Meta-Modeling for urban noise mapping
Antoine LESIEUR(1) , Pierre AUMOND(2) , Vivien MALLET(3) , Arnaud CAN(4)(1) INRIA, Sorbonne Université, CNRS, LJLL, France, antoine.lesieur@inria.fr(2) IFSTTAR, Cerema, UMRAE, France, pierre.aumond@ifsttar.fr(3) INRIA, Sorbonne Université, CNRS, LJLL, France, vivien.mallet@inria.fr(4) IFSTTAR, Cerema, UMRAE, France, arnaud.can@ifsttar.fr

Abstract
Urban noise maps are usually computed by noise software which couples emission and attenuation rules like
CNOSSOS. These models can require several hours to produce a map over a city for a single set of input
data. This computational cost makes the models unusable for applications like uncertainty quantification where
hundreds of simulations may be required. One solution is to replace the physical model with a meta-model
which is extremely fast and yet fairly reproduces the results of the physical model. The strategy is first to reducethe dimension of both inputs and outputs of the physical model, which leads to a reduced model. This reduced
model is then replaced by a statistical emulator. The emulator is trained with calls to the reduced model for aset of chosen inputs. The emulator relies on interpolation between the training output values. We applied thisapproach to the NoiseModelling software over the city of Lorient (France), using Kriging at the emulation step.
It required a training set of 2000 calls to the physical model, but then the meta-model is 25 000 times faster thanthe physical model, while preserving the main behavior of the physical model, with 1.58 dB of mean quadraticerror.
Keywords: Noise maps, Computing

INTRODUCTION
Noise pollution is associated with multiple psychological and physical disorders such as sleep disturbance, nervousness, cognitive impairment or hypertension, and is a major challenge to tackle for urban areas. While theWorld Health Organization (WHO) recommends, in its guideline, an outdoor night value of 40 dB(A) [4], 8
million European adults suffer sleep disturbance due to environmental noise [2].
The European Noise Directive (END) 2002/49/EC [1] defines a common approach for European cities intended
to avoid, prevent or reduce the harmful effects of noise. In particular, European cities with more than 100 000
inhabitants are required to produce noise maps at least every 5-year term. These maps should present the distribution of a given indicator over the studied area, in such a way that the exposure of dwellers to environmental
noise can be estimated.
A regulatory noise map is a 2D representation of the equivalent average noise level field over an urban area.
In software such as NoiseModelling [3], the noise propagation algorithm used for the generation of the noiselevel field is based on a path finding algorithm which computes the wave amplitude attenuation along theacoustic path between the sources and the receptors, taking into account the reflections and the diffractions.
The attenuation rules obey the European standard CNOSSOS [5] which has become the reference since theapplication of the directive 2002/49/EC in 2015. When this method is applied at a city level, the high numberof sources and receptors makes this method computationally expensive. It is not adapted to the generation ofhourly or even daily noise maps. The computation time would exceed the time step. An annually averaged
equivalent noise level field which relies on annual traffic and weather data is usually represented in standard
noise maps.
Recent works [9] have tried to enrich noise maps with data from sensor networks. Yet they have been confronted to a computation time limitation. Also, too much computation time leads to the intractability of large

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Monte Carlo simulations and prevents us to apply advanced methods for uncertainty quantification, data assimilation, inverse modeling or network design which can require tens of thousands of calls to the model.
The objective of this paper is to circumvent this limitation by building a meta-model that reproduces the mainfeatures of the noise map simulator and requires negligible computational resources. This work relies on previous studies that have been done for urban air quality models [6]. In this paper, we explain the construction ofa meta-model based on dimension reduction and interpolation methods applied to each projection on the basisof the reduced output space. The interpolation method we selected in this paper is the Kriging interpolation, it
is trained by a Latin Hypercube Sampling (LHS) in the input space and their corresponding simulations at the
sample points. Explanations about the reduction and emulation strategy are provided in section 2. In section 3,
the case study is described. The meta-model is generated for an area of the city of Lorient (France) using themean traffic data (speed and flow rate) over a year. The performance of the dimension reduction and emulation
are analyzed in dB. The paper concludes on how the meta-model opens the way to several applications.

