Water table fluctuation with a random initial condition

The nonlinear Boussinesq equation is used to understand water table fluctuations in various ditch drainage problems. An approximate solution of this equation with a random initial condition and deterministic boundary conditions, recharge rate and aquifer parameters has been developed to predict a transient water table in a ditch-drainage system. The effects of uncertainty in the initial condition on the water table are illustrated with the help of a synthetic example. These results would find applications in ditch-drainage design.Notation A / tanh t - a lower value of the random variable representing the initial water table height at the mid point - a+b Upper value of the random variable representing the initial water table height at the midpoint - B tanh t - C 4/ - h variable water table height - h mean of the variable water table height - h _m variable water table height at the mid point - h _m mean of the variable water table height at the mid point - K hydraulic conductivity - L half spacing between the ditches - m ₀ initial water table height at the mid point - N Uniform rate of recharge - S specific yield - t time of observation - x distance measured from the ditch boundary - (4/SL)(NK)^1/2 - (L/4)(N/K)^1/2 - dummy integral variable     

Water table fluctuation in response to transient recharge from a rectangular basin

A problem of water-table fluctuation in a finite two-dimensional aquifer system in response to transient recharge from an overlying rectangular area is studied. An analytical solution is obtained by using the method of finite Fourier transform to predict the transient position of the water-table. The solution for constant rate of recharge is shown as a special case of the present solution. Effects of variation in the rate of recharge on the growth of two-dimensional groundwater mound is illustrated with the help of a numerical example.Notation A half width of the aquifer [L] - B half length of the aquifer [L] - D half width of the recharge basin [L] - e specific yield - h varying water-table height [L] - h ₀ initial water-table height [L] - h weighted mean of the depth of saturation [L] - K hydraulic conductivity [LT^–1] - L half length of the recharge basin [L] - P(t) time varying rate of recharge [LT^–1] - P ₁ +P ₀ initial rate of time varying recharge [LT^–1] - P ₁ final rate of time varying recharge [LT^–1] - t time of observation [T] - x, y coordinate axes - decay constant [T^–1]     

An analytical solution for water-table fluctuation in a finite aquifer due to transient recharge from a strip basin

Two cases of water-table fluctuation in a finite aquifer in response to transient recharge from a strip basin are investigated. In the first case the aquifer is bounded by open water-bodies, whereas in second one the aquifer is bounded by impermeable boundaries on both sides. Analytical solutions are presented to predict the transient position of the water-table. The solutions are obtained by using finite Fourier sine and cosine transforms.Notations A width of the aquifer [L] - e specific yield - h variable water-table height [L] - h ₀ initial water-table height [L] - weighted mean of the depth of saturation [L] - K hydraulic conductivity [LT^–1] - m,n integers - P ₁ +P ₀ initial rate of transient recharge [LT^–1] - P ₁ final rate of transient recharge [LT^–1] - P constant rate of recharge [LT^–1] - x ₁ distance of left boundary of the strip basin [L] - x ₂ distance of right boundary of the strip basin [L] - t time of observation [T] - decay constant [T^–1]     

Forecast model of water consumption for Naples

The data refer to the monthly water consumption in the Neapolitan area over more than a 30 year period. The model proposed makes it possible to separate the trend in the water consumption time series from the seasonal fluctuation characterized by monthly peak coefficients with residual component. An ARMA (1,1) model has been used to fit the residual component process. Furthermore, the availability of daily water consumption data for a three-year period allows the calculation of the daily peak coefficients for each month, and makes it possible to determine future water demand on the day of peak water consumption.Notation j numerical order of the month in the year - i numerical order of the year in the time series - t numerical order of the month in the time series - h numerical order of the month in the sequence of measured and predicted consumption values after the final stage t of the observation period - Z _ji effective monthly water consumption in the month j in the year i (expressed as m³/day) - T _ji predicted monthly water consumption in the month j in the year i minus the seasonal and stochastic component (expressed as m³/day) - C _ji monthly peak coefficient - E _ji stochastic component of the monthly water consumption in the month of j in the year i - Z _i water consumption in the year i (expressed as m³/year) - Z _j (t) water consumption in the month j during the observation period (expressed as m³/day) - evaluation of the correlation coefficient - Z _j (t) water consumption in the month j during the observation period minus the trend - Y _t transformed stochastic component from E _t : Y _t =ln Et - Y _t+h measured value of stochastic component for t+h period after the final stage t of the observation period - Y _t (h) predicted value of stochastic component for t+h period after the final stage t of the observation period - _j transformation coefficients from the ARMA process (m, n) to the MA () process     

