Observed data are not always as informative as expected and may be inconsistent with other data sources. Hydrologists typically adopt the Bayesian framework to update hydrological parameters, which provides a generalized formalism that integrates prior probability representing prior knowledge with likelihood that reflects how accurately a model can reproduce observations to form a posterior probability. Suppose we have several versions of a hydrological model, each with different sets of parameters. Then, the purpose of the Bayesian updating procedure adopted in this study is to assign a posterior probability to every hydrological parameter set as new taxi data become available.

Two components are critical for this Bayesian updating procedure: one is the prior probability and the other is the likelihood function. Regarding the prior probability, for their famous calibration model called generalized likelihood uncertainty estimation, Beven and Binley (1992) stated that all parameter combinations are considered equally probable before additional information is introduced. After the first update, the prior probability of each updating iteration can be replaced by the posterior probability of the latest updating iteration. Likelihood, which is a measurement of how well a given model conforms to the observed taxi behavior, is not as easy to compute as the prior probability because the parameter set to be estimated is hydrology related, whereas the observed evidence is taxi related. Therefore, we must determine how to construct a taxi-based proxy whose probability is equal to the associated hydrological parameter and construct a function enabling the transformation from hydrological parameters to taxi-related proxies.

The proxy selected in this study was the time series of the no-taxi-passing probability. Figure 1 presents a generalized procedure for converting a rainfall time series into a time series of no-taxi-passing probabilities for each hydrological parameter. This procedure consists of three steps. First, a hydrological model is used to convert a rainfall time series into a hydrograph. Second, a runoff-disruption function that relates runoff to the probability that a road is blocked is used to transform the hydrograph into a time series of road disruption probabilities. Third, the taxi arrival rate is combined with the time series of road disruption probabilities to derive a time series of no-taxi-passing probabilities. The hydrological model and taxi arrival rate are considered to be unique for every road and are invariable within a short period, whereas the runoff-disruption function is identical for all roads.

Integrating this three-step process with the Bayesian equation enables us to compute the posterior probability of a parameter set based on taxi data. For a specific road, suppose there are N hydrological parameter sets to be estimated. Because the runoff-disruption function and taxi arrival rate are assumed to be fixed for the road, we can construct a composite function converting the ith parameter set, which is denoted as θ(i), into a time series of no-taxi-passing probabilities, which is denoted as Ω(i). Therefore, the probability of θ(i) being optimal is equal to the probability of Ω(i) being true, which can be expressed as follows: 1P(θ(i))=P(Ω(i)), where P(θ(i)) and P(Ω(i)) are the prior probabilities of θ(i) and Ω(i), respectively. As taxi observations become available, P(θ(i)) (or P(Ω(i))) can be updated using the Bayes theorem as 2P(θ(i)|X)=P(Ω(i)|X)∝P(θ(i))L(X|θ(i)), where X is the taxi observation, and P(θ(i)|X) and P(Ω(i)|X) are the posterior probabilities of θ(i) and Ω(i), respectively, conditional on the taxi observation. L(X|θ(i)) is the likelihood of X given θ(i). The optimal parameter set is that which yields the Ω(i) that most closely fits the observed taxi data.

Solving Eq. (2) involves the calculation of P(θ(i)) and L(X|θ(i)). The derivation of P(θ(i)) depends on prior knowledge regarding the parameter distribution, which is typically obtained using traditional hydrological methods. However, this prerequisite knowledge may not always be readily accessible based on limited data availability. In such cases, Beven and Binley (1992) suggested that any parameter set combination could be considered to be equally likely. This implies that the parameter set is drawn from a uniform distribution as follows: 3P(θ(i))=1/N. In this study, we compared the effects of two types of prior parameter distributions, namely a uniform distribution and a distribution derived from digital elevation model (DEM) data, on the resulting posterior distributions.

L(X|θ(i)), which is a likelihood function, describes the joint probability of the observed taxi data X as a function of the chosen θ(i). Consider a rainfall event that is divided into T 5 min intervals. From the taxi data, we can obtain a sequence of taxi-related observations, which are denoted as X={x1,x2,…,xT}, where xt=1 if the observed road has at least one taxi pass during the tth interval, and xt=0 otherwise. Ω(i)={ω1(i),ω2(i),…,ωT(i)} is also a T-dimensional vector, where ωt(i) is the no-taxi-passing probability in the tth interval with θ(i) as the parameter set. Note that Ω(i) is only determined by the chosen hydrological parameter and rainfall time series, and is not measured from observed data. Considering that the arrival of taxis is independent of time, L(X|θ(i)) can be formulated as 4L(X|θ(i))=L(X|Ω(i))=∏t=1T(1-ωt(i))xt(ωt(i))1-xt. By substituting Eq. (4) into Eq. (2), the following equation can be obtained: 5P(θ(i)|X)∝P(θ(i))∏t=1T(1-ωt(i))xt(ωt(i))1-xt. Equation (5) is the proposed Bayesian updating model for calibrating hydrological parameters based on taxi data, where X can be directly measured and ωt(i) is calculated through the three-step process illustrated in Fig. 1, which will be discussed in detail in the following section. Having selected an updating model, the optimal parameter for one period of observations may not be optimal for another period. Because the model may have continuing inputs of new taxi observations, the posterior probability for θ(i) should be updated as new evidence becomes available. For the second update, the posterior probability from the first observation becomes the prior probability for the second observation and the posterior probability for θ(i) is recursively updated as 6P(θ(i)|X2)∝L(X2|θ(i))P(θ(i)|X1), where X1 and X2 are the first and the second taxi observations.

Section 2.1 presented a generalized three-step procedure for converting a rainfall time series into a time series of no-taxi-passing probabilities. In this section, we specialize this process by integrating existing theories with our model. The three conceptualized steps illustrated in Fig. 1 were replaced with three concrete submodels. First, a Soil Conservation Service (SCS) unit hydrograph was used to convert rainfall excess into a hydrograph of the target road. Second, an empirical runoff-disruption function based on data extracted from various experimental, observational, and modeling studies was applied to convert the hydrograph into a time series of road disruption probabilities. Third, a Poisson distribution representing the distribution of taxi arrival rate was combined with the road disruption probability time series to derive a no-taxi-passing probability time series.

The proposed method was validated on flood-prone roads located in Shenzhen, China, which is a coastal city frequently hit by extreme storms during summer. To the best of our knowledge, Shenzhen is the only city that has shared runoff-related data with the public in China. Three data sources, namely taxi GPS data, rainfall data, and authoritative water level data, were used to validate our parameter calibration method. Hydrological parameters were calibrated using the first two data sources and the water level data acted as the ground truth to validate the proposed method. Taxi GPS data were anonymized and aggregated in 5 min intervals. Rainfall data, which were also collected in 5 min intervals, were measured at 115 gauging stations citywide and mapped to the road network throughout Shenzhen using the ordinary Kriging spatial interpolation algorithm. The water level data were only measured at certain flood-prone points with a dynamic sampling interval ranging from 5 min when the weather was rainy to 1 h when the weather was clear. The proposed calibration method was validated by analyzing the hydrographs derived from the calibrated hydrological models against the authoritative water levels for 10 selected roads. Detailed information on the three data sources is provided in Table 4.

The two storm events on 9 and 23 May 2015 were treated as calibration events, and a storm on 11 June 2019 was retained for testing. Clearly, there is a 4 year gap between the calibration data and validation data based on data availability. The hydrological environments of flood-prone roads may have changed during these years, which could render the parameters calibrated based on data from 2015 inaccurate for analysis in 2019. To reduce the validation error caused by this time gap, the roads to be validated should have been vulnerable to flooding in both 2015 and 2019 so that the hydrological parameters of these roads would have a higher chance of remaining unchanged. Therefore, a total of 10 flood-prone roads that were labeled as such in both the List of 2015 Flood-prone Roads in Shenzhen (Water Authority of Shenzhen Municipality, 2015) and the List of 2019 Flood-prone Roads in Shenzhen (Water Authority of Shenzhen Municipality, 2019) were carefully selected (Fig. 8).