REDUCTION AND EMULATION OF THE NOISE MAP SIMULATOR
2.1 Description of the framework
Noise mapping simulators use a large amount of input data: flow rate and mean speed in road sections, buildings
distribution, general topography, weather data, ground and building absorption and the receptors grid.
Statistical emulation is usually applied to models with a relatively small number of inputs. If we consider eachbuilding and road section separately, the Noisemodelling simulator has hundreds of inputs. A common way to
reduce the large dimension of input data into a low dimension vector p = (p1 . . . pk )T ∈ Rk of k parameters
is to apply multiplicative coefficients to given input sets. For instance, p1 can be the multiplicative coefficient
applied to the yearly averaged traffic data in all roads of the study area, so that we can change the total traffic
by changing p1 —but we cannot change the traffic in a single specific road. We can define through p all the
inputs that we want to vary, e.g., all uncertain inputs if we carry out uncertainty quantification. See section 3.3
for the list of parameters pi that we considered in our case.
Let xb ∈ Rn be the output of our model, i.e., the noise levels at the n receptors. Then, we mathematicallydescribe the simulator M as a multivariate function
M

: Rk
p

→
Rn
b
7
→
x = M (p)(1)

It takes several hours to get M (p) from p. Assume we need to compute a large amount of simulations byvarying
p in Io⊂ Rk . The meta-modeling strategy is to call the modelS=nn for a restricted amount of inputs o
(1)
(r)
(1)
(r)
b
(1)
b
(r)
= M (p ), . . . , x= M (p ) ⊂p ,..., p⊂ I which are optimally distributed in I. With S and xnc
R , we try to learn the dependency between the inputs and the outputs, which is translated in a meta-model M
which is computationally cheap and has an acceptable performance loss compared to the original simulator:c(p) = x̂b ' xb
∀p ∈ I, M(2)

2.2 Dimension Reduction
Before the emulation step, which applies to scalar functions, we need to represent the model outputs into a few
scalars to be emulated. Indeed, a direct application of the emulation to the simulator output would imply todesign an emulator for each receptor. With tens of thousands of receptors, the overall computation time would
Ψi )i∈J1,dK with d  n.
be too high. The strategy is to project xb onto a subspace spanned by a reduced basis (Ψ
1 r (i)Ψ1 , . . . , Ψ d ] ∈ Rn×d and x̄b = ∑ xb , we would expect that
If Ψ = [Ψr i=1
xb ' x̄b + Ψ ΨT (xb − x̄b )

(3)

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Since x̄b and Ψ are known, one would only have to generate d emulators for Ψ T M (p) ∈ Rd instead of nemulators for M (p) ∈ Rn .
The reduced basis is chosen so as to represent the variability of the noise level field. It is computed with the(1)(r)
training set X = [xb , . . . , xb ] obtained with the LHS strategy, and it is determined by a principal component
analysis (PCA). The components are sorted so that the i-th component accounts for the i-th largest variability
λi of the data. The total quadratic error of the approximations of the training set is
r

∑ kxb
h=1

(xb

(h)

(h)

d

− x̄b − ∑ ((xb(h)

n

Ψ j k2 =− x̄b )T Ψ j )Ψ

∑ λ`

(4)

`=d+1

j=1

− x̄b )T Ψ j is the projection of xb − x̄b in the direction Ψ j .
(1)

(r)

If we call X̄ the centered training set X̄ = [xb − x̄b , . . . , xb − x̄b ] then for any ` ∈ J1, nK, the variability ofthe `-th components satisfies X̄X̄T Ψ ` = λ` Ψ ` . As we will see in the case study, we can get a high explained
variance, like 97%, with only a few vectors in the reduced basis (about five).
2.3 Emulation
For any p ∈ I, we wish to emulate Ψ T M (p) ∈ Rd . We denote αi = Ψ Ti M (p) ∈ R, and α̂i its emulator which
α = (α̂1 , . . . , α̂d ) thenshould satisfy ∀p ∈ I, α̂i (p) ' αi (p). If we denote α̂α (p) ' Ψ T M (p)∀p ∈ I, α̂(5)2.3.1 Kriging
The Kriging emulator is a statistical interpolator which assumes that αi is a centered stationary stochastic process, follows a Gaussian distribution with a fixed variance σi2 : ∀p ∈ I, αi (p) ∼ N (0, σi ), and for any m ∈ N
and (p1 , . . . , pm ) ∈ Rk×m , (αi (p1 ), . . . , αi (pm )) follows a Gaussian distribution of dimension m.
With these assumptions, it is possible to build a unique estimator [7] with the following properties:• linear, it is a linear combination of the training data;• unbiased, the prediction matches the data at the training points;hi• optimal, in the sense that it minimizes the variance of the prediction error E (α̂i (p) − αi (p))2 .
A key offline computation of the emulator consists in estimating the covariance kernel of the Gaussian process.
We assume the second-order stationarity of the Gaussian process, which implies the covariance between two
points only depends of the distance separating them. In addition, for computing considerations, we choose aseparable kernel, i.e., a tensor product of k univariate functions, each depending on only one parameter. Finally, for regularity considerations, we choose the so-called Matern 5/2 kernel φ (r, θ ) which is a good trade-off
between the rough exponentially decreasing covariance function and the smooth Gaussian covariance function.
This choice is compatible with the assumption that the process is one time differentiable.
The covariance kernel has the following formk∀(p1 , p2 ) ∈ I 2 , Cov(αi (p1 ), αi (p2 )) = ∏ φ (|p1i − p2i |, θi )

(6)

i=1

The covariance kernel is differentiable with respect to θ , hence it is possible to infer θopt under the maximum
likelihood estimation with a gradient descent algorithm.

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2.4 Complete Meta-Model
Finally, the complete meta-model readsc(p) = x̄b + Ψ(α̂α (p) − ΨT x̄b )x̂b = M(7)

The meta-model now has the same outputs as NoiseModelling. Its performance can be compared with NoiseModelling itself or with the physical measurements after interpolation at the observed locations.
only implies
Since x̄b and Ψ Ψ T x̄b are fixed and already known, a call to Mˆ for a new set of parameters p,(j)
α (p), which consists in computing the covariance vector Cov αi (p), αi (p )the computation of α̂∈ Rrj∈J1,rKand a sequence of matrix multiplications of a rather cheap computational cost.

META-MODELING OF NOISEMODELLING APPLIED TO LORIENT (FRANCE)
3.1 NoiseModelling
NoiseModelling1 [3] is an open source software designed to produce noise maps for evaluating the noise impact
on urban mobility plans. It is implemented in a GIS software (OrbisGIS2 ) and uses traffic, topographic andmeteorological data to generate noise maps. The noise calculation method implemented within NoiseModellingis based on the standard European method CNOSSOS, as a reference method to be used under the Directive
2002/49/EC relating to the assessment and management of environmental noise [5]. In the version used for thispaper, the noise calculation method implemented is a slightly simplified version of the standard.
The noise level field is represented by receptors organized as a regular grid in our study. NoiseModelling
follows the rules defined by CNOSSOS to compute the noise map.
3.2 Case study
The CENSE project is an interdisciplinary project funded by the French research national agency (ANR) andaims to characterize urban landscapes combining numerical predictions and local observations. The city ofLorient has been chosen to conduct the experiment, and a network of noise sensors will be deployed across thecity. The need for a flexible model is crucial to check the compatibility of the observations with the simulation,
hence a meta-model is an adequate tool for this project.
The receptors grid is a regular grid with a space step of 10 m where we removed the points inside the buildings.
They are located 4 m above the ground, which is approximately the height of the physical sensors (with somevariations though). The total number of receptors is 9780.
The traffic inputs are derived from the annual average traffic data gathered by the CEREMA and the city ofLorient traffic department, the French national study center for territory development, for the year 2016. Soil,
topography and buildings data come from the IGN3 , the French national geographical institute and the city ofLorient.
3.3 Input parameters
Table 1 shows the list of the k = 12 coefficients p, split into two types of parameters, condition parameters anduncertainty parameters.
3.4 Sampling
Since the dimension of the input space is relatively large (k = 12), a regular grid to sample I would include too
many points. For instance, a grid with 10 discretization points in each direction would require 1012 simulations,
1 See http://noise-planet.org/noisemodelling.html
2 See http://orbisgis.org/
3 See http://professionnels.ign.fr/

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Table 1. Input parameters and their input range for the meta-model. They are all dimensionless multiplicative
coefficients applied to the reference values, except the altitude variation which is an additive value, and thetemperature and absorption/reflection coefficient which are the actual values.
Parameter
Condition parameters
Light vehicle speed
Heavy vehicle speed
Light vehicle flow
Heavy vehicle flow
Temperature (◦C)
Uncertainty parameters
Buildings height
Receptors height
Slope amplitude
Distance to junction
Wall absorption coefficient
Ground absorption – G value
Altitude variation (m)Minimum value

Maximum value

0
0
0
0
-5

2
2
2
2
30

0.5
0.5
0.9
0.9
-0.2
0
-0.5

1.6
1.6
1.2
1.2
0.2
0.7
0.5

which is computationally intractable. Instead, we make use of a Latin hypercube sampling (LHS) which allows
to explore I with a fairly low number of points. The parameters boundaries are slightly enlarged so that theeffective boundaries of the input space lives within the interior of the LHS.
In order to evaluate the meta-model performance, we picked a testing sample whose points are chosen to be
the farthest from the initial LHS points. The size of the testing sample is 90.