Analytical solutions of unsteady drainage problems for soils subjected to variable recharge from a semi-confined aquifer

In drainage of agricultural lands, the upward vertical recharge from a semi-confined aquifer depends on the difference of the piezometric heads on the two sides of the semi-impermeable layer through which this recharge takes place. This means that the recharge through the semi-impermeable base depends on the unknown height of the unsteady water table. In the nonhomogeneous Boussinesq equation, which describes the drainage problems, the downward uniform rate of the recharge from rain or irrigation and the recharge from the semiconfined aquifer are expressed by two terms. By solving the Boussinesq equation expressions for the nondimensional height of the water table and the nondimensional discharge of the drains per unit drained area are obtained for three different initial conditions. Some known solutions are shown as special cases of the present solutions. Variation of nondimensional water table heights at half distance of the drain spacing and the nondimensional discharge of the drains with nondimensional time have been graphically illustrated with the help of synthetic examples.Notation B _s thickness of the semi-impervious layer [L] - c hydraulic resistance of the semi-impervious layer [T] - D depth of the drains from the base [L] - d _e equivalent depth [L] - h=h(x, t) height of the water table [L] - h ₀ initial height of the water table [L] - h _t water table height at mid-distance of drains att [L] - h _j ,h _k water table height at mid-distance of drains at timej andfk, respectively [L] - H ₀ piezometric head in the semi-confined aquifer [L] - K hydraulic conductivity of the soil [LT^–1] - K _s hydraulic conductivity of the semi-impervious layer [LT^–1] - k ₀,k ₁,k ₂ nondimensional constants - L distance between the drains [L] - q ₀ upward recharge per unit surface area through the semi-impervious layer [LT^–1] - q _t discharge per unit drainable area of drains at timet [LT^–1] - R,R ₀ recharge per unit surface area from rain or irrigation during the unsteady and steady-state, respectively, [LT^–1] - S specific yield of the soil - t time of observation [T] - x distance measured from the drain [L] - leakage factor [L] - nondimensional distance - nondimensional time     

A real-time flood forecasting model based on maximum-entropy spectral analysis: I. Development

