Assessing the Internal Variability of Large-Eddy Simulations for Microscale Pollutant Dispersion Prediction in an Idealized Urban Environment

Abstract

This study aims at estimating the inherent variability of microscale boundary-layer flows and its impact on air pollutant dispersion in urban environments. For this purpose, we present a methodology combining high-fidelity large-eddy simulation (LES) and a stationary bootstrap algorithm, to estimate the internal variability of time-averaged quantities over a given analysis period thanks to sub-average samples. A detailed validation of an LES microscale air pollutant dispersion model in the framework of the Mock Urban Setting Test (MUST) field-scale experiment is performed. We show that the LES results are in overall good agreement with the experimental measurements of wind velocity and tracer concentration, especially in terms of fluctuations and peaks of concentrations. We also show that both LES estimates and the MUST experimental measurements are subject to significant internal variability, which is therefore essential to take into account in the model validation. Moreover, we demonstrate that the LES model can accurately reproduce the observed internal variability.

Appendix I: Validation of the Stationary Bootstrap Approach

In this appendix, several convergence tests of the stationary bootstrap approach applied to the MUST trial are shown to verify the robustness and plausibility of the confidence intervals predicted by bootstrap as a complement to Sect. 4.

1.1 Convergence with the Number of Bootstrap Replicates

With bootstrap methods, such as the stationary bootstrap, bootstrap replicates of one original sample are used to compute Monte Carlo estimates of several statistics of the physical quantities of interest (time-averages or fluctuations), such as their variance (Eq. 12) or confidence intervals. The convergence of the estimator error is therefore in $\mathcal {O}(1/\sqrt{B})$ with B the number of bootstrap replicates.

Table 5 Values of 2.5th percentiles evaluated with stationary bootstrap for different numbers of replicates B. Estimations are given for one example of simulated and observed mean concentration (at tower B at $z=2\,$m), as well as for the air quality metrics (Sect. 3.6.2) and flow validation metrics (Sect. 3.6.1)

We assess the convergence for the 2.5th and 97.5th percentiles as it requires more bootstrap replicates than for bias or variance estimation (Davison and Hinkley 1997). Table 5 shows the evolution of the 2.5th percentile of the mean concentration at tower B at $z=2\,$m and of the model validation metrics evaluated according to the bootstrap procedure (Eq. 14) for different values of B. The bootstrap estimations of the 2.5th percentiles show some variability for very low numbers of bootstrap replicates (between 100 and 500), but then quickly converge for all the considered quantities. The same analysis was carried out for the 97.5th percentile and gave similar results. We conclude that B = 5000 bootstrap samples are more than sufficient to achieve convergence. This result is in line with the literature, which recommends between 1000 and 10,000 replicates (Davison and Hinkley 1997; Chang and Hanna 2005).

1.2 Effect of the Sub-Averaging Period

Since both simulated and measured time series are well sampled in time, it is possible to change the sub-averaging period $\delta _t$ to adjust the number of sub-average samples $N_t$ (Eq. 10). To assess the effect of the sub-averaging period on the internal variability estimation, the estimated percentiles obtained with a stationary bootstrap of sub-averages over $\delta _t =10\,\textrm{s}$ and $\delta _t =5\,\textrm{s}$ are compared. By reducing the sub-averaging period, the samples are getting more dependent. It is therefore mandatory to adapt the mean block length parameter $\ell$ of the stationary bootstrap method. For $\delta _t =5\,\textrm{s}$, it results to new values of $\ell _{sim}=1.38$ and $\ell _{obs}=2.62$ for the time-averaged concentrations. This is consistent since it means that, for more dependent data, the blocks should be larger than the ones used for $\delta _t =10\,\textrm{s}$ (Table 2, page 21). Table 6 shows the 2.5th percentile estimates for the main quantities of interest for the two different values of $\delta _t$. Results indicate that changing the sub-averaging period has a very limited impact on the stationary bootstrap estimations. This is because changing the sub-averaging period only changes the division of the original sample (Eq. 10) and so does not provide any additional information on the underlying distribution of the time-averaged quantities.

Table 6 Values of 2.5th percentiles evaluated with stationary bootstrap for different sub-averaging period $\delta _t$. Estimations are given for one example of simulated and observed mean concentration (at tower B at $z=2\,$m), as well as for the air quality metrics (Sect. 3.6.2)

1.3 Convergence with the Number of Sub-average Samples

As mentioned in Sect. 4.4.3, it is essential to have a sufficient number of sub-average samples $N_t$ in the original sample. In particular, too few samples may result in internal variability underestimation. To increase $N_t$, the LES simulation acquisition time is increased from 200 s to 400 s, and then 600 s (Fig. 4). With $\delta _t = 10$ s the resulting number of sub-averages is 40, and 60 respectively, against 20 for the reference sample. In any case, the bootstrap replicates are obtained by resampling only 20 sub-averages over the $N_t$ available, even if $N_t = 60$. Indeed, the objective is still to quantify the variability over the 200-s analysis period and not over 600 s. Two additional realizations of 200-s averages (in cyan and magenta in Fig. 4) are used for validation purposes.

Fig. 13

Box plots of the air quality metrics distributions obtained with stationary bootstrap with resampling of 20 sub-averages among 20, 40 and 60, in orange, green and red, respectively. Point estimations corresponding to the reference and two independent realizations of 200-s simulation are represented as orange squares, cyan circles and magenta triangles. Results are given for a FB, b NMSE, c FAC2, d MG, and e VG metrics (Sect. 3.6)

The distributions of the air quality metrics estimated by stationary bootstrap with resampling of 20 sub-averages among $Nt=20$, 40 and 60 are shown in Fig. 13. The bootstrap ensemble averages slightly change because the time-averaged quantities over 200, 400 and 600 s are different. Nevertheless, increasing the number of samples for stationary bootstrap gives similar estimations of the metrics dispersion. For FAC2, MG and VG metrics, which are nonlinear, the tails of the distributions seem more dependent on the number of samples. In all cases, the three bootstrap estimates cover the two independent realizations, once again supporting the validity of the stationary bootstrap.

As the orders of magnitude of the three estimates are overall consistent with each other, we conclude that $N_t = 20$ samples of sub-averages are sufficient for the stationary bootstrap method to converge. This implies that it is not required to run longer simulations to capture internal variability. The convergence with the number of samples is similarly verified for the envelopes of tracer concentration and wind velocity statistics presented in Sect. 4.