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An Autocorrelation Approach for Parameter Estimation of Fractional Order Equal-root Autoregressive Models Using Hyper-geometric Functions
Abstract
A method of estimating the parameters of an autoregressive model with real and equal roots in its characteristic equation is developed. The proposed method uses the serial autocorrelation function in the estimation process. Unlike solution of Yule-Walker equations, this method does not require an a priori knowledge of the order of the model and is not restricted to integer order models. The possible use of this method in estimating the parameters of an autoregressive cum moving average model and those of the continuous cascade model of linear and equal reservoirs is indicated.
... Following Peiris (2003), the fractional noise process by Andèl (1986) can be seen as a special case of the Fractional equal root Autoregressive of order 1 (FerAR(1, d)) process introduced by Spolia et al. (1980). In a forthcoming paper, we discuss this process and its estimation in the frequency domain. ...
... The Fractional equal root Autoregressive Moving Average (FerARMA) processes have been considered in the literature (see Peiris (2003), Peiris et al. (2005) and Pillai et al. (2012)) to model some features in time series, which cannot be fully described by standard ARIMA models. The FerAR(1, d) process was initially introduced by Spolia et al. (1980) via the following equation: ...
Electricity load forecasting is a crucial issue for power market players, since good forecasts can avoid wasting energy and prevent system failure. In this paper, we introduce a sinusoidal waveform process to forecast electricity loads, which adds to the standard statistical methods already discussed in the literature. For a suitable parameter's set, the process displays either short or long memory. We find that our model yields lower Root Mean Square Forecast Errors in both short and medium-term prediction horizons, if compared with standard methods in linear time series analysis. MSC: 60G10 (stationary processes), 62M10 (time series). KEYWORDS: Cyclical long memory, Fractional sinusoidal waveform process, Whittle estimation, Forecasting, Electricity loads.
... The fractional noise process (obtained by assuming x t as a white noise sequence) can be seen as a special case of the Fractional equal-root Autoregressive of order 1 (FerAR(1,d)) process, introduced by Spolia et al. (1980) and also mentioned in Hosking(1981). ...
... The FerAR(1,d) process is an interesting special case of (2) with d 1 = d, d 2 = 1, φ ∈ (0, 1) and ψ = 0. It was initially introduced by Spolia et al. (1980), and also mentioned by Hosking(1981). More recently, it has been formalized by Peiris (2003). ...
Long memory models have been widely proposed in the literature to study strong dependence in time series. Their spectral density is unbounded in at least one frequency and the autocovariance function decays hyperbolically to zero. These models can be seen as a special case of the Fractional equal-root Autoregressive Moving Average (FerARMA) model, which displays short memory for a suitable parameter's set. We provide a Central Limit Theorem for the Whittle estimator of this process. Moreover , we find a better forecasting performance in the long horizon predictions with respect to long memory models. MSC: 60G10 (stationary processes), 62M10 (time series), 60F05 (central limit theorems). KEYWORDS: Long memory, Fractional processes, Whittle estimation, Predictions, Realized volatility.
... Des modèles qui suivent ce type de relation sont présentés dans la littérature depuis que Sherman (1932) a introduit le concept d'hydrogramme unitaire. Ils sont également utilisés par , , Spolia et al. (1980), , , et . ...
... a présenté une méthode basée sur l'expansion de la réponse impulsionnelle dans une séquence de polynômes orthogonaux unitaires pondérés. Spolia et al. (1980) ont représenté la réponse impulsionnelle par une distribution Gamma à deux paramètres. La méthode de déconvolution a été utilisée par Dreiss (1982) et plus récemment par Padilla et Pulido-Bosch (1995). ...
One of the main problems of the socio-economic development of Amazonia is the lack of electrical energy for the numerous small isolated communities of the region, which impedes the value addition in their agricultural production, the growth of revenues and the job creation. This constraint can motivate, maybe to impel, the migration of the populations of these communities towards large cities, which results in grave social problems in these.