RESULTS
4.1 Dimension reduction
We applied the principal component analysis to a training set which is a Latin hypercube sampling of size
2000. With 4 principal components, the remaining unexplained variance goes below 3 %. Table 2 displays the
RMSE of the projection against the real simulation under the number of selected principal components. The
RMSE decreases fast with the first vectors, and slowly goes to zero as we increase the number of principal
components.
Table 2. Scores of the projection against the size of the basis. The simulation is compared to its projection on
the testing sample, at all the receptors
Basis size
RMSE (dB)
NRMSE (%)

3
2.54
4.20

4
2.43
4.01

12
2.01
3.31

20
1.17
1.94

50
0.91
1.50

100
0.81
1.33

4.2 Comparison of the meta-model with the direct model
The Kriging emulator has been generated with the R package Dicekriging 1.5.5 [8]. The evaluation part consistsin comparing the meta-model outputs with the actual outputs of the NoiseModelling simulator, and in observing
the performance with the scores described in table 3.

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Table 3. Scores for the performance evaluation of a model. (ci )i is the simulated sequence. (oi )i is thecorresponding observed sequence. n is the total number of elements in the sequence. c̄ and ō are respectively
the mean of (ci )i and (oi )i
Score

Formula1 n∑ (ci − oi )
n i=1

Bias

∑n (ci − c̄)(oi − ō)pp n i=1∑i=1 (ci − c̄)2 ∑ni=1 (oi − ō)2Correlations
Root mean square error (RMSE)n1n∑ (ci − oi )2i=1

Normalized mean square error (NRMSE)

RMSE
ō

The objective is to show the spatial distribution of the scores in order to observe what spatial organizationinfluences the loss of performance, and then to see the distribution of the scores, which is the most intuitive
way to check the validity of the meta-model.
4.2.1 Spatialized scores
Figure 1 shows the spatial repartition of the bias, correlation and RMSE scores for all the receptors of the study.
The results show that the meta-models perform best in area with low buildings density where the influence ofreflections and diffractions is lower. The value of the noise sources has little impact on the performance.

(a) Spatialized Bias (dB)(b) Spatialized Correlation (dimensionless)(c) Spatialized RMSE (dB)Figure 1. Maps of bias, correlation and RMSE between the Kriging meta-model results and the complete
simulation.

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4.2.2 Scores distribution and local evolution
The scores distribution, over all the simulations of the testing sample, is shown in figure 2. It reflects the
distribution of bias, correlation and RMSE over the parameters space. We see on the bias histogram (2a) thatthe majority of the error is centered around 0 dB and it has a very narrow error variation range. More than
95 % of the simulations have a bias lower than 1 dB, and 93 % of the simulations have a RMSE below 2 dB.
The average RMSE across all simulations is 1.58 dB.

(a) Bias distribution (dB)(b) Correlation distribution(c) RMSE distribution (dB)Figure 2. Distribution of the scores (per simulation) of the meta-model results compared to the simulation.

4.3 Computational costs : building and applying the meta-model
The 2000 simulations have been fully run in parallel during 3 days on 192 cores. The outputs allowed us togenerate the reduced basis Ψ and the interpolation parameters for the Kriging emulator. The offline computationconsists in the building of the meta-model. The online time consists in the calls to the meta-model whose
computation time are hugely negligible compared to the offline computation time: 65 ms for each call. A
simulation with NoiseModelling takes 27 min on 3 cores.

CONCLUSIONS
The generation of a NoiseModelling meta-model required 2000 simulations. The meta-model results remainvery close to the results of NoiseModelling with an average RMSE of 2 dB. The computational cost of themeta-model to generate one noise map is 65 ms, compared to 27 min for NoiseModelling, which corresponds toa speed-up of 25 000.
The two stages of the meta-model, dimension reduction and interpolation, have been carried out with a LHS
training sample. This sample sweeps the whole input space with a limited size, even in high-dimension spaces.
With this training data, we have generated a reduced basis of size 4, which explains 97 % of the variance. We
then emulated the projections of the noise maps on each basis vector. With a training sample of 2000 simulations, we obtained satisfactory results with respect to the bias, correlation and RMSE scores, when comparing
the meta-model with NoiseModelling on a testing sample.
The computation time saving is the main benefit of the meta-model. With an acceptable loss of performancecompared to the original NoiseModelling simulator, we gained an extraordinary amount of computation time.
The loss of performance against observed values from local sensors should be carried out in future work. The
CENSE project will provide these data since a network of sensors will be installed in the area we presented inthis paper.
Several applications can emerge from the meta-model design. Uncertainty quantification is the first that comesto mind, since it requires a very large amount of calls to the simulator which would be intractable with Noise-

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Modelling. Data assimilation [10] is also a major investigation area where meta-modeling can be used, by
quickly generating a simulation map in real time and correcting it with observations.

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