We introduced two types of prior distributions to demonstrate the effects of prior distributions on calibrated parameters. The first prior distribution was determined based on prior knowledge and DEMs from Shenzhen, which were obtained from ASTER GDEM V3, which is a product of NASA and Japan's Ministry of Economy, Trade, and Industry (METI) (Ministry of Economy, Trade, and Industry (METI) of Japan and the United States National Aeronautics and Space Administration (NASA), 2023). This global DEM covers the entire land surface of the earth with a 30 m resolution, exhibiting notable improvements in horizontal and vertical accuracy while reducing anomalies compared with previous versions. We inputted the DEMs from Shenzhen into the hydrological software PCSWMM to delineate catchments and calculate the catchment area. Subsequently, we computed the time of concentration using the watershed lag method (Natural Resources Conservation Service, 2010b). As suggested by Zhang and Huang (2018), we used the average curve number for Shenzhen in 2015, which was assessed to be 60, as the estimated curve number for each road under validation.

We then constructed a discretized parameter space for the three parameters for each road as follows: for the curve number, we examined eight possible values centered on 60 with steps of five. For the catchment area, we considered 20 possible values centered on the estimated value with steps of 0.01 km2. For the time of concentration, we considered 30 possible values centered on the estimated value with steps of 5 min. After constructing the parameter space for the parameters, we assigned a triangular prior distribution to each, which assumed the maximum probability at the estimated value and linearly decreased to zero at the parameter space boundaries, as depicted in Fig. 9.

The second prior distribution assumed that the three parameters all follow uniform distributions. The parameter spaces for the second prior distribution were the same as those for the first. As a result, the joint probability of each parameter set was equal to (1/20)×(1/30)×(1/8). To facilitate comparisons, we present the detailed information on the two types of prior distributions in Table 5.

We first calibrated the parameters based on the prior distributions calculated according to the DEMs and other prior knowledge. The resulting posterior distributions are presented in Fig. 10. Each row in Fig. 10 represents a different road, and each column represents a curve number. Each subplot presents the joint probability distribution of the catchment area and time of concentration for a given curve number. The color intensity in Fig. 10 represents the magnitude of the probabilities. Following two iterations of updating, the posterior probability distributions for both the catchment area and time of concentration converge around the optimal parameter sets for most flood-prone roads. This demonstrates that incorporating taxi observations significantly reduces the uncertainty associated with catchment area and time of concentration. The probability typically achieves its maximum value when the curve number is either 55 or 60. Furthermore, each subplot contains a salient cluster with higher probability than other regions, suggesting that there may be multiple acceptable parameter sets.

Furthermore, the optimal catchment area under a given curve number decreases as the curve number increases, whereas the optimal time of concentration under a given curve number increases with the curve number. This is logical, because a higher curve number corresponds to increased rainfall excess under identical rainfall conditions, requiring a reduction in catchment area to maintain the runoff that best aligns with the taxi observations. Similarly, an increase in the time of concentration diminishes the peak runoff produced by the additional runoff generated by a higher curve number, thereby preserving the optimal runoff status.

We also present the marginal distributions of the three parameters for 10 roads before and after calibration in Fig. 11. In Fig. 11, the marginal posterior distributions of the curve number appear relatively similar to the marginal prior distributions. It seems that the proposed method employing taxi data provides limited information regarding the distribution of curve numbers compared with the catchment area and time of concentration. This outcome may be a result of the range and discretization granularity of the parameter spaces. Catchment area and time of concentration encompass 20 and 30 possible values, respectively, whereas the curve number has only 8 potential values. The smaller parameter space of the curve number reduces the search space, and its impact on the no-taxi-passing probability is comparatively lower than that of the catchment area and time of concentration.