Energy supply by large hydropower is not possible from the economic point of view; because of the construction costs of transmission lines and the voltage change for small-scale supply. A way to solve this problem is the setting up of isolated generation systems, such as the micro-hydro power, by taking advantage of Amazonia's small rivers network.
The purpose of this thesis is to propose a methodology for the setting up of micro-hydro power plants that are a renewable source of energy. This methodology has to take into account hydro-logical and topographic characteristics of the small catchments of the region. Moreover, it has to take into account the economy and culture of the small isolated communities that live in the small catchments and this in a perspective of sustainable development.
The methodology is based on the decision support system that starts in the energy demand of the small communities. This energy is necessary to generate jobs and revenues; therefore, encourag-ing the settlement of the local population and decreasing its migration. Additionally, the energy generated by micro-hydro power plants will help to improve the life-style of the population of these small communities.
The decision support system consists mainly of the hydrological, topography, energy, economic, geotechnical and environmental aspects. Nevertheless, this system takes into account social as-pects, which make part of the base of the sustainable development, such as energy demand and community commitment; and preliminary evaluations on the field. The social, hydrological, topography, energy and economic aspects; and preliminary evaluations on the field are focused on details. The environmental aspects are not developed because micro-hydro powers are low-head and run-of-river, this scheme does not favour the formation of large accumulation lakes upstream of the small rivers. Furthermore, the geotechnical aspects are not considered because of the size of the micro-hydro powers.
Hydrological aspects of small catchments are very important for the design of the micro-hydro powers, in particular because the sites are ungauged. This design is determined for the flow-duration curves that are simulated with a hydrological rainfall-runoff model or its transferability. The rainfall-runoff model is used because the rainfalls are the only available data in the small catchments of the region.
Energy aspects indicate that the peak period of the energy demand, represented by the peak of the agricultural production, and the peak period of the energy production, represented by the floods of the region occur simultaneous in time, between January and June. This serendipity helps planning the energy production and minimizing the environmental impacts, because the highest stream flow of the region's hydrological regimen is able to supply the peak of energy demand. The serendipity justifying also the choice of run-of-river schemes, which are the least harmful to the environment among the possible hydroelectric power plants, these schemes are designed with 1 or 2 turbines to supply the energy demand.
Economics aspects are based on the comparison of the estimations of hydro-micro power’s costs, diesel generators and price of the rural grid energy. These comparison shows that the hydro-micro powers are a viable economic option compared to the diesel generators that are often used in isolated regions. Nevertheless, this comparison indicates that the energy of rural grid is cheap-er than hydro-micro powers energy.
The use of the hydro-powers resources of the small catchments of Amazonia combined with the minimization of the environmental impacts to the energy supply of energy demand of the small communities of region, of an economically viable way, along with jobs and revenues generation, supports the perspective of sustainable development of the thesis methodology.
... Ceci laisse de côté les innombrables petits et microbassins non jaugés qui pourraient être exploités par les petites communautés isolées avec l'implantation de micro-centrales hydro-électriques (Mesquita et al. 1999 ). Celles-ci ont une puissance installée inférieure à 100 kW (Johansson et al. 1993 Papazafiriou (1976), Spolia et al. (1980), Goring (1984), Kachroo (1992) et Labat et al. (2000. Les hypothèses du modèle sont la linéarité et l'invariabilité dans le temps de la relation entre la pluie et le débit. ...
... Les séries de pluie et de débit étant connues, la première étape consiste à déterminer la réponse impulsionnelle du système linéaire. Papazafiriou (1976) a présenté une méthode basée sur l'expansion de la réponse impulsionnelle dans une séquence de polynômes orthogonaux unitaires pondérés. Spolia et al. (1980) ont représenté la réponse impulsionnelle par une distribution Gamma à deux paramètres. La méthode de déconvolution aété utilisée par Dreiss (1982) et plus récemment par Padilla et Pulido-Bosch (1995). Le principe de cette méthode est de partir de l'équation de la convolution pour déterminer la réponse impulsionnelle en utilisant l'entré ...
