(a) 2(b) 2(c) Coefficient of efficiency UIRF-ARMA(1, 1) UIRF-ARMA(1, 2) 0.707460 0.767933 0.978702 0.455501 0.98170 0.95487 0.97784 0.96396 UIRF-ARMA(2, 1) −0.575257 0.980513 0.978702 0.124667 UIRF-ARMA(2, 2) 0.95325 0.99790 0.97892 0.94675 1838 R.K. Rai, et al. 0.16 Fig. 1 Unit impulse response function derived from ARMA (1, 2) and Ojha et al. (1999) for single storm runoff 0.14 0.12 UIRF (1/h) 0.1 0.08 0.06 0.04 0.02 0 0 20 40 60 80 100 Time (h) QT ¼ ½2:4712; 1:3407; 9:1251; . . . . . . . . . . . . ; 0:3349; 0:2377; 0:1667Š where QT is the transpose matrix of matrix Q. Using Eq. 9 the estimated values of the parameters a1, b0, b1, and b2 of the ARMA(1, 2) process are 0.8251784, 0.0856885, 0.0505143, and 0.0473072, respectively. The derived UIRF for the considered storm runoff event is hðt Þ ¼ 0:130692δt 0:057329δt 1 2 þ 0:216380ð0:8251784Þt 1 ð36Þ where t is the unit step time. As the unit step is of 3 h, for t=1 the Eq. 36 results UIRF, h(t) at time 3 h. UIRFs using Eq. 36 are used to convert into the unit pulse response function or unit hydrograph by multiplying the area of the watershed (km2) and the conversion factor of 2.78. The UIRF for storm event is shown in Fig. 1, and the computed and observed direct runoff hydrograph in Fig. 2, which shows the present methodology to perform satisfactorily (Table 3). Computed:UIRF-ARMA(1,2) observed DRH 16 Computed: ARMA(1,2) 14 Computed: Optimized Nash 12 Discharge (m3/s) Fig. 2 Comparison of observed and computed direct surface runoff hydrographs from UIRFARMA (1, 2) and other existing approaches for the storm event used by Ojha et al. (1999) 10 8 6 4 2 0 0 20 40 60 Time (hour) 80 100 Another look at Z-transform technique 1839 Computed: UIRF-ARMA (2,2) Fig. 3 Comparison of observed and computed direct surface runoff hydrographs for the event of June 30, 1996 of Arki watershed of Satluj river of India 14 Observed DRH Computed:ARMA(2,2) Discharge (m3/s) 12 Computed:Optimized Nash 10 8 6 4 2 0 0 1 2 3 4 5 Time (hour) 6 7 8 9 6.2 Case 2 (a) The evaluation of ARMA (1, 2) and ARMA (2, 2) on the storm event of June 30, 1996 of Arki watershed indicates that ARMA (2,2) has the least calibration error (Table 2). The estimated values of the parameters a1, a2, b0, b1, and b2 of ARMA(2, 2) are 1.16792, −0.33943, 0.130324, 0.137048, and 0.071952 respectively and the derived UIRF-ARMA(2, 2) is hðt Þ ¼ 0:2119789δt 1 þ 4:198467ð0:62370Þt 1 4:280123ð0:54422Þt 1 ð37Þ where t corresponds to the unit time of 0.5 h. The computed direct runoff hydrograph by UIRF-ARMA (2, 2) is shown in Fig. 3 along with the observed direct runoff hydrograph. The proposed method reproduces approximately same DRH. (b) For storm event of August 18, 1987 of Catsop watershed (Table 4), ARMA (2, 2) has the least calibration error (Table 2). With ARMA (2, 2) parameters a1, a2, b0, b1, and b2 of 0.626467, −0.031622, 0.307294, 0.4288235 and −0.330808, respectively, the resulting UIRF is hðtÞ ¼ (c) 10:46134δt 1 þ 0:048603ð0:571096Þt 1 þ 10:720172ð0:05537Þt 1 ð38Þ It is derived for unit