Abstract
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.
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Abbreviations
- 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
- ℓ :
-
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
- \(\bar x\) :
-
mean of the rainfall data
- y t :
-
direct runoff data (discharge)
- Y :
-
Y() = runoff process
- \(\bar y\) :
-
mean of the runoff data
- (x, y) t :
-
rainfall-runoff data (att ε T)
- (x, y, z) t :
-
rainfall-runoff-sediment yield data (att ε T)
- z :
-
complex number (in spectral analysis)
- Δ k :
-
coefficient of the LB algorithm atkth step
- λ nj :
-
Lagrange multiplier atjth location in the Λ n matrix
- Λ n :
-
Λ n = matrix of the Lagrange multiplier atkth step
- ρ X (k),ρ Y (k):
-
autocorrelation function of rainfall and runoff processes atkth lag
- ρ 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
- 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
- \(\bar x\) :
-
forecast ofx
- \(\bar x(\ell )\) :
-
forecast ofx ℓ-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
References
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Krstanovic, P.F., Singh, V.P. A real-time flood forecasting model based on maximum-entropy spectral analysis: I. Development. Water Resour Manage 7, 109–129 (1993). https://doi.org/10.1007/BF00872477
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DOI: https://doi.org/10.1007/BF00872477