This paper, the first of two, develops a real-time flood forecasting model using Burg's maximum-entropy spectral analysis (MESA). Fundamental to MESA is the extension of autocovariance and cross-covariance matrices describing the correlations within and between rainfall and runoff series. These matrices are used to derive the model forecasting equations (with and without feedback). The model may be potentially applicable to any pair of correlated hydrologic processes.Notation a _k extension coefficient of the model atkth step - B _k backward extension matrix forkth step - B _ijk element of the matrixB _k (i,j=1, 2) - c _k coefficient of the entropy model atkth step in the LB algorithm - e _k (e _x ,e _y )k = forecast error vector atkth step - E _k error matrix atkth step - E _ijk element of theE _k (i,j=1, 2) - f frequency - F _k forward extension matrix atkth step - F _ijk element of theF _k matrix (i,j=1, 2) - H(f) entropy expressed in terms of frequency - H _X entropy of the rainfall process (X) - H _Y entropy of the runoff process (Y) - H _XY entropy of the rainfall-runoff process - I identity matrix - forecast lead time - m model order, number of autocorrelations - R correlation matrix - S _x standard deviation of the rainfall data - S _y standard deviation of the runoff data - t time - T ₁ rainfall record - T ₂ runoff record - T rainfall-runoff record (T=T ₁ T ₂) - x _t rainfall data (depth) - X X() = rainfall process - mean of the rainfall data - y _t direct runoff data (discharge) - Y Y() = runoff process - mean of the runoff data - (x, y) _t rainfall-runoff data (att T) - (x, y, z) _t rainfall-runoff-sediment yield data (att T) - z complex number (in spectral analysis) - _k coefficient of the LB algorithm atkth step - _nj Lagrange multiplier atjth location in the _n matrix - _{n n} = matrix of the Lagrange multiplier atkth step - _X (k), _Y (k) autocorrelation function of rainfall and runoff processes atkth lag - _XY (k) cross-correlation function of rainfall and runoff processes atkth lag - W ₁(f) power spectrum of rainfall or runoff - W ₂(f) cross-spectrum of rainfall or runoff Abbreviations acf autocorrelation function - ARMA autoregressive moving average (model) - ARMAX ARMA with exogenous input - ccf cross-correlation function - det() determinant of the (...) matrix - E[...] expectation of [...] - FLT forecast lead time - KF Kalman filter - LB Levinson-Burg (algorithm) - MESA maximum entropy spectral analysis - MSE mean square error - SS state-space (model) - STI sampling time interval - forecast ofx - forecast ofx -step ahead - x _F feedback ofx-value (real value) - |x| module (absolute value) ofx - X ^–1 inverse of the matrixX - X* transpose of the matrixX     

Optimization-simulation models for the design of an Irrigation project

Optimization-simulation models were used for the systems analysis of a water resources system. The Karjan Irrigation reservoir project in India was taken as the system. Two types of optimization models, i.e., linear programming, and dynamic programming (continuous and discontinuous) were used for preliminary design purposes. The simulation technique was used for further screening. The linear programming model is most suitable for finding reservoir capacity. Dynamic programming (continuous and discontinuous models) may be used for further refining the output targets and finding the possible reservoir carry-over storages, respectively. The simulation should then be used to obtain the near optimum values of the design variables.Notations a ₁ Unit irrigation benefit [Rs.10⁵ L^–3] - B ₁ Gross annual irrigation benefit [Rs.10⁵] - B _1,t Gross irrigation benefit in periodt [Rs.10⁵] - C ₁ Annual capital cost of irrigation [Rs.10⁵] - C ₁ Annual capital cost function for irrigation [Rs.10⁵ L^–3] - C _1,t Fraction of annual capital cost for irrigation in periodt [Rs.10⁵] - C ₂ Annual capital cost of reservoir [Rs.10⁵] - C ₂ Annual capital cost function for reservoir [Rs.10⁵ L^–3] - C _2,t Fraction of annual capital cost for reservoir in periodt [Rs.10⁵] - El _t Reservoir evaporation in timet [L³] - f _t Optimal return from staget [Rs.10⁵] - g _t The return function for periodt [Rs.10⁵] - I _t Catchment inflow into the reservoir in periodt [L³] - I _t Water that joins the main stem just above the irrigation diversion canal in timet [L³] - _t Local inflow to the reservoir from the surrounding area in timet [L³] - Ir Annual irrigation target [L³] - K _t Proportion of annual irrigation targetIr to be diverted for irrigation in timet - K _t Amount by whichK _t exceeds unity is the fraction of the end storage which is assigned to reservoir evaporation losses - L Loss in irrigation benefits per unit deficit in the supply [Rs.10⁵ L^–3] - L ₁ Lower bound on annual irrigation target,Ir [L³] - L ₂ Lower bound on reservoir capacity,Y [L³] - N Number of time periods in the planning horizon - O _t Total water release from the reservoir in periodt [L³] - O _t ^* The optimal total water release from the reservoir in timet [L³] - _t Secondary water release from the reservoir in timet [L³] - O _t Reservoir release to the natural channel in timet [L³] - Od _t Irrigation demand in timet [L³] - Om ₁ Annual OM cost of irrigation [Rs.10⁵] - Om ₁ Annual OM cost function for irrigation [Rs.10⁵ L^–3] - Om _1,t Fraction of annual OM cost for irrigation in periodt [Rs.10⁵] - Om ₂ Annual OM cost of reservoir [Rs.10⁵] - Om ₂ Annual OM cost function for reservoir [Rs.10⁵ L^–3] - Om _2,t Fraction of annual OM cost for reservoir in periodt [L³] - Omin_t Lower bound onO _t in timet [L³] - Omax_t Upper bound onO _t in timet [L³] - P _t Precipitation directly upon reservoir in timet [L³] - S _t Gross/live reservoir storage at the end of timet (gross storage in the linear program and live storage in the dynamic program) [L³] - S _t–1 Gross/live reservoir storage at the beginning of timet [L³] - t Any time period - Tr_t Transformation function - U ₁ Upper bound onIr [L³] - U ₂ Upper bound onY [L³] - Y Total capacity of reservoir at maximum pool level [L³] - Ya Fixed active (live) capacity of the reservoir (Y-Yd) [L³] - Ya _t Active (live) capacity (Ymax_t –Ymin_t) of the reservoir in timet [L³] - Yd Dead storage of the reservoir [L³] - Ymax_t Capacity up to the normal pool level of the reservoir in timet [L³] - Ymax_t Live capacity up to the normal pool level of the reservoir in timet [L³] - Ymin_t Capacity up to the minimum pool level of the reservoir in timet [L³] - Ymin_t Live capacity up to the minimum pool level of the reservoir in timet [L³]     