In Amazonia, hydroelectric power production has only been developed on large basins, which are the only ones that have been gauged. This excludes innumerable small catchments, for which only rainfall data are available. Therefore, the objective of the work presented in this paper is to develop a hydrological rainfall–runoff model to simulate flow duration curves for hydro power production planning. The model is based on a linear and time-invariant system (input–output). The impulse response of the system is calculated from the cross-spectral analysis between the rainfall and runoff series. This response is optimized successive approximations to minimize the root mean square of the error. The test catchment area has 7 years of rainfall and runoff data; 4 years are considered for the calibration and 3 years for the validation of the model. A sensitivity analysis of the model to the sample size is carried out to determine the shortest possible data period that produces results comparable to those of the model validation. Key words: model rainfall–runoff, small catchments, Amazonia, impulse response, sensitivity analysis.
... An interesting special case of (3) is the FerAR(1,d) process with d 1 = d, d 2 = 1, φ ∈ (0, 1) and ψ = 0. It was initially introduced by Spolia et al. (1980) and also mentioned by Hosking (1981). More recently, it has been formalized in Peiris (2003). ...
Long memory models can be generalised by the Fractional equal-root Autoregressive Moving Average (FerARMA) process, which displays short memory for a suitable parameter's set. Consequently, the spectrum is bounded, ensuring stationarity also for values of the memory parameter d larger than 0.5. The FerARMA generalization is proposed here to forecast highly persistent time series, as climate records of tree rings and paleo-temperature reconstructions. The main advantage of a bounded spectrum allows for more accurate predictions with respect to standard long memory models, especially if a long prediction horizon is considered.
... It is also not well known that a continuous time counterpart of a linear ARX model is expressed in the form of a linear differential equation including terms of fractional derivatives (Spolia et al., 1980), which well reproduce the effect of hysteresis, or memory effect, as in the autoregressive part. Differential equations including terms of fractional derivatives have been employed for modelling different practical phenomena such as population dynamics (Bushnaq et al., 2018a), HIV/AIDS infection (Bushnaq et al., 2018b), and infiltration of water into soil (Fernández-Pato et al., 2018). ...
A new concept is developed to mathematically understand the dynamics of the rainfall-runoff events in a barren catchment of the Jordan Rift Valley. Time series data of rainfall and runoff have been acquired at an observation point in the catchment. Due to the extreme arid environment, water current as the runoff from the catchment is ephemeral, and the rainfall-runoff events are clearly distinguishable from each other. Firstly, a pair of linear autoregressive models with exogenous input (ARX models) is identified to tightly bound each runoff time series using the simplex method of linear programming. The exogenous input part is compatible with the conventional unit hydrograph method, while the autoregressive part is regarded as a discretized differential operator of fractional orders. Then, a linear fractional differential equation is determined to approximate each linear ARX model, which restricts the perturbation of the actual causal relationship between rainfall intensity and runoff discharge. The resulting lower and upper bounding rainfall-runoff models with fractional derivatives are examined in the system-theoretic framework. Finally, a nominal model from which actual nonlinear and stochastic phenomena perturb is arranged to envelope the all upper bounding rainfall-runoff models in the frequency domain, leading to the formulation of a challenging fractional optimal control problem involving stochastic processes.
... Οι στατιστικές αναλύσεις των χρονοσειρών βασίζονται στις εργασίες των Jenkins and Watts (1968), Hannan (1970), Brillinger (1975), Box and Jenkins (1976) και εφαρμόστηκαν αρχικά στην υδρολογία του καρστ από τους Delleur (1971), Yevjevich (1972), Spolia and Chander (1973), Spolia et al. (1980), Ledolter (1978), Lettenmaier (1980) και άλλους, με στόχο τη συμπλήρωση των χρονοσειρών αλλά και την εκτίμηση των παραμέτρων των στοχαστικών μοντέλων. Οι Mangin (1981), Mangin and Pulido-Bosch (1983), Mangin (1984), , Larocque et al. (1998), Lambrakis et al. (2000), Panagopoulos & Lambrakis (2006), κ.ά., βασιζόμενοι στις αναλύσεις των χρονοσειρών εφάρμοσαν μεθοδολογίες για την περιγραφή και τη λειτουργία των καρστικών υδροφόρων. ...
