Streamflow data from the “Trancura River before the Llafenco River” gauging station was selected (ID: 09414001) due to the near-natural hydrologic regime and data monitoring frequency. Its catchment is located in Southern Chile (lat: 39.33° S, lon: 71.77° W), and considers only near-natural flow conditions, neither altered through hydropower plants nor lake regulation or water transfer. The catchment area is 1380 km², the mean annual precipitation is 2853 mm, and the aridity index is 0.3. The maximum, mean, and outlet catchment elevations are 3746, 1147, and 363 m a.s.l., respectively. The gauging station covers several decades of monitoring, but the period from 23 November 2001 to 31 January 2024 was selected for analysis due to its hourly measurement frequency (resulting in a total of 181 038 non-empty values). This temporal scale is crucial to characterise hydrological dynamics in terms of magnitude, frequency, duration, timing, and rate of change. Additionally, the computing of the effective flow work parameter (W*, see Sect. 2.3.1) is very sensitive to this time discretisation, in which a more accurate hydrograph representation leads to a more accurate calculation. Figure 1 shows the analysed catchment and two hydrological years of stream flows, whereas Table 1 shows 12 hydrological signatures that characterise hydrological dynamics.

The case study consists of a two-span bridge (60 m long) with one submerged pier (the bridge span is 30 m). The bridge pier is cylindrical with a diameter of 2.5 m. The river cross-section has an average width of 50 m. The riverbed was composed of non-uniform gravels, with a d50 of 41.6 mm and a d90 of 100.2 mm. Table 2 shows the grain size distribution of the riverbed sediments.

The dependence structure of the river streamflow process was modelled in terms of the climacogram stochastic tool (see Dimitriadis et al., 2021; Dimitriadis and Koutsoyiannis, 2015; Koutsoyiannis, 2010), defined as the variance γk of the averaged process z¯t depending on the averaging time scale k. The climacogram is useful for detecting and evaluating LTP dynamics. For STP and ITP, the climacospectrum stochastic tool is used. Such a stochastic tool was recently introduced by Koutsoyiannis (2016, 2025), having, in most cases, the same asymptotic behaviour and properties as the power spectrum, but is computed only in terms of variances. Consequently, it presents less biased results, making it more convenient for model fitting and identification (Koutsoyiannis, 2016). Table 3 presents the mathematical expressions for the climacogram and climacospectrum stochastic tools

A parsimonious stochastic model was used to fit the dependence structure of the river streamflow process in terms of the climacogram (Filtered Hurst-Kolmogorov model with a generalised Cauchy-Dagum-type climacogram, FHK-CD). The FHK-CD model is general and flexible for any process, regardless of whether the process is rough or smooth, persistent or antipersistent. It has six fitting parameters and can model the dependence structure from small to large temporal scales:

4γk=λ121+k/α12H-1+λ221-1+k/α2-2-M where λ1 and λ2 are scale parameters with dimensions of [x2], α1 and α2 are scale parameters with dimensions of [t], and M and H are the fractal and Hurst parameters, respectively. It is worth mentioning that M and H are dimensionless, ranging between 0 and 1, and determine the local and global dependence structure. Eq. (4) can be simplified, when parsimony is sought, by setting λ1=λ2=λ and α1=α2=α. Therefore, the FHK-CD model with four parameters is presented in Eq. (5): 5γk=λ21+k/α2H-1+λ21-1+k/α-2-M The variance estimator for the climacogram and climacospectrum can be computed as: 6γ^¯k=1n/κ∑i=1n/κz¯iκ-μ^¯2 where κ=k/Δ is the dimensionless time scale, Δ is the time resolution at each sample time series, n/κ is the integer part of n/κ, and z¯iκ is the ith element of the averaged sample of the process at scale κ, i.e.: 7z¯iκ=1κ∑j=i-1κ+1iκzj It is worth mentioning that the bias of variance estimator presented in Eq. (6) can be easily computed by Eq. (8). Additionally, and for fitting purposes, the standardised climacogram (γ^¯k/γ^¯1) and climacospectrum (ψ^¯k/ψ^¯1) are used. The effect of the annual cycle was removed following Koutsoyiannis (2019, 2020), i.e. multiplying 12 coefficients (one per month and summing 1 in total) by the streamflow values. The value of the coefficients was found by minimising the variance of the transformed time series. 8Eγ^¯k=γk-γn/kk For synthetic generation purposes, the asymmetric-moving-average (AMA) scheme is adopted (see Koutsoyiannis, 2019, 2020). The AMA scheme can explicitly preserve any arbitrary second-order dependence structure, as long as a mathematical expression for it exists. The marginal distribution function of the generated data approximates the original one through the statistical moments of the process. In this study, the first four moments are preserved, enabling a good approximation of the original marginal distribution function as was shown in Dimitriadis and Koutsoyiannis (2018). Therefore, the approximation can be carried out with controlled accuracy. The AMA scheme generates z¯i by 9z¯i=∑j=-JJajv¯i-j where aj are coefficients (weights) computed from the autocovariance function and v¯i is white noise. Notice that J is theoretically infinite, although for simulation purposes J is taken as a large integer providing negligible truncation errors (Koutsoyiannis, 2000). A generic solution for aj was proposed by Koutsoyiannis (2020): 10aj=∫-1/21/2e2πθω-jωARωdω where θω is any real function having the following property θ-ω=-θω and ARω=2sdω. Time asymmetry is handled through the preservation of the skewness ratio between the differenced process z̃¯τ=z¯τ-zτ-1 and the original one z¯τ, 11C̃sCs=∑j=-JJaj-aj-13∑j=-JJaj-aj-123/2∑j=-JJaj23/2∑j=-JJaj3 where C̃s is the skewness of the differenced process z̃¯ and Cs is the skewness of the original process z¯τ. Note that the skewness ratio depends only on the aη sequence, and as a consequence, it depends on the (a)symmetric parameter θω. Interesting the fact that θω=0 implies time symmetry and AMA converges to the symmetric moving average (SMA) scheme. However, 0<θω≤1/4 preserves time asymmetry. As a consequence, θω must be chosen to preserve the skewness ratio. It is worth mentioning that time asymmetry is an important feature of streamflow at time scales up to four days (see Koutsoyiannis, 2019; Vavoulogiannis et al., 2021) and consequently, it should be preserved when working at the hourly scale. Figure 2 shows an example of synthetically generated streamflow using AMA with two different θ values. Notice that θ=0.00 generates symmetric-to-the-peak time series, whereas θ=0.10 generates positively asymmetric time series.

The bridge-pier scour entropic (BRISENT) model is considered for time-dependent scour analysis because of its flexibility and parsimonious modelling within time-dependent hydraulic conditions (Pizarro et al., 2017a). BRISENT relies on the principle of maximum entropy and on the effective flow work parameter, W* (Pizarro et al., 2017b), which accounts for the influence of flood waves on the local scour process. W* is mathematically defined as follows: 12W*tcon=∫0tcon1tRut-0.5ucuR4δdt where tcon is the considered time, tR is a reference time defined as zR/uR, zR=D2/2d50 is a reference length (D is the pier-diameter and d50 the sediment-size), ut is the time-dependent section averaged flow velocity, uR is a reference velocity (=ρ′gd50, where ρ′=ρS-ρW/ρW is the relative density between sediment “s” and water “w”, and g is the gravitational acceleration). uc is the critical flow velocity for incipient motion of sediment particles and δ is an on-off step function defined as: 13δ=0,ut/uc<0.51,ut/uc≥0.5 The BRISENT model has three model parameters (λ,Wmax*, and S) and their empirical expressions rely on a large laboratory dataset with a wide range of key dimensionless parameters which control the scour process. BRISENT is presented in Eq. (14), while from Eq. (15) to (17) are the empirical expressions for the fitting parameters. 14Z*=1λln1+W*Wmax*expS-115λ=4.237Dd500.95716Wmax*=exp21.84lnDd50-0.29517S=21.185D/d500.9570.4D/d501.2+10.6D/d50-0.13. The three parameters defining the BRISENT response have distinct physical and statistical roles. The dimensionless effective flow work, W*, represents the cumulative hydraulic forcing the flow applies on the sediment bed around the pier. It incorporates both the magnitude and duration of the flow. Its limiting value, Wmax*, represents the cumulative effective flow work associated with the maximum relative scour depth Zmax*, ideally corresponding to equilibrium scour. It normalises the process between its initial and final states. The dimensionless parameter λ originates as a Lagrange multiplier in the maximum-entropy derivation and controls the exponential form of the relative-scour probability distribution and, consequently, the curvature of the scour-work relationship. Finally, S is the dimensionless entropic-scour parameter linking λ to the maximum relative scour depth. It controls the degree of non-linearity in the response, with larger values indicating more rapid initial scour development followed by progressively slower development to its maximum value.