Optimization of the water chemistry of the primary coolant at nuclear power plants with VVÉR

Results of the use of automatic hydrogen-content meter for controlling the parameter of hydrogen in the primary coolant circuit of the Kola nuclear power plant are presented. It is shown that the correlation between the hydrogen parameter in the coolant and the hydrazine parameter in the makeup water can be used for controlling the water chemistry of the primary coolant system, which should make it possible to optimize the water chemistry at different power levelsTranslated from Élektricheskie Stantsii, No. 12, December 2004, pp. 31 – 33.     

Pre-posterior analysis as a tool for data evaluation: Application to aquifer contamination

This paper deals with the frequently encountered problem of pre-posterior data evaluation, i.e., assessment of the value of data before they become available. The role of data is to reduced the risk associated with decisions taken under conditions of uncertainty. However, while the inclusion of relevant data reduces risk, data acquisition involves cost, and there is thus an optimal level beyond which any addition of data has a negative net benefit. The Bayesian approach is applied to construct a method for updating decisions and evaluating the anticipated reduction in risk following consideration of additional data. The methodology is demonstrated on a problem of management of an aquifer under threat of contamination.Notation L matrix of losses for all combinations of states and decisions - l, m, h possible salinity levels from the proposed borehole - N, M, F possible decisions - P(·) vector of prior probabilities of states - P(.|l), P(.|m), P(.|h) conditional (updated) probability vectors of the different states given the salinity levels - P(.|), P(.|), P(.|) probability vectors of the different salinity levels given the true states (likelihood function) - P(l), P(m), P(h) probabilities of the salinity levels, irrespective of the true state - R(.|l), R(.|m), R(.|h) posterior risk vectors of the different decisions given the salinity levels - R(N), R(M), R(F) prior risk associated with different decisions - , , possible true states     

