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Matalas (1967) Multi-Site MAR(1)
Type Parametric
Resolution Monthly
Sites Multisite
Class MATALASGenerator

Overview

The Matalas MAR(1) model is the classical parametric baseline for multi-site stochastic streamflow generation. It extends the univariate Thomas-Fiering model to n sites by fitting a matrix autoregressive model to standardized monthly flows. A separate pair of coefficient matrices is estimated for each of the 12 calendar-month transitions.

Algorithm

Preprocessing

  1. Validate input; resample to monthly if daily data are provided.
  2. Clip values to 1e-6 to avoid log of zero.
  3. Optionally apply log(Q + 1) transformation to reduce skewness.

Fitting

  1. Standardize observed flows by monthly means and standard deviations: 
    Z(t) = (Q(t) - mu_m) / sigma_m,    m = month(t)
  2. Estimate cross-correlation matrices for each transition m to m+1: 
    S0(m) = (1/(n-1)) * Z(m)^T * Z(m)         # lag-0 covariance
S1(m) = (1/(n-1)) * Z(m+1)^T * Z(m)        # lag-1 cross-covariance
  3. Solve for coefficient matrices: 
    A(m) = S1(m) * S0(m)^{-1}                   # AR coefficient matrix
M(m) = S0(m+1) - A(m) * S0(m) * A(m)^T      # innovation covariance
B(m) = cholesky(M(m))                         # lower Cholesky factor
    If M(m) is not positive semi-definite, project to nearest PSD matrix via eigenvalue clipping before Cholesky decomposition.

The December to January transition wraps across the year boundary.

Generation

  1. Initialize: Z_0 ~ N(0, I)
  2. Recurse for each subsequent month: 
    Z(t+1) = A(m) * Z(t) + B(m) * epsilon(t+1),    epsilon ~ N(0, I)
  3. Back-transform to flow space: 
    Q(t) = sigma_m * Z(t) + mu_m
  4. If log transform was applied: Q = exp(Q) - 1

Parameters

Parameter Type Default Description
Q_obs pd.DataFrame or pd.Series - Observed streamflow with DatetimeIndex
log_transform bool True Apply log(Q+1) before standardization
name Optional[str] None Optional name identifier for this generator instance
debug bool False Enable debug logging

Properties Preserved

  • Monthly means and standard deviations at each site
  • Lag-1 serial correlation at each site
  • Contemporaneous cross-site correlations

Not preserved: - Higher-order autocorrelation (lag > 1) - Non-Gaussian marginal distributions

Limitations

  • First-order memory only (lag-1)
  • Normality assumption after transformation
  • Requires sufficient record length for stable covariance estimation
  • Covariance matrices may require PSD repair with limited data

References

Primary: Matalas, N.C. (1967). Mathematical assessment of synthetic hydrology. Water Resources Research, 3(4), 937-945. https://doi.org/10.1029/WR003i004p00937

See also: - Salas, J.D., Delleur, J.W., Yevjevich, V., and Lane, W.L. (1980). Applied Modeling of Hydrologic Time Series. Water Resources Publications.

Implementation: src/synhydro/methods/generation/parametric/matalas.py Tests: tests/test_matalas_generator.py