THE HELLENIC KARST. PROPOSITIONS FOR RESEARCH METHODOLOGIES AND APPLICATION EXAMPLES
LAMBRAKIS N., Professor of Hydrogeology,
Panepistimioupoli Patron, 265 04, nlambrakis@upatras.gr
Abs tract
The carbonate rocks of the Hellenic geotectonic zones are mostly karstified and host important aquifers with hydraulic transmissivity values varying between 10 -3 και 10 -2
m2/s. The Hellenic karst is described as mature and morphological characteristics like karents, dolines, polges, caves etc. are abudant in such areas. The epikarstic zone, where present, controls the groundwater infiltration and thus the aquifer recharge. Difficulties in research due to inaccessibility of mostly of the karst areas can be compensated satisfactorily by applying statistical analysis to the spring’s time series, and by evaluating the groundwater’s quality. For a ll available springs,s and the corresponding aquifers that are presented in the current manuscript, data processing clearly showed that karst aquifers display an important storage that could be explained by the presence of a dense network of small voids characterized by slow laminar flow or/and the presence of the epikarst zone that controls the water infiltration into the phreatic zone. Although karst displays all elements of a telogenetic formation, yet the data processing indicated that it has not reached maturity. Properties such as the complexity, the homogeneity, the degree of the karst structure development are also obvious. Individual karst units may be simple or complex, some exhibiting the characteristics of homogeneous formations, however for all of them, the phreatic zone is relatively well drained. Moreover, the hydrochemical and isotopic approach decisively contributed to the delimitation of karst aquifers, the emergence of their relationship with others, and the presence of the dominant minerals and hydrochemical processes that regulate the groundwater quality.
2
/s. The Hellenic
... The simple linear rainfall-runoff model was presented by Sherman (1932), who introduced the unit hydrograph concept. Since then, linear models have been used over the years, for example, by Rodriguez (1967), Papazafiriou (1976), Spolia et al. (1980), Goring (1984), Kachroo et al. (1992), Labat et al. (2000), Rajurkar et al. (2002), Blanco et al. (2005), Stedinger et al. (2008) and Udomboso and Amahia (2011). ...
The objective of this study is to analyse three rainfall–runoff hydrological models applied in two small catchments in the Amazon region to simulate flow duration curves (FDCs). The simple linear model (SLM) considers the rainfall–runoff process as an input–output time-invariant system. However, the rainfall–runoff process is nonlinear; thus, a modification is applied to the SLM based on the residual relationship between the simulated and observed discharges, generating the modified linear model (MLM). In the third model (SVM), the nonlinearity due to infiltration and evapotranspiration is incorporated into the system through the sigmoid variable gain factor. The performance criteria adopted were a distance metric (δ) and the Nash-Sutcliffe coefficient (R 2) determined between simulated and observed flows. The good results of the models, mainly the MLM and SVM, showed that they could be applied to simulate FDCs in small catchments in the Amazon region.Editor D. Koutsoyiannis; Associate editor A. MontanariCitation Blanco, C.J.C., Santos, S.S.M., Quintas, M.C., Vinagre, M.V.A., and Mesquita, A.L.A., 2013. Contribution to hydrological modelling of small Amazonian catchments: application of rainfall–runoff models to simulate flow duration curves. Hydrological Sciences Journal, 58 (7), 1–11.
... Time series analyses, as developed principally by Jenkins and Watts (1968), Hannan (1970), Brillinger (1975) and Box and Jenkins (1976) have been applied in hydrology by Delleur (1971), Yevjevich (1972), Spolia and Chander (1973), Spolia et al. (1980), Ledolter (1978), Lettenmaier (1980) and others. These works have been oriented essentially towards forecasting, completion of data and estimation of parameters of stochastic models. ...
Correlation and cross-spectral analysis can be applied to the study of karstic aquifers to characterize the transformations in these systems between the input function (precipitation) and the output function (discharge). The parameters that can be deduced are the response time, the distinction between quickflow, intermediate flow and baseflow, and the mean delay. This method offers quantifiable and objective criteria for differentiation and comparisons of karstic aquifers.
... Correlation and spectral analyses were first applied, in surface hydrological systems orientated mainly towards forecasting, completion of data and estimation of parameters for stochastic models (e.g. Delleur, 1971;Spolia et al., 1980). Mangin (1981aMangin ( ,b, 1984 adapted this methodology to study the discharge from karstic aquifers. ...
The most complete historical series of instrumental data available, spanning more than a century, on rainfall, temperature and outflow of a karst spring obtained from gauging stations in the south of the Iberian peninsula were analysed by means of spectral and correlation analyses and continuous wavelet analyses. Annual periodicity of the rainfall and temperature distributions was constant over more than 100 years, although weaker (6-month) periodicities have also been observed, as well as rainfall and temperature periodicities of 5 and 2.5 years, which have also been recorded in other areas of Europe. These multiannual scale components can be explained by climatic variations or effects described in the literature in connection with the North Atlantic Oscillation (NAO) and are likely to be the same as the climate variability at decadal to annual scale detected in several proxy data from geological records. No long-term trends in the distribution of precipitation and temperature were detected.
The main objective of this paper is to review and promote the usefulness of generalized fractionally differenced Gegenbauer processes in time series and econometric research endeavours. In particular, theoretical and computational aspects centered around fractionally differenced Gegenbauer processes with long memory together with a number of interesting and elegant extensions will be discussed. In-depth conceptual developments and large scale simulation study results are presented for clarity and completeness. This survey highlights a number of gaps in the existing literature of this subject area and becomes a valuable reference source for time series practitioners.
It is becoming increasingly recognized that some long series of data can be adequately and parsimoniously modelled by stationary processes with long-range dependence. Some new discrete-time models for long-range dependence or slow decay, defined by their correlation structures, are discussed. The exact power-law correlation structure is examined in detail.
We study six algorithms to select a regularization parameter for deconvolution problems appearing in high-speed communication measurement applications. We investigate these algorithms in the presence of unspecified noise correlation analyzing their performance as components of a multivariate random variable and study their joint distribution by Monte Carlo. We find that several parameter selection algorithms, despite their widespread use, are not robust to unspecified noise correlations. Specifically, the discrepancy principle fails to return adequate regularizations for rough noise, while the generalized cross validation (GCV), unbiased predictive risk, and information complexity selectors can fail for smooth noise. For some experimental configurations, GCV failed completely, returning zero successful inversions out of 500 noise instantiations. These parameter selection algorithms share in the characteristic that they do not contain mechanisms to monitor quantities derived from the parameter-dependent solution vector. By contrast, the L-curve and quasi-optimality criteria do contain such mechanisms, and furthermore exhibited significantly fewer failures and correlated highly with the optimal inversion across all noise levels and correlations.
The objective of this paper is to study the stochastic structure of weekly irrigation requirement of a crop. The irrigation requirement time series is assumed to be represented by an additive model with trend, periodic and stochastic as its components. Each component is identified and, if found, removed from the original series. The turning point test and Kendall's rank correlation test are applied for detecting the trend. In the analysis of series, the correlogram technique is used to detect the periodicity, which is then analyzed by Fourier series method. Harmonic analysis is done for identifying the number of significant harmonics. The series is then tested for stationarity and the dependent part of the stochastic component is found to be well expressed by the second order autoregressive model. Therefore, as a result, the developed model superposes a periodic-deterministic process and a stochastic component. The adequacy of fit is judged by the insignificant correlation and normal distribution of the obtained residuals. The developed periodic-stochastic model may be used for representing the time based structure of the irrigation requirement time series of a paddy crop.