time step=0.5 h. Using Eq. 38, the computed and observed runoff hydrograph is shown in Fig. 4. Evidently, both the generated and observed direct runoff hydrographs fairly match each other. For the single storm runoff event of August 18, 1995 (Table 4) of Banha watershed, the higher value of CE (Table 2) for ARMA (1, 2) suggests derivation of UIRF for the event. Eventually, the parameters a1, b0, b1, and b2 of ARMA(1, 2) are 0.710873, 0.000281424, 0.08662033, and 0.16457237, respectively, and the derived UIRF using Eq. 51 is hðtÞ ¼ 0:447517δt 1 0:231507δt 2 þ 0:447798ð0:710873Þt 1 ð39Þ where t is the unit time of 0.5 h. As seen from Fig. 5, The direct runoff hydrograph is reproduced well. 1840 R.K. Rai, et al. Table 3 Observed (Ojha et al., 1999) and computed direct runoff hydrograph for single storm runoff event Time (h) Effective rainfall intensity (cm/h) Observed DRH (cumec) UIRFARMA(1,2) (1) (2) (3) (4) RE(1) (5)a RE(2) (6)a RE(3) (7)a 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 0.0000 0.0295 0.0000 0.0996 0.000000 0.085689 0.121223 0.147337 0.121580 0.100325 0.082786 0.068313 0.056371 0.046516 0.038384 0.031673 0.026136 0.021567 0.017797 0.014685 0.012118 0.010000 0.008251 0.006809 0.005619 0.004636 0.003826 0.003157 0.002605 0.002150 0.001774 0.001464 0.000000 2.071400 2.930387 3.561677 2.939019 2.425215 2.001235 1.651376 1.362680 1.124454 0.927875 0.765662 0.631808 0.521354 0.430210 0.355000 0.292939 0.241727 0.199468 0.164596 0.135821 0.112077 0.092483 0.076315 0.062974 0.051965 0.042880 0.035384 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000000 0.000000 0.000000 6.993613 9.893788 12.02520 9.922934 8.188191 6.756718 5.575498 4.600780 3.796465 3.132761 2.585086 2.133157 1.760235 1.452508 1.198578 0.989041 0.816135 0.673457 0.555722 0.458570 0.378402 0.312249 0.257661 0.212617 0.175447 a 0.0000 2.4712 1.3407 9.1251 12.2212 13.9808 12.4169 9.6054 7.1567 6.0866 5.1878 4.1271 3.6428 3.6661 1.8749 2.5165 1.9227 1.7632 1.4934 1.1445 0.8772 0.7659 0.663 0.5567 0.4557 0.3349 0.2377 0.1667 DRHs DRH (cumec) (8) 0.000000 2.071400 2.930387 10.55529 12.83281 14.45041 11.92417 9.839566 8.119398 6.699952 5.528655 4.562127 3.764569 3.106441 2.563368 2.115236 1.745447 1.440305 1.188509 0.980732 0.809279 0.667799 0.551053 0.454717 0.375223 0.309626 0.255497 0.210830 RE=Col.(2)×2.78×295 Col.(5)=RE×Col.(4) Col.(8)=Col.(5)+Col.(6)+Col.(7) Thus, the estimated CE values for Cases 1 and 2a–c are 0.987, 0.998, 0.964 and 0.979, respectively, and the results presented in Figs. 2, 3, 4 and 5 suggest the applicability of the proposed methodology to both UIRF-ARMA(1,2) and UIRF-ARMA(2,2) and, in turn, the computation of direct runoff hydrograph and UIRF. Furthermore, the proposed approach is simple and easy to apply, better surrogate of the Argand diagram based approach. Further, to test the advantage in using the present approach, the direct runoff hydrographs has been computed using the simple ARMA (p, q) process of same order used for the derivation of UIRF and the two-parameter Nash model (Nash 1957) and is also compared with the results obtained from the UIRF-ARMA (p, q) model. The simple ARMA (p, q) model has been fitted using the least-squares method (Eqs. 7, 8, 8a, 8b, 8c, 8d and 9) and the parameters of the Nash model (i.e. n and K) have been optimized using the Another look at Z-transform technique 1841 Computed:UIRF-ARMA(2,2) Fig. 4 Comparison of observed and computed direct surface runoff hydrograph for Catsop watershed, South Limburg, The Netherlands for the event of August 18, 1987 0.2 Observed DRH Computed:ARMA(2,2) 0.18 Computed:Optimized Nash Discharge (m3/s) 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 1 2 3 4 5 Time (hour) Marquardt optimization algorithm (Marquardt 1963). The resulting direct runoff hydrographs from the simple ARMA as well as from Nash model are shown in Figs. 2, 3, 4 and 5, respectively, for the single storm event of four watersheds. Besides the visual comparisons, the performance of the model has been tested using the statistical measures, viz. root mean square error (RMSE) and mean absolute percent error (MAPE). The computed values of these test statistics for simple ARMA and Nash model along with the UIRF-ARMA model are given in Table 5. It is clear from Figs. 2, 3, 4 and 5 as well as from Table 5 that the performance of UIRF-ARMA (p, q) is better than the optimized Nash model in reproduction of direct surface runoff. Though, the UIRF has been developed from the response function of 120 Fig. 5 Comparison of observed and computed direct surface runoff hydrograph for the event of 18 August, 1995 of Banha watershed of India Computed: UIRF-ARMA(1,2) Observed DRH Computed: ARMA(1,2) Computed:Optimized Nash Discharge (m3/s) 100 80 60 40 20 0 0 5 10 Time (hour) 15 1842 R.K. Rai, et al. Table 4 Observed storm runoff event Banha watershed of Hazaribagh, Bihar, India and Catsop watershed of South Limburg, The Netherlands Banha Watershed(17.17 km2) (18 August, 1995) Catsop Watershed (0.4156 km2) (18 August, 1987) Time (h) Rainfall Intensity (cm/h) DRH (m3/s) Time (min) Rainfall Intensity (cm/h) ERH (cm/h) Time (min) DRH (m3/s) 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 570 600 630 660 690 720 750 780 0.00 0.90 3.24 5.34 0.24 0.00 0.000 0.000 0.000 0.119 36.712 95.687 84.756 44.768 24.987 15.995 11.144 8.999 7.714 6.512 5.552 5.100 4.962 4.799 3.862 3.562 2.846 2.329 1.751 1.593 0.986 0.494 0.000 1,280 1,310 1,560 0.000 3.620 0.000 0.000 0.244 0.000 1,280 1,295 1,310a 1,320 1,340 1,370 1,395 1,400a 1,430a 1,440 1,460a 1,490a 1,520a 1,550a 1,560 1,580a 0.0000000 0.0241600 0.0864760 0.1280200 0.1748500 0.0137100 0.0032900 0.0030600 0.0014800 0.0009300 0.0007850 0.0005675 0.0003500 0.0001325 0.0000600 0.0000000 a Interpolated data for making equal time interval of 30 min from 1,280 min Table 5 Values of RMSE and MAPE for different models used for comparison Cases RMSE UIRF-ARMA Case Case Case Case 1 0.1041 2(a) 0.0706 2(b) 1.49E−05 2(c) 1.006   Ph bi MAPE ¼ 100  ABS Q i MAPE (%) ARMA Optimized nash 0.1041 0.0402 0.0180 1.006 0.128 0.348 0.0061 1.3460 Qi i UIRF-ARMA ARMA 38.8 38.8 21.9 12.6 0.0034 2.55 388.28 388.28 , 2 1=2 P b i Qi n; RMSE ¼ Q n i Optimized nash 52.61 94.91 1.56 582.06 Another look at Z-transform technique observed DRH predicted UIRF-ARMA(1,1) 10.00 DRH ORDINATES (M3/SEC) Fig. 6 Comparison of observed and regenerated direct surface runoff hydrographs from different combination of UIRF-ARMA (p, q) for the storm event of July 20, 1994 1843 9.00 predicted UIRF-ARMA(1,2) predicted UIRF-ARMA(2,1) predicted UIRF-ARMA(2,2) 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00 0 2 4 6 TIME IN HOURS 8 10 the simple ARMA (p, q) process using the Z-transform technique, the resulting direct runoff hydrograph obtained from the UIRF-ARMA (p, q) is more or less similar to the simple ARMA (p, q) model. Thus, it is clear that the Z-transform technique provides a good tool for deriving the unit impulse response function from the simple ARMA process. Considering the fact that other ARMA process can be utilized for the development of unit impulse response functions, the performance of UIRF-ARMA(1, 1) and UIRF-ARMA (2, 1) are also examined in line with UIRF-ARMA(1, 2) and UIRF-ARMA(2, 2). The coefficients of efficiency of all the processes for all the considered cases are given in Table 2. The Table 2 indicates that ARMA (1, 2) and ARMA (2, 2) have very high value of coefficient of efficiency in all the cases and thus, have the potential for the derivation of UIRF. To examine the effect of the orders of the ARMA (p, q) process in estimating the direct runoff hydrograph the storm event of July 20, 1994 of Arki watershed of Satluj river catchment was used and all the four UIRFs were fitted. The computed direct runoff hydrograph of the event is shown in Fig. 6 which shows that the q is effective in reproduction of the rising limb and crest segments of the hydrograph, and p in the recession segment. 7 Conclusions The following conclusions can be derived from this study: 1. The derived equations for UIRFs of ARMA process can be used as general expressions in discrete time domain for determination of time distribution of direct runoff. 2. The application results of UIRF-ARMA (1, 2) and (2, 2) processes are in close agreement with the observed DRH. The results obtained from UIRF-ARMA(p, q) model are superior to optimized Nash model and mostly similar to simple ARMA(p, q) model. Thus, the Z-transform technique provides a good tool for deriving the unit impulse response function from the simple ARMA process and can be a useful substitute to the commonly practiced Argrand diagram based approach. 3. In general, the UIRF-ARMA (1, 2) and (2, 2) processes have indicated a better performance than UIRF-ARMA (1, 1) and (2, 1) processes. 