Anne Lacroix

Federico García Lorca

Primer Gabriel Sevilla

Romancero Gitano

t. I Obras Completas

Ed De Miguel

García Posada

Jacques Véase

Fernando Issorel

Villalón

Keywords:

una gitanización del nombre de María si se considera que "Reyes" o "de los Reyes" es un patronímico.

andaluz. Además de la alusión directa a los gitanos 8 y de la hispanización del nombre de María, llamada "Anunciación" 9 , el poeta usa casi exclusivamente los diminutivos en -illo 10 , menciona las flores de su tierra natal 11 y la Giralda, a la que convierte en la bisabuela del arcángel 12 .

El poema de Lorca no le hace mucho caso a la fuente evangélica de la cual se inspira.

Al mensaje original se añaden por ejemplo requiebros que la Virgen y el ángel se dirigen en una mutua admiración. A la salutación "Dios te salve, Anunciación. / Morena de maravilla", María contesta con la exclamación "¡Ay San Gabriel de mis ojos! / ¡Gabrielillo de mi vida!" 13 "San Gabriel. Sevilla" se caracteriza por una abundancia de imágenes y una escritura exuberante, que enlaza de manera brillantísima personificaciones 14 , antítesis 15 , hipérbatos 16 , comparaciones 17 y metáforas de una riqueza asombrosa 18 .

El poema ofrece una descripción muy detallada del arcángel anunciador cuya apariencia, aunque excepcionalmente bella, es humana. La asimilación del ángel al hombre se hace a la vez patente y humorística en el verso 6, donde el poeta le pone zapatos de charol 19 .

Federico García Lorca alude cinco veces a la coquetería de los ángeles en sus poemas, reveladores todos de una misma inclinación en ellos: el gusto por la indumentaria y el deseo de gustar por su porte. Además de los zapatos de charol de San Gabriel, podemos encontrar en "San Rafael. Córdoba", "Nocturnos de Marzo" y "La oración brota de la torre vieja…" estas precisiones:

El Arcángel aljamiado de lentejuelas oscuras 20 .

Miro y encuentro a un muchacho melancólico, todo vestido de blanco, ………. "¡Yo soy el diablo!" 21

Las campanadas son como ramos de flores ………. Como ruidos de faldas que los ángeles llevan22 .

Todas estas descripciones indican una preocupación por la elegancia y el refinamiento por parte de los ángeles lorquianos, incluso una verdadera coquetería. Es sin duda alguna el arcángel San Miguel el que muestra la mayor afectación, multiplicando encajes, bordados y perfumes, desvelando una belleza ostentativa, hasta tal punto que su preciosidad lo hace andrógino, casi femenino: Los ángeles de Lorca serían por lo tanto superficiales y falsos si no tuvieran también una gracia natural. Siete veces en su obra poética, el poeta de Granada exalta la venustidad de estas criaturas y se hace su cantor, inspirándose en una doble fuente: la estatuaria griega clásica y su Andalucía natal. Así es como crea un nuevo modelo, que sirve de pauta para edificar todos sus ángeles: el efebo gitano24 . El poema más representativo al respecto, y sin duda también el más magistral, es "San Gabriel. Sevilla". Se trata de un romance octosilábico en dos partes: la primera es una descripción del arcángel y la segunda una reelaboración del mensaje divino a la Virgen.

No nos equivoquemos. El propósito del poeta, manifestado en el título de la composición, es centrar toda su atención en el arcángel y arraigar a éste en su tierra andaluza.

El papel de María, personaje secundario de la historia, es ser el interlocutor y realce del ángel. Su belleza, saludada brevemente por San Gabriel 25 , no es nada comparada con la de su visitante celeste. La muchacha, cuyo pobre atavío contrasta con la magnificencia del ángel: 25 "San Gabriel. Sevilla", Primer Romancero Gitano, ibid., pp. 427-429, vv. