Hydrologic theory and practice has rapidly moved away from a deterministic view of the design storm and toward an appreciation of joint probability and the potential for variance of almost all hydrologic inputs to design hydrograph estimation (Babister et al. 2016, Rahman et al. 2002, Nathan et al. 2016, Sharafati et al. 2020). The joint probability approach mimics mother nature in that the influence of all probability distributed inputs are explicitly considered, thereby providing a more realistic representation of the flood generation processes (Nathan and Weinmann 2013). The joint probability approach has been heavily promoted in recent years due to its holistic ability to consider the probabilistic nature of all flood-producing input variables (Charalambous et al. 2013, Hill and Mein 1996, Kuczera et al. 2006). Climate change is also a contributor to more variance and uncertainty of temporal and spatial patterns of rainfall (Mamo 2015, Wasko and Sharma 2015, Fadhel et al. 2018, Hettiarachchi et al., 2018, Nguyen et al. 2010).

In the original study by Ronalds and Zhang (2019) a monte carlo simulation was utilised with random temporal patterns, generated using the multiplicative cascade technique. The uncertainty of temporal patterns is one of the highest attention elements in current hydrologic practices. As described by Bhuiyan et al. (2010), hydrographs developed by using the same rainfall depth but with different temporal patterns can result in design flood variances of as much as 250%. Temporal patterns are also one of the variables shown to exhibit a wide variability in their observed values and the use of mean or median values of these variables has been shown to be insufficient to represent real-world hydrograph predictions in modelling (Caballero and Rahman 2014a, 2014b).

The spatial variance of rainfall is another significant uncertainty that was not factored into the previous study, despite it being known to have significant impacts on hydrologic model results (Mejía and Moglen 2010, Cho et al. 2018, Lin et al. 2022). In response to this, a new way of accounting for the spatial variance of rainfall between the upper and lower portions of a regional catchment in a hydrologic model was developed to assess the performance of the new equations, using the framework in Fig. 6.

In this model framework, a single rainfall depth is disaggregated over time using the multiplicative cascade method with a random variable (x) to create an artificial temporal pattern. The temporal pattern is then applied unevenly to the sub catchments using a second random spatial pattern, also generated using a multiplicative cascade.

The example regional catchment used to test this model is located in Nambour, Queensland (Bureau of Meteorology Catchment ID 141003). As per the methodology of Ronalds and Zhang (2019), the monte carlo simulation is repeated 1,000 times for each hypothetical development site location along the regional catchment length (*R**L*) and for scenarios with and without detention.

The results of the additional element of random probability modelling have showed that the original equations not only remain reliable, but that the margin of error is reduced between the mean of the monte carlo results and the predications made by the equations. Figure 7 shows the model results in comparison to the predictions of the equations.