The scour hole refilling process was considered, following Link et al. (2020), i.e., sediment deposition occurring during the falling limb of the hydrograph. The time-dependent sediment deposition rate, dt, was computed as 18dt=αdwsqtgst-gs*t where, αd is considered a constant (=0.5 according to Link et al., 2020; Pizarro et al., 2022a), ws is the sediment falling velocity, qt is the specific discharge per unit width, gst is the sediment supply rate (kg m s⁻¹), and gs*t is the sediment transport capacity (kg m s⁻¹) estimated using the Meyer-Peter and Müller (1948) equation: 19gs*=8ρsρ′gds30.5CRCR′1.5Θ-Θc1.5 where CR is the total Chézy coefficient due to effective bed roughness ks (=18log12h/ks), CR′ is the Chézy coefficient due to particle roughness d90 (=18log12h/d90), and Θ and Θc are the Shields and threshold Shields parameters.

The presented deposition modelling framework considers sediment deposition if the hydrograph is in the falling limb and gst>gs*t. gst is estimated as gs*t evaluated in the previous instant, i.e. gsi=ξgs*i-1, where ξ is an exceedance sediment supply coefficient and i reports the discrete time instant. In consequence, the sediment deposition depth is calculated through a fractional transport formulation as follows: 20di=αdρs1-p∑j=1nwsjPjqiξgs*i-1,j-gs*i,jΔt where j is a counter for the considered sediment sizes in the granulometric curve, ρs is the sediment density, p is the sediment porosity, and Pj is the percentage of the soil which contains the sediment size dsj. Therefore, the scour depth considering the refilling scour process is computed as zi=zscour[i]-d[i] and Wi*=Wscour[i]*-ΔWd[i]*. The scour and deposition models were implemented in a software package, and a sensitivity analysis of their parameters was presented by Pizarro et al. (2022a). Figure 3 shows an example of scour depth time series computed with the scour and fill models. It is worth mentioning that erosional and sediment (re)deposition dynamics are captured by the simulation.

Equilibrium scour (zeq) is computed with the widely used scour equation by Sheppard et al. (2014): 21zeqa*=2.5f1f2f3for0.4≤uuc<1.022zeqa*=f12.2uuc-1u1puc-1+2.5f3u1p-uucu1puc-1for1.0<uuc<u1puc23zeqa*=2.2f1foruuc>u1puc24u1p1=5uc25u1p2=0.6gy126u1p=u1p1foru1p1≥u1p2u1p2foru1p1<u1p227f1=tanhha*0.428f2=1-1.2lnuuc229f3=a*d500.4a*d501.2+10.6a*d50-0.13

To assess the correspondence between the detection of extreme events in streamflow and scour, the critical success index (CSI, see Eq. 30) was used. The CSI computed the number of hits (H), misses (M) and false alarm (F) events. Hits are defined as matching events identified for both streamflow and scour, whereas misses are events identified only for streamflow. False alarm events are those identified only for scour. CSI ranges between 0 and 1, with the latter the optimum value. We defined hits as the scour events that overlap with the streamflow events by at least 70 %. A tolerance of 2 h before and 2 h after the original event identification was considered, accounting for possible uncertainty in the event identification procedure. The threshold for streamflow and scour events identification was the 40th percentile of the annual maxima series. 30CSI=HH+M+F Return periods of scour depths were computed directly from the scour time series using annual maxima series and fitting the Generalised Extreme Value (GEV) probability distribution function. A low CSI indicates poor temporal correspondence between detected streamflow and scour events, but it could not by itself demonstrate that conventional streamflow-based design produces biased scour return levels.

