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WARM - Wavelet Auto-Regressive Method (Nowak et al. 2011)
Type Parametric
Resolution Annual
Sites Univariate
Class WARMGenerator

Overview

WARM combines continuous wavelet transforms with autoregressive modeling to generate synthetic annual streamflow. Its key innovation is the Scale Averaged Wavelet Power (SAWP), which captures time-varying spectral characteristics. This makes WARM particularly suited for non-stationary flows with regime shifts or low-frequency variability (e.g., climate oscillations, multi-decadal droughts).

Algorithm

Preprocessing

  1. Validate input (univariate time series, minimum ~20 years recommended).
  2. Resample to annual frequency if needed (sum monthly or daily).

Fitting

  1. Continuous Wavelet Transform - decompose observed flows into time-frequency components using mother wavelet (default: Morlet): 

    W(s, t) = CWT(Q_obs, scales, wavelet)
    Result: coefficient matrix of shape (n_scales, n_years).

  2. Scale Averaged Wavelet Power - compute time-varying power across all scales: 

    SAWP(t) = (1/S) * sum_s |W(s, t)|^2
    High SAWP indicates high energy/variability at time t.

  3. Normalize by SAWP - remove time-varying power to produce stationary components: 

    W_norm(s, t) = W(s, t) / sqrt(SAWP(t) + epsilon)

  4. Fit AR models - for each scale s, fit an AR(p) model to normalized coefficients via Yule-Walker equations. Store parameters: mean, AR coefficients, innovation variance.

Generation

  1. Generate normalized coefficients - for each scale, run the fitted AR(p) process forward with Gaussian innovations.
  2. Resample SAWP - bootstrap SAWP values from the historical record with replacement.
  3. Rescale coefficients - restore time-varying power: 
    W_syn(s, t) = W_norm_syn(s, t) * sqrt(SAWP_syn(t) + epsilon)
  4. Inverse wavelet transform - reconstruct streamflow via weighted summation across scales. Adjust mean and variance to match observed, then clip negatives to zero.

Parameters

Parameter Type Default Description
Q_obs pd.Series or pd.DataFrame - Observed streamflow with DatetimeIndex
wavelet str 'morl' Mother wavelet; any pywt continuous wavelet (e.g., 'morl', 'morlet')
scales int 64 Number of frequency scales (rule of thumb: T/2 to T)
ar_order int 1 AR model order (1 or 2 typically sufficient)
name Optional[str] None Optional name identifier for this generator instance
debug bool False Enable debug logging

Properties Preserved

  • Mean and variance (exactly, via post-reconstruction adjustment)
  • Lag-1 autocorrelation (approximately, via AR models)
  • Multi-year persistence and low-frequency variability (via wavelet decomposition)
  • Non-stationary variance structure (via SAWP)
  • Power spectrum (via wavelet amplitude preservation)

Not preserved: - Exact marginal distribution shape (Gaussian innovations) - Skewness and higher moments (not explicitly modeled) - Spatial correlations (univariate method)

Limitations

  • Annual frequency only - for finer resolution, couple with a disaggregation method
  • Univariate (single site) - no native multi-site support
  • Inverse CWT is approximate (PyWavelets); corrected by mean/variance adjustment
  • Edge effects in CWT for short records (< 20 years)
  • Gaussian AR innovations may underrepresent extremes

References

Primary: Nowak, K., Rajagopalan, B., and Zagona, E. (2011). A Wavelet Auto-Regressive Method (WARM) for multi-site streamflow simulation of data with non-stationary trends. Journal of Hydrology, 410(1-2), 1-12. https://doi.org/10.1016/j.jhydrol.2011.08.049

See also: - Erkyihun, S.T., Rajagopalan, B., Zagona, E., Lall, U., and Nowak, K. (2016). Wavelet-based time series bootstrap model for multidecadal streamflow simulation using climate indicators. Water Resources Research, 52(5), 4061-4077. https://doi.org/10.1002/2016WR018696 - Kwon, H.-H., Lall, U., and Khalil, A.F. (2007). Stochastic simulation model for nonstationary time series using an autoregressive wavelet decomposition. Water Resources Research, 43(5). - Torrence, C., and Compo, G.P. (1998). A practical guide to wavelet analysis. Bulletin of the American Meteorological Society, 79(1), 61-78.

Implementation: src/synhydro/methods/generation/parametric/warm.py Tests: tests/test_warm_generator.py