An Aquifer Management Case Study – The Chalk of the English South Downs

The Chalk aquifer of the English South Downs is very heavily utilised. The groundwater resources have enjoyed a formal programme of management which started in the 1950s, although a number of actions had been taken earlier in order to deal with saline intrusion and potential risk to groundwater quality from urbanisation. In the late 1950s the policy of leakage/storage boreholes was first adopted, whereby the leakage boreholes along the coast were pumped in winter to intercept fresh water discharge to the sea and to maximise the recharge potential inland, and inland storage boreholes were used, as much as possible, in the summer months only. A comprehensive monitoring programme supported by aquifer modelling has enabled a gradual increase in overall abstraction to take place without increasing groundwater degradation due to saline intrusion. There have been various pollution prevention strategies over the years, and these have been effective in protecting the groundwater despite the high population density and widespread agricultural activity within the South Downs. The management of the aquifer has clearly been successful; there are many lessons from this experience that can be applied to other regions and other aquifers.     

Errors of kinematic-wave and diffusion-wave approximations for time-independent flows

Time-independent (or steady-state) cases of planar (overland) flow were treated. Errors of the kinematic-wave and diffusion-wave approximations were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth-gradient at the downstream end. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with error in the range of 1–2% for values ofKF ₀ ² (7.5). Even for small values ofKF ₂ ⁰ (e.g.,KF ₂ ⁰ =0.75), the errors were typically in the range of 11–15%. The accuracy of the diffusion wave approximation was greatly influenced by the downstream boundary condition. The error of the kinematic wave approximation was found to vary from 7 to 13% in the regions 0.05x0.95 forKF ₀ ² =0.75 and was greater than 30% forKF ₀ ² =0.75.     

Determination of optimal unit hydrographs by linear programming

A unit hydrograph (UH) obtained from past storms can be used to predict a direct runoff hydrograph (DRH) based on the effective rainfall hyetograph (ERH) of a new storm. The objective functions in commonly used linear programming (LP) formulations for obtaining an optimal UH are (1) minimizing the sum of absolute deviations (MSAD) and (2) minimizing the largest absolute deviation (MLAD). This paper proposes two alternative LP formulations for obtaining an optimal UH, namely, (1) minimizing the weighted sum of absolute deviations (MWSAD) and (2) minimizing the range of deviations (MRNG). In this paper the predicted DRHs as well as the regenerated DRHs by using the UHs obtained from different LP formulations were compared using a statistical cross-validation technique. The golden section search method was used to determine the optimal weights for the model of MWSAD. The numerical results show that the UH by MRNG is better than that by MLAD in regenerating and predicting DRHs. It is also found that the model MWSAD with a properly selected weighing function would produce a UH that is better in predicting the DRHs than the commonly used MSAD.Notations M number of effective rainfall increments - N number of direct runoff hydrograph ordinates - R number of storms - MSAD minimize sum of absolute deviation - MWSAD minimize weighted sum of absolute deviation - MLAD minimize the largest absolute deviation - MRNG minimize the range of deviation - RMSE root mean square error - P _m effective rainfall in time interval [(m–1)t,mt] - Q _n direct runoff at discrete timent - U _k unit hydrograph ordinate at discrete timekt - W _n weight assigned to error associated with estimatingQ _n - _n ⁺ error associated with over-estimation ofQ _n - _n ^– error associated with under-estimation ofQ _n - _max ⁺ maximum positive error in fitting direct runoff hydrograph - _max ^– maximum negative error in fitting direct runoff hydrograph - _max largest absolute error in fitting obtained direct runoff - E _r,1 thelth error criterion measuring the fit between the observed DRHs and the predicted (or reproduced) DRHs for therth storm - E ₁ averaged value of error criterion overR storms     