Monte Carlo analysis was performed with N=12000 simulations, each of them with 60 years at an hourly temporal scale of the synthetic generated streamflow time series. These simulations were used as input for the time-dependent scour and fill model, generating the same number of scour time series. Additionally, and with the intention to assess the effect of upstream sediment supply on scour (and, therefore, the sediment deposition effect), three different cases were investigated: (i) ξ=1.0; (ii) ξ=1.5; and, (iii) ξ=2.0. The latter cases range from sediment rates equal to the transport capacity (i.e., ξ=1.0) to sediment supply exceeding the transport capacity (i.e., ξ=1.5 and ξ=2.0). In total, 36 000 simulations (12 000 for each ξ case) were performed.

The methodology employed in this work encompasses the stochastic analysis of the Trancura River gauging station and the synthetic generation of river streamflow time series, which preserves time asymmetry, the second-order dependence structure, and the probability density function (through the first four statistical moments). Synthetic streamflow time series were replicates of the historical record and the inputs for the time-dependent scour and fill model. Monte Carlo analysis was performed with N=12000 simulations, each of them with 60 years at an hourly temporal scale of the synthetic generated streamflow time series and scour time series. Three different cases of upstream sediment supply having a sediment rate equal to the transport capacity (i.e., ξ=1.0) and sediment supply 50 % and 100 % higher than the transport capacity (i.e., ξ=1.5 and ξ=2.0, respectively). The stochastic properties of the scouring process were quantified in terms of the second-order dependence structure, the Hurst and Mandelbrot parameters, and time asymmetry through the skewness of the differenced process. The non-stochastic properties of the scouring process were quantified in terms of the CSI index and scour return periods following Rifo et al. (2022).

Codes for streamflow generation and scour computation were written in MATLAB and are available at Pizarro et al. (2025). In particular, the scripts consider the analysis and modelling of the river streamflow process, its synthetic generation (using the AMA scheme, which preserves time asymmetry, the second-order stochastic properties, and the first four statistical moments), and the scour assessment following the methodology described above.

Figure 4 shows a representative example of the synthetically generated hourly streamflow series (orange) and the corresponding scour-depth response (blue) over a 60-year simulation period (one of the 36 000 simulations performed). As shown in Fig. 4, the simulated streamflow time series exhibits pronounced variability, with recurrent peaks associated with high-flow events and recessions during lower-flow periods. These dynamics are consistent with the hydrological behaviour of the catchment. In this specific realisation, the maximum and median streamflow values reached 2299 and 94 m³ s⁻¹, respectively, reflecting both the extreme events and the central tendency captured by the stochastic generator.

The blue curve shows the evolution of scour depth, computed by coupling the AMA-generated streamflow with the erosional and fill model under the assumption of ξ=1.0 (i.e., sediment supply equal to the transport capacity). The resulting scour-depth fluctuations showed deeper scour during high-magnitude streamflow peaks. Maximum and median scour depths for this simulation were 5.63 and 3.29 m, demonstrating the model's capacity to translate hydrological variability into adjustments in sediment dynamics. Overall, Fig. 4 highlights the value of integrating stochastic flow generation with process-based scour modelling, enabling the exploration of long-term scour response under a wide ensemble of hydrological conditions.

Deterministic properties of the scour process were assessed through the Critical Success Index (CSI) – between river streamflow and bridge scour – and the ratio between the magnitude of scour with a particular return period (computed from the scour simulations) and the equilibrium scour depth computed with the scour formula by Sheppard et al. (2014). Figure 5 shows that the three analysed cases (colours blue, red, and yellow are related to ξ values of 1.0, 1.5, and 2.0, respectively) yielded similar CSI values, ranging from approximately 0.15 to 0.60. A value of 0.15 means that only 15 % of all identified events occurred simultaneously for both river streamflow and scour. A trend from symmetric to positively asymmetric distribution of CSI's values for ξ equal 1.0 to 2.0 (CSI median values are 0.33, 0.32, and 0.30 for ξ equal 1.0, 1.5, and 2.0, respectively) is observed. The resulting low CSI values indicate poor temporal correspondence between detected streamflow and scour events. However, this does not necessarily imply that streamflow-based design produces biased scour return levels.