Using the correlogram and variance spectral density, the time series of rainfall and discharge of a hydrological system are analysed in a descriptive way differing from the traditional method. From a methodological viewpoint some computational working choices are made. Applied to three karstic systems in the Pyrenean Mountains (France-Spain), these analyses permitted a comparison and a study of the different reactions of these three systems. They showed the presence of memory effects which modulated the input rainfall for short, medium and long term. These modulations characterize each system. They reflect the storages and the manner in which these fill and empty, thus indicating the karstification of the aquifer. The examples discussed indicate that these analyses provide an excellent method for the investigation of a hydrological system.RésuméLa structure des séries chronologiques de pluie et de débit des systèmes hydrologiques est analysée, de façon descriptive, à partir du corrélogramme et du spectre de densité de variance. Cette approche descriptive diffère de l'emploi traditionnel des analyses corrélatoire et spectrale. Du point de vue méthodologique certains choix ont été réalisés dans la mise en oeuvre des calculs. Appliquées à trois systèmes karstiques pyrénéens, ces analyses ont permis de les comparer et d'étudier le comportement de chacun d'exuc d'un cycle à l'autre. Elles ont montré la présence d'effets mémoires qui modulent, à court, à moyen et à long terme, le signal d'entrée correspondant à la pluie. La modulation de ce signal est caractéristique de chaque système; elle traduit l'importance des réserves et la manière dont celles-ci se constituent et se vidangent. Elle fournit par conséquent un indice de l'état de karstification de l'aquifère. Les exemples traités montrent que ces analyses repreésentent un excellent moyen d'investigation des systèmes hydrologiques.
A Markov autocorrelation pulse (MACP) model for two sites daily streamflow generation is proposed in this study. By using two-component pulses with a one-order Markov autocorrelation deterministic part and a skewed Weibull random part as inputs to the hydrologic system, a simplified model for daily streamflow generation is formulated. Pulses occurring concurrently at two neighboring gauges are assumed to be cross-correlated processes. The mean, standard deviation, auto- and crosscorrelation of daily, monthly, and annual streamflows are computed for observed and generated flow sequences. The performance of the stochastic streamflow model is evaluated. Results show that the MACP model may relatively and satisfactorily reproduce both linear and nonlinear dependence of daily streamflow processes. The effectiveness of the model is demonstrated through its practical application to the generation of daily streamflow for the Wupper River in Germany. The significance of the findings reported in this paper is that the MACP model provides a flexible and adaptive method for reproducing the historical frequency distribution of daily streamflow sequence.
. A class of models for one dimensional time series is presented. The spectrum of such a model is obtained by raising the spectrum of a known parameterized model to an exponent, allowed to attain arbitrary real values. For a moving average model this for example means that the roots of the moving average operator are allowed to have any real order. This method adds a further flexibility to the model which for example allows us to model long memory time series using only a few parameters. The exponent is parameterized in a special way to make the estimation of the parameter determining the exponent asymptotically independent of the estimation of the other model-parameters. The asymptotic distribution of the estimators is derived. The idea is also used for multiplicative models with an exponent for each seasonal factor. In this case the estimators are only approximately independent for a large season length. Finally an application of the model is given using the Beveridge wheat price index.
A stochastic model for the temporal and spatial description of daily streamflow at multiple sites is presented. It is an extension of a chain-dependent process commonly fitted to daily precipitation series. The extension involves allowing daily pulses to be auto- and cross-correlated at temporal and spatial scales. The chain-dependent Markov correlation pulse (CDMCP) model, consisting of a two-state, first-order Markov chain for occurrences and a Markov correlation model for nonzero pulse height, is extended to simultaneous pulse generation at multiple sites. The pulse model coupled with a variable response function was applied to the daily streamflow generation for Wuper River in Germany, and was satisfactory in preserving the short-term structure of the occurrence process, as well as part of the statistics of the seasonal streamflow. The use of Markov model yields good agreement of autocorrelation in the synthetic pulse and streamflow series.