Time-dependent soil-water distribution under a circular trickle source

Soil-water distribution in homogeneous soil profiles of Yolo clay loam and Yolo sand (Typic xerorthents) irrigated from a circular source of water, was measured several times after the initiation of irrigation. The effect of trickle discharge rates and soil type on the locations of the wetting front and soil-water distribution was considered. Soil-water tension and hydraulic conductivity, as functions of soil-water content, were also measured. The theories of time-dependent, linearized infiltration from a circular source and a finite-element solution of the two-dimensional transient soil-water equation were compared with the experimental results. In general, for both soils the computer horizontal and vertical advances of the wetting front were closely related to those observed. With both theories, a better prediction of the wetting front position for the clay loam soil than for the sandy soil is shown. The calculated and measured horizontal vertical advances did not agree over long periods of time. With the linearized solution, overestimated and underestimated vertical advances for the clay and sandy soils, respectively, were shown. The finite-element model approximate in a better way the vertical advances than the linearized solution, while an opposite tendency for the horizontal advances indicated, especially in sandy soil.Notation k constant (dK/d) - K hydraulic conductivity - K ₀ saturated hydraulic conductivity - J ₀,J ₁ Bessel functions of the first kind - h soil water tension - q Q/r ₀ ² - Q discharge rate - r cylindrical coordinate; also horizontal distance in soil surface - R dimensionless quantity forr - r ₀ constant pond radius - R ₀ dimensionless quantity forr ₀ - t time - T dimensionless quantity fort - x, y Cartesian coordinates - z vertical coordinate; also vertical distance along thez axis chosen positively downward - Z dimensionless quantity forz - empirical soil characteristic constant - dummy variable of integration - volumetric soil water content - matrix flux potential - dimensionless quantity for      

Analysis of the relationship for determining the bed load in sand channels

Conclusion The results of the analysis showed that in flatland rivers with a sand channel withu _*u _*0 where movement of the bed load is realized mainly in the form of steep waves and ripples [5], for determining the bed loadq _bd GGI's relationship (1), in whichq _bd is represented as a function of the height of the bed formsh _w, should be used. This relationship gives results substantially differing from the results calculated by relationships in whichq _bd is expressed as a function of the size of the sediment particlesd (8), (9) and valid for conditions not characteristic of sand channels of flatland rivers. Whenu _*t u _*<u _*0, when there is still no mass traction of all fractions of the channel deposits, the Einstein (8) or Paintal (15) relationships are more substantiated for all channels, but for practical calculations for sand channels of flatland rivers one can assume thatq _bd=0 whenu _*<u _*0.Translated from Gidrotekhnicheskoe Stroitel'stvo, No. 5, pp. 47–50, May, 1989.     

Efficient steam turbines produced by the “Ural Turbine Plant” company

Design features and efficiency of some steam turbines produced at present by a plant formed as a result of division of the Turbine Motor Plant Company into several enterprises are presented.Translated from Élektricheskie Stantsii, No. 11, November 2004, pp. 27 – 32.     

Dam-Break flood forecasting in Piemonte region,northwest Italy

Six major reservoirs in Piemonte region, northwest Italy, have been examined in order to assess the possible flood damages to the downstream area. In this paper, some results of the hydraulic study are presented. The floods are simulated by computer models with the input data which describe the imagined dam-break events as well as other facts. Some important practical aspects of the work are extensively discussed, i.e. the problems concerning determination of the dam-breaches, the influence of the breach parameters, and estimation of the hydraulic resistance factors.Notation A = cross sectional area of water flow - C = Chezy roughness coefficient - C = discharge coefficient - g = acceleration due to gravity - H = height of dam-breach - H = height of dam - h = water surface level above the datum plane - L = width of dam - Q = flow discharge - Q_e, Q_u = inflow and outflow discharges respectively - q = lateral inflow discharge - R_* = hydraulic radius - T = formation time of dam-breach - t = time - V = volume of reservoir storage - W = width of dam-breach - X = downstream distance from a dam along the river - = velocity distribution factor     

Use of parameter estimation in determining soil hydraulic properties from unsaturated transient flow

This paper proposes a model for determining the parameters given by the closed-form equations of van Genuchten. An objective function is made by the observed data from vertical drainage, and the solutions of optimization show that less computation and more accurate estimates are made as head profiles are taken into account rather than cumulative drainage. Sensitivity analysis of the error vector to parameters interprets this reason. The convergence and stability of solutions are evaluated with different magnitudes of measured errors in the head, and the results show good estimates will be obtained if a sufficient pressure head at the soil bottom is applied. A variable _k is introduced to avoid the estimations of and n being affected by the uncertainties of Ks and _s .     