Figure 6 shows the frequency distribution of the computed scour depth with T=100 (blue), 150 (orange), 200 (yellow), and 300 (cyan) years for each scenario of upstream sediment supply with ξ=1.0, 1.5, and 2.0. The above-mentioned return periods were selected due to their direct relationship with risk levels and design methods. Notice that zeq was computed with the equilibrium scour equation by Sheppard et al. (2014) (i.e., from Eq. 21 to 29) and assuming hydraulic variables associated with a 100-year flood event. Therefore, if zT=100=zeq(i.e., the median of scour depths with T=100 years computed from scour time series is equal to the equilibrium scour depth), the same return period will be related to both processes. In all cases, the scour depth to equilibrium scour depth ratio (z/zeq) increased with ξ. In most analysed cases, scour depths computed with scour time series were different and higher than the equilibrium scour depth (especially for T=200 and T=300 years). As a consequence, and considering that the median for zT=100/zeq is between 0.93 and 0.95 (for all analysed ξ), a given return period of streamflow does not necessarily translate into the same return period of scour. Exceedance probabilities for zT=100/zeq were lower than 0.0076 for all analysed cases. Table 4 shows the exceedance probability for z/zeq>1.0.

The second-order dependence structure was quantified in terms of the climacogram and climacospectrum adapted for bias. All computations were standardised by dividing them by their first value. Only 90 % of all available scales were used for model fitting purposes, mainly motivated by the high calculation uncertainty in small samples (i.e., with less than 10 values. See Dimitriadis and Koutsoyiannis, 2015). Figure 7 shows the envelope standardised climacograms as a function of ξ=1.0 (red), 1.5 (blue), and 2.0 (black). It is worth noticing that the higher the ξ value, the narrower the envelopes; and, climacograms for ξ=1.5 and ξ=2.0 are similar.

Figure 8 shows the mean standardised climacograms and climacospectrums for ξ equals to 1.0 (red), 1.5 (blue), and 2.0 (black). Mean standardised stochastic tools were considered as representative of the 12 000 simulations. The continuous green line corresponds to the FHK-CD model. Interestingly, asymptotic dynamics at both small (assessed through the Mandelbrot parameter, M) and large (assessed through the Hurst parameter, H) temporal scales yielded similar values (M≈0.40 and H≈0.60). The latter is indicative of rough and (weak) persistent dynamics. Table 5 summarises these asymptotic behaviours for the means and envelopes. Mean values are related to the mean standardised stochastic tool, while min and max correspond to the lower and upper envelopes.

Interestingly, and focusing on the envelopes, the strength of the Hurst dynamics of the scouring process depends on ξ, ranging from H≈0.00 (strong antipersistence dynamics) to H=0.86 (strong LTP dynamics). On the other hand, the Mandelbrot parameter took values from 0.10 to 0.41 (i.e., having less variability than H). It is worth mentioning that rough dynamics (i.e., M<0.5) instead of a smooth one (i.e., M>0.5) is usual in many hydrological processes at the analysed hourly scale such as the cases of near-surface temperature, relative humidity, dew point, sea level pressure, wind speed, and precipitation (see, e.g., Dimitriadis and Koutsoyiannis, 2018; Pizarro et al., 2022b).

In terms of time asymmetry, the scour process showed negative skewness coefficients at short time scales, followed by a transition to positive values and convergence towards zero at longer time scales (see Fig. 9). This behaviour indicates that scour increments are not time-symmetric: large negative fluctuations dominate at fine resolutions, while positive dynamics emerge as aggregation increases. A systematic dependence of the scour depth on ξ, i.e. on the upstream sediment supply, was observed. For ξ=1.0, 1.5, and 2.0, the transition from negative to positive skewness occurred at approximately 195 (h), 80 (h), and 54 (h), respectively. The subsequent return from positive to near-zero values occurred at around 20 000 (h), 6500 (h), and 5000 (h) for the same ξ values. These thresholds reveal that higher sediment availability reduces the persistence of asymmetry, leading to faster stabilisation of scour fluctuations.

The observed scale-dependent dynamics confirm that bridge scour cannot be described solely by equilibrium formulations, which ignore time asymmetry and stochastic persistence. Instead, our findings highlighted that skewness provides an additional diagnostic tool to assess the temporal structure of scour variability, especially in the presence of fill in the scour hole due to upstream sediment supply. Incorporating time asymmetry into stochastic frameworks, therefore, enhances the reliability of probabilistic scour hazard assessments.