Fig. 1
Illustration of the double penalty effect: a simulated Gaussian plume (left panel) where the source has been shifted to the left compared to the true Gaussian plume (middle panel). The squared difference is plotted on the right panel and it is characterised by two maxima: on the right, the simulation is penalised for not predicting the plume where it should be and on the left the simulation is penalised for predicting a plume where there is none.
Fig. 2
Illustration of the traditional linear interpolation (left) and of the optimal transport interpolation (right) between two normalised densities defined in the domain [0,1]. Interpreting the densities as mass distributions, the traditional linear interpolation appears as a non-physical displacement.
Fig. 3
Gaussian plume (left) and the pdf derived from the plume colour palette (right).
Fig. 4
A typical one-dimensional setup (d=1), where . The centred grid where is defined and the staggered grid where is defined are both represented (bullets and squares respectively). The reservoir variables extends the domain on the left and on the right (darker squares on the lateral boundaries of the staggered grid).
Table 1. Wall-clock time spent in seconds on 103 iterations of several optimal transport problems in d=1 and d=2 for the DR and PD solvers and for several resolutions
ResolutionDRPDResolutionDRPD
3221.41.132331.223.86423.02.6643295.5258.0128213.711.112832970.52420.6
Fig. 5
Cesium-137 activity concentration at ground level at 7:00 UTC on 25 March 2011. The concentrations values are displayed in logarithmic scale using max(log10(c/c thr),0), where c is the activity concentration and c thr=10−3 Bq.m−3 is the threshold. The plot of the simulation indexed by 3×i+j is shown in row i and in column j. Hence, simulations in the same row have the same source location and simulations in the same column have the same time shift.
Fig. 6
Tables of indicators between two simulations. The mean square error (MSE) and the Pearson correlation coefficient (PCC) are displayed in the first column. The Wasserstein distance using the normalisation method Wn , using high resolution (hr) or coarse resolution (cr) is displayed in the second column. The Wasserstein distance using the reservoir method Wr , using high resolution (hr) or coarse resolution (cr) is displayed in the third column. The darker shades correspond to the highest matches (smallest values for the mean square error and the Wasserstein distance, and highest values for the correlation) and the dimmer shades correspond to the lowest matches (highest values for the mean square error and the Wasserstein distance, and smallest values for the correlation). The cell in the k-th row and l-th column compares simulation k to simulation l following to the simulation numbering described in Fig. 5.
Fig. 7
Illustration of the deformation induced by optimal transport of cesium-137 plumes at ground level at 7:00 UTC on 25 March 2011, between simulations 1 and 3 using normalisation (top panels), between simulations 1 and 6 using normalisation (middle panels), and simulations 1 and 6 using the reservoir technique (bottom panels). The deformation of the plumes are shown at five time steps, the first one being the concentration field of the first simulation, and the last one being the concentration field of the second simulation. The concentrations values are displayed in logarithmic scale using max(log10(c/c thr),0), where c is the activity concentration and c thr=10−3 Bq.m−3 is the threshold.
Fig. 8
Concentration histograms of simulation 4, 6 and 8. The concentrations values are in logarithmic scale using max(log10(c/c thr),0), where c is the activity concentration and c thr=10−3 Bq.m−3 is the threshold.
Table 2. Table of statistical indicators between simulations 4, 6 and 8: the Wasserstein distance using normalisation (Wn ), the Wasserstein distance using reservoirs (Wr ), the Wasserstein distance between the colour palettes (Wp ), the mean square error (MSE) and the Pearson correlation coefficient (PCC)
4 vs 64.473.840.03831.04−0.034 vs 82.465.900.6980.69−0.026 vs 81.111.400.5440.560.32
Fig. 9
Observations of the cesium-137 activity deposited on the ground in logarithmic scale max(log10(c/c thr),0), where c is the activity concentration and c thr=104 Bq.m−2 is the threshold. The dark grey pixels correspond to grid cells where the deposition value is known but lower than c thr. The light grey pixels correspond to grid cells where the deposition value is below the limit of detection.
Fig. 10
Masked deposition maps of the cesium-137 activity in logarithmic scale max(log10(c/c thr),0), where c is the activity concentration and c thr=104 Bq.m−2 is the threshold for the observations, and five model simulations.
Table 3. Configuration setups for the four simulations from Quérel et al. (2016)
IRSN 1WRF (Winiarek et al., 2014)Saunier et al., 20131982; maritimeIRSN 2radar (Saito et al., 2015a)Saunier et al., 2013Jaenicke, 1982; urbanIRSN 3WRF (Winiarek et al., 2014)Saunier et al., 2013Jaenicke, 1982; continentalIRSN 4radar (Saito et al., 2015a)Saunier et al., 2013Jaenicke, 1982; urbanSimulationWet deposition velocityIn-cloud scavengingBelow-cloud scavengingIRSN 1(Zhang et al., 2001)1991NeglectedIRSN 2(Zhang et al., 2001)1999Quérel et al., 2014, and Blanchard, 1953IRSN 30.2 cm·s−1Roselle and Binkowski, 1999IRSN modelIRSN 40.2 cm·s−1NeglectedLaakso et al., 2003
Fig. 11
Zoomed masked deposition maps of the cesium-137 activity in logarithmic scale max(log10(c/c thr),0), where c is the activity concentration and c thr=104 Bq.m−2 is the threshold for the observations, and five model simulations.
Table 4. Mean squared error (MSE), Pearson correlation coefficient (PCC) and Wasserstein distance (W) of the comparison between the deposition values of one of the five simulations and the observation values, for the whole map of the zoomed region where most of the deposited activity lies
SimulationMSEPCCW [km]MSE [×106 Bq.m−2]PCCW [km]
Winiarek et al.0.0170.7369.00.5220.7620.1IRSN 10.0200.7724.17.720.4039.1IRSN 20.0230.7828.818.00.3938.3IRSN 30.0260.7930.03.790.4143.2IRSN 40.0240.6936.020.80.3741.3