For example, for road ID = 6, the optimal parameter set consists of a catchment area of 0.19 km2, time of concentration of 0.9 h, and curve number of 55. To investigate the effects of these parameters on the hydrograph and time series of no-taxi-passing probabilities, we held two parameters constant at their optimal values and observed the impact of changing the third parameter. Our findings are presented in Fig. 12. One can see that when the catchment area varies from 0.04 to 0.23 km2 the maximum no-taxi-passing probability increases from 20 % to 100 % and the duration for which the no-taxi-passing probability exceeds 0.5 increases from 0.0 to 1.3 h. Similarly, when the time of concentration fluctuates from 0.1 to 1.9 h, the peak time of the no-taxi-passing probability varies from 0.5 to 1.8 h. In contrast, when the curve number varies from 40 to 75, the maximum no-taxi-passing probability is fixed at 100 %, the duration for which the no-taxi-passing probability exceeds 0.5 extends from 1.1 to 1.3 h, and the peak time of the no-taxi-passing probability remains fixed at 1.1 h. These small fluctuations in the time series of no-taxi-passing probabilities are representative of why the distribution of curve numbers remains relatively stable after calibration compared with the catchment area and time of concentration.

The posterior distributions calibrated based on the uniform prior distribution are presented in Fig. 13. When comparing two posterior distributions derived from two prior distributions, it is clear that the posterior distributions of the catchment area and time of concentration are very similar, indicating that the impact of prior distributions on these parameters rapidly diminishes after taxi-related knowledge is added. As stated by Beven and Binley (1992, p. 286), “as soon as information is added in terms of comparisons between observed and predicted responses then, if this information has value, the distribution of calculated likelihood values should dominate the uniform prior distribution when uncertainty estimates are recalculated”.

After the parameter sets were calibrated, they were combined with an SCS unit hydrograph to construct an SUH, which was combined with the rainfall data from 11 June 2019 to produce the predicted hydrograph. Because the posterior probability associated with each parameter set can be regarded as a fuzzy measure reflecting the degree of belief that the parameter set is true, the weighted runoff values for each parameter set were summed to calculate the final predicted runoff: 17Q=∑i=1NP(θ(i)|X)Q(i). Here, Q is the final predicted runoff, Q(i) is the simulated runoff derived from the ith parameter set, and P(θ(i)|X) is the posterior probability of the ith parameter set, which acts as a weight.

The output of the calibrated hydrological model is runoff (with units of m3 s-1), whereas the validation data are water level data (with units of m). Because the calibration data and validation data came from multiple sources and have different units, conventional error-based statistics, such as the mean absolute error, were not suitable for this study. The discharge of a stream is rarely measured directly. Instead, streamflow is typically determined by converting measured water depth (i.e., water stage) into discharge through a rating curve, which provides a functional relationship between the water stage and discharge at a specified point (Le Coz et al., 2014). Inspired by the application of the rating curve, we validated our method by estimating the goodness of fit between the water level which was measured in the field and the corresponding runoff which was predicted based on the proposed calibration method. A higher goodness of fit indicates synchronous trends between the runoff and water level, which indirectly demonstrates the feasibility of the proposed method.

Because the posterior distributions derived from the two types of prior distributions were very similar, we only considered the posterior distribution calibrated based on prior distributions derived from DEMs and other prior knowledge for validation. Comparisons between the observed water depth and simulated runoff for 10 selected roads are presented in Fig. 14, and corresponding scatter plots are presented in Fig. 15. We use the Pearson correlation coefficient, which measures the linear correlation between two variables, as a goodness of fit indicator. One can see that 8 of 10 roads are characterized by significant positive Pearson coefficients, indicating that the runoff and water have similar and consistent variation trends.

It is noteworthy that goodness of fit simply describes the degree of correlation between the observed and simulated data, and may contain validation bias. As suggested by Legates and McCabe (1999), correlation-based statistics are insensitive to additive and proportional differences between simulations and observations. Therefore, the fitting of a rating curve is only a partial validation and the usefulness of the proposed calibration method requires further analysis.