Spectral Reflectance,Growth and Chlorophyll Relationships for Rice Crop in a Semi-Arid Region of India

Relations among spectral reflectance, chlorophyll a, and growth of rice plants grown on irrigated light textured soil in a semi arid region are presented here. There was a linear relation between spectral reflectance and rice plant height (r = 0.97), for band 1 (0.45–0.52 m) reflectance values. On the other hand, in bands 2 (0.52–0.60 m) and 3 (0.63–0.69 m), reflectance values decreased until 70 days after planting (DAP) and then increased during the reproductive phase of the crop. The near infrared band 4 (0.76–0.90 m) showed a maximum reflectance at 59 DAP (panicle initiation stage) and a decline in reflectance thereafter through maturity. The peak value of IR/R ratio was 16.39 at 62 DAP during the early reproductive phase; thereafter, it declines gradually with the maturity of the crop. Chlorophyll a concentration was high during early growth (vegetative and early reproductive stages) and decreased during the flowering and maturity stages. The rice plant canopy show a high chlorophyll a concentration at 64 and 59 DAP for sites A and B, respectively. Chlorophyll a concentration is higher in site A plant canopies than it is in site B during the entire crop cycle. A good inverse correlation (r = 0.91) has been found between chlorophyll a and band 1, while the IR/R ratio and the normalised difference vegetation index (NDVI) showed a relationship (r = 0.78) with the chlorophyll a concentration during the crop cycle. Band 2, 3 and 4 radiance values show a biphasic linear relationship with chlorophyll a concentrations, negative for early growth and positive for flowering and maturity stages. Results indicate that the period between 66 to 70 DAP is most suitable for the assessment of rice crop yield, based on chlorophyll a concentration.     

Balance of the rotors of hydroelectric generators

Conclusions The described method of finding the heavy point of a turbine rotor permits balancing without the use of complex apparatus and cumbersome graphic plots; the plane of imbalance is determined with sufficient accuracy during one start.Translated from Gidrotekhnicheskoe Stroitel'stvo, No. 9, pp. 54–55, September, 1981.     

A real-time flood forecasting model based on maximum-entropy spectral analysis: II. Application

The MESA-based model, developed in the first paper, for real-time flood forecasting was verified on five watersheds from different regions of the world. The sampling time interval and forecast lead time varied from several minutes to one day. The model was found to be superior to a state-space model for all events where it was difficult to obtain prior information about model parameters. The mathematical form of the model was found to be similar to a bivariate autoregressive (AR) model, and under certain conditions, these two models became equivalent.Notation A _k parameter matrix of the bivariate AR model - B backshift operator in time series analysis - eT forecast error (vector) at timet = T - _t uncorrelated random series (white noise) - F _k forward extension matrix of the entropy model forkth lag - I identity matrix - m order of the entropy model - N number of observations - P order of the AR model - Q _p peak of the direct runoff hydrograph - R correlation matrix - t _p time to peak of the direct runoff hydrograph - ₁ coefficient of variation - ₂ ratio of absolute error to the mean - forecasted runoff - x _i observed runoff - mean of the observed runoff - X ^–1 inverse ofX matrix - X* transpose of theX matrix Abbreviations AIC Akaike information criterion - AR autoregressive (model) - AR(p) autoregressive process of thepth order - ARIMA autoregressive integrated moving average (model) - acf autocorrelation function - ccf cross-correlation function - FLT forecast lead time - MESA maximum entropy spectral analysis - MSE mean square error - STI sampling time interval