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Kirsch (2013) Monthly Bootstrap Generator
Type Nonparametric
Resolution Monthly
Sites Multisite
Class KirschGenerator

Overview

The Kirsch method generates synthetic multi-site monthly streamflow by bootstrapping standardized residuals and imposing fitted correlation structure via Cholesky decomposition. A cross-year shifted matrix preserves December-to-January correlations. An optional normal score transform prevents bias when working in log-space.

Algorithm

Preprocessing

  1. Aggregate to monthly - group by (year, month) and sum.
  2. Optional log transform - if generate_using_log_flow=True, apply log(Q) (clipped at 1e-6).

Fitting

  1. Monthly statistics - for each month m and site s, compute mean and standard deviation.
  2. Standardized residuals: 
    Z_h[y, m, s] = (Q[y, m, s] - mean[m, s]) / std[m, s]
  3. Normal score transform (if log-flow) - for each month-site pair:
  4. Rank residuals, map to normal quantiles via Hazen plotting positions
  5. Store mapping for inverse transform during generation
  6. Result: Y in standard normal space
  7. Cross-year shifted matrix Y_prime - preserves inter-year correlations: 
    Y_prime[:, 0:6, :]  = Y[:-1, 6:12, :]   # Jul-Dec of year i
Y_prime[:, 6:12, :] = Y[1:, 0:6, :]      # Jan-Jun of year i+1
  8. Cholesky decomposition - for each site, compute 12x12 correlation matrix of Y (and Y_prime), repair if not PSD (spectral method), then Cholesky factor.

Generation

  1. Bootstrap - sample random year indices for each (year, month) position.
  2. Cholesky mixing - multiply bootstrap samples by Cholesky factor to impose correlation: 
    Z[:, :, s] = X[:, :, s] @ U[s]
  3. Combine Z and Z_prime to preserve intra-year correlations: 
    ZC[i, 0:6, :]  = Z_prime[i, 6:12, :]    # first half from shifted
ZC[i, 6:12, :] = Z[i+1, 6:12, :]         # second half from regular
  4. Inverse normal score transform (if log-flow) - map back using stored mappings with linear tail extrapolation.
  5. Destandardize: Q_syn = ZC * std[m] + mean[m]
  6. Back-transform from log space if applicable: Q_syn = exp(Q_syn)

Parameters

Parameter Type Default Description
Q_obs pd.DataFrame - Observed multi-site streamflow with DatetimeIndex
generate_using_log_flow bool True Log-transform before processing (recommended for skewed data)
matrix_repair_method str 'spectral' Method for repairing non-PSD correlation matrices
name Optional[str] None Optional name identifier for this generator instance
debug bool False Enable debug logging

Properties Preserved

  • Monthly means and standard deviations (per site)
  • Spatial cross-site correlations (via Cholesky decomposition)
  • Intra-annual temporal correlation (within and across year boundaries)
  • Empirical marginal distributions (nonparametric)

Not preserved: - Values outside observed range (bootstrap limitation, except via NST tail extrapolation)

Limitations

  • Requires complete years (all 12 months present per year)
  • Y_prime construction loses one year of data
  • Sample correlation matrices may require PSD repair, which can inflate correlations
  • Bootstrap resampling bounded by historical range

References

Primary: Kirsch, B.R., Characklis, G.W., and Zeff, H.B. (2013). Evaluating the impact of alternative hydro-climate scenarios on transfer agreements: A practical improvement for generating synthetic streamflows. Journal of Water Resources Planning and Management, 139(4), 396-406. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000287

Implementation: src/synhydro/methods/generation/nonparametric/kirsch.py Tests: tests/test_kirsch_generator.py