Development of stochastic models for streamflow synthesis based on colored-noise–dominated hydrologic systems: textural pattern modelling for black- and pink-noise systems and Brownian Motion modelling for brown-noise systems

Abstract

Civil engineering structures are often designed based on flood design criteria; however, there is no guarantee that these design floods represent worst-case scenarios. It is because streamflow has been recorded for only a few hundred years, while rivers have been running for over the past thousands of years. Consequently, the available records may not include the most extreme flood events. This limitation creates a need for synthesizing additional possible streamflow scenarios. Over the decades, hydrologists have developed various methods to synthesize streamflow, ranging from early stochastic models in the 1960s to AI-based approaches of today. The evolution of streamflow synthesis methods relies on two equally important considerations: a deeper understanding of streamflow characteristics and advances in computational techniques. In some cases, however, the first consideration is overlooked under the assumption that more complex models can better synthesize possible scenarios. Therefore, this research focuses equally on these two crucial considerations: (1) improving the understanding of monthly streamflow characteristics, and (2) developing models according to the identified characteristics. To improve the understanding of streamflow characteristics, this research quantifies scaling behaviour as a representation of the memory of hydrological systems using an approach based on identifying jumps in the power spectral density of the system being analyzed. Using this framework, 143 hydrometric stations in Ontario were analyzed to identify pink-, brown-, and black-noise–dominated monthly streamflow behaviour. Of these, two stations were characterized by the dominance of brown noise, 39 by pink noise, and the remaining 102 by black noise. Based on the identified dominant noise, the development of modelling approaches is then initiated in this research. For hydrometric stations dominated by the brown noise, an Extended Geometric Brownian Motion (EGBM) model has been developed. Compared with a multiplicative ARIMA model, the EGBM better reproduces the statistical properties of historical streamflow and yields residuals with statistically insignificant lag-1 autocorrelation. For stations exhibiting pink- or black-noise dominance, a textural pattern recognition system (TPRS) has been developed. The comparison of TPRS and ARIMA/SARIMA models shows that for pink-noise–dominated watersheds, although TPRS performs slightly better, the improvement is not sufficient to justify the added model complexity in cases where high accuracy gains are not required. In contrast, black-noise–dominated watersheds exhibit stronger temporal dependence, which is more appropriately captured using the proposed TPRS framework. Further comparison of the proposed TPRS with traditional pattern recognition methods demonstrates its effectiveness in reproducing autocorrelation structures and higher-order statistical features. The results show that the TPRS improves the representation of the autocorrelation function (ACF) up to 100 lags, compared with 24 lags for traditional pattern recognition methods. Moreover, the Hurst coefficient analysis confirms that the TPRS model provides a slightly enhanced representation of statistical characteristics exhibited by the historical time series. These findings support more informed model selection by reducing the mismatch between model structure and underlying hydrological behaviour, ultimately improving the reliability of synthesized streamflow scenarios for flood frequency analysis and flood risk assessment.