4. Within the premise of the present study, q is effective in reproduction of the rising limb and crest segments of the hydrograph, and p in the recession segment 1844 R.K. Rai, et al. Acknowledgements The authors wish to sincerely thank anonymous reviewers whose comments greatly improved the quality of this paper. Appendix Derivation of UIRF of other ARMA (p, q) Process UIRF for ARMA (1, 1) The UIRF for ARMA(1, 1) (UIRF-ARMA(1, 1)) can be written in the form of transfer function as: ð1 a1 BÞQðt Þ ¼ ðb0 þ b1 BÞI ðtÞ ð40Þ Its Z-transform is H ðZ Þ ¼ QðZ Þ b0 þ b1 Z ¼ I ðZ Þ 1 a1 Z 1 1 ¼ b0 Z þ b1 Z a1 ð41Þ a1 ފ ð42Þ Dividing Eq. 41 by Z gives H ðZ Þ=Z ¼ ðb0 Z þ b1 Þ=½Z ðZ in which Q(Z) and I(Z) are the Z-transforms of Q(t) and I(t) sequences, respectively, and H(Z) is the Z-transform of the transfer function. The inverse Z-transform of H(Z) yields the UIRF as: H ðtÞ ¼ Aδt þ Bat1 where and ð43Þ  b0 Z þ b1  A¼ ¼ Z a1 Z¼0 B¼ b1 a1  b0 Z þ b1  b0 a1 þ b1 ¼  Z a1 Z¼a1 ð44Þ ð45Þ Here, UIRF is zero at t=0. Thus, substituting (t−1) in place of t for t=1, 2, ....., n+1 in Eq. 43, UIRF can be expressed as: hðt Þ ¼ Aδt 1 þ Ba1t 1 ð46Þ in which h(t) is UIRF at discrete time t and δt −1 is the Dirac delta function:  1; for t ¼ 1 δt 1 ¼ 0; for t 6¼ 1 ð47Þ UIRF of ARMA (1, 2) As above, the ARMA (1, 2) process can be written as:   Qðt Þ ¼ b0 þ b1 B þ b2 B2 ð1  a1 BÞ I ðt Þ ð48Þ Another look at Z-transform technique 1845 and the following can be derived H ðZ Þ ¼ A þ B½1=Z Š þ C ½Z=ðZ a1 ފ ð49Þ The inverse Z-transform of Eq. 49 results in H ðtÞ ¼ Aδt þ Bδt 1 þ Cat1 2 þ Cat1 ð50Þ and the UIRF is: hðtÞ ¼ Aδt 1 þ Bδt 1 ð51Þ where A¼ ½b1 þ b2 =a1 Š=a1 ð52Þ ðb2 =a1 Þ ð53Þ B¼ C ¼ b0 þ ½b1 þ ðb2 =a1 ފ=a1 δt δt ð54Þ 1 ¼  1 0 ; for t ¼ 1 ; otherwise ð55Þ 2 ¼  1 0 ; for t ¼ 2 ; otherwise ð56Þ UIRF of ARMA (2, 1) Similar expressions for ARMA (2, 1) process can be given as: QðtÞ ¼ 1 b0 þ b1 B I ðt Þ a1 B a2 B2 Þ ð57Þ and H ðZ Þ ¼ QðZ Þ b0 Z 2 þ b1 Z ¼ 2 I ðZ Þ Z a1 Z a2 ð58Þ If r1 and r2 are the roots of the denominator of Eq. 58, its inverse Z-transform can be derived for cases if r1 and r2 are (a) real and unequal and (b) complex conjugate. If r1 and r2 are real and unequal, Eq. 58 can be written as: H ðZ Þ b0 Z þ b1 ¼ ðZ r1 ÞðZ r2 Þ Z ð59Þ 1846 R.K. Rai, et al. where r1 ¼ 1 2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 þ a21 þ 4a2 ð60Þ r2 ¼ 1 2  a1 ð61Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a21 þ 4a2 The partial fraction expansion of Eq. 59 is H ðZ Þ=Z ¼ A=ðZ r1 Þ þ B=ðZ r2 Þ ð62Þ The inverse Z-transform of Eq. 62 is H ðtÞ ¼ Ar1t þ Br2t ð63Þ and, therefore, hðtÞ ¼ Ar1t 1 þ Br2t 1 ð64Þ where  b0 Z þ b1  A¼ Z r2 Z¼r1  b0 Z þ b1  B¼ Z r1 Z¼r2 ð65Þ ð66Þ If the roots are complex conjugate (r1 =b+ic and r2 =b−ic), the inverse Z-transform of Eq. 58 will be H ðt Þ ¼ at M Sinðt θ þ φÞ c ð67Þ UIRF can, therefore, be expressed as: hðtÞ ¼ Mat c 1 Sinfðt 1Þθ þ φg ð68Þ where a¼ pffiffiffiffiffiffiffiffiffiffiffiffi ð a2 Þ b ¼ a1 =2 c¼ 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ffi a1 þ 4a2 θ ¼ tan 1 ðc=bÞ ð69Þ ð70Þ ð71Þ ð72Þ Another look at Z-transform technique 1847 M ¼ jW j ð73Þ φ ¼ ArgðW Þ ð74Þ W ¼ ðb0 Z þ b1 ÞjZ¼bþic ð75Þ References Blank D, Delleur JW (1968) A program for estimating runoff from Indiana watersheds: 1. 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