The family of autoregressive integrated moving-average processes, widely used in time series analysis, is generalized by permitting the degree of differencing to take fractional values. The fractional differencing operator is defined as an infinite binomial series expansion in powers of the backward-shift operator. Fractionally differenced processes exhibit long-term persistence and antipersistence; the dependence between observations a long time span apart decays much more slowly with time span than is the case with the more commonly studied time series models. Long-term persistent processes have applications in economics and hydrology; compared to existing models of long-term persistence, the family of models introduced here offers much greater flexibility in the simultaneous modelling of the short-term and long-term behaviour of a time series.
The linkage between stochastic and parametric hydrology models is investigated. Using autocorrelation analysis, it is shown how a conceptual unit hydrograph model consisting of n linear reservoirs in cascade leads to an autoregressive time series which may be used for generating synthetic streamflow. The theory is verified using Nash's data on the Ashbrook catchment and two floods in a Pennsylvania watershed.
It is suggested that the physical characteristics of a watershed exert two simultaneous and distinct influences upon the resultant unit hydrograph: one, whereby the runoff is brought to the valley; the other, in getting the runoff through the mouth. An empirical equation based on this thesis results in two parameters which are readily expressible in terms of the peak discharge and the time of peak. Between these two sets of parameters, the functional relationship is sufficiently complex to rob the latter set of any direct tie-in with the watershed characteristics. Herein lies a possible explanation of the general failure encountered in correlating watershed characteristics with peak discharge and time of peak. It is hoped that the proposed parameters may be more successful in bringing about this much sought-after correlation.
The paper presents a critical review of current methods employed for the establishment of quantitative relationships between precipitation and runoff in hydrology. These procedures, which are classified under the general categories of ‘parametric’ and ‘stochastic’ hydrologies, include the methods of correlation analysis, partial and general synthesis, general nonlinear analysis, and statistical simulation. They are studied as mathematical models of the hydrologic systems, having specific theoretical properties and limitations. The structures of these models are examined with the aid of flow charts, and the areas of their optimum application are suggested.
By the single assumption that the reservoir action in a catchment can be separated from translation, the general equation of the unit hydrograph is shown to be This is simplified by two further simple assumptions to give which can be conveniently calculated.
In unit hydrograph correlations the number of degrees of freedom which it is useful to maintain in the form of the instantaneous unit hydrograph is shown to be limited by the number of significant independent correlations with the catchment characteristics. The moments of the instantaneous unit hydrograph are suggested as a series of parameters of the response, for which correlations should be sought. A simple method of obtaining these moments is evolved, and a method of choosing, in any given case, between several two-parameter forms is demonstrated.
The Wiener-Hopf theory of optimum linear systems is applied to the determination of the stable pulse response of a monotone hydrologic system from coincident records of input and output in the form of discrete time series. In application to the rainfall runoff system, linear programming methods are used in the solution of the Wiener-Hopf equations to obtain physically realizable unit hydrographs.
An autoregressive cum moving average (ARMA) formulation for modelling of surface runoff systems is proposed. It emerges from this study that Nash's model, which has been used successfully in representing British catchments, is a special case of this formulation. The structural relationship between the parameters of the general ARMA and the conceptual formulations has been established. A particular case of the general ARMA formulation has been proposed for the modelling of surface runoff behaviour of the system. The proposed formulation corresponds to a conceptual model of two linear reservoirs in series. This model has been investigated for its stability, nature of response function and range of parameter values.
The discrete analogies of the classical Muskingum and the equal-reservoir cascade models are compared with the ARMA-type models used in the generation of synthetic time series. Applying the cascade concept to the ARMA-type difference equation and including a pure translation parameter, a family of discrete linear parametric models is developed. Expressions are derived for the direct estimation of model parameters by cumulants for the general input-output case and for the special case of a white-noise input. The family of models can be applied in unit-hydrograph analysis, in flood routing, and in the generation of synthetic time series.
The characteristic of a linear time invariant system can be found uniquely from any two outputs that have no error: it is not necessary to know the inputs or the time duration of the system response. In hydrologic terms, the unitgraph of a catchment with linear characteristics may be found from records of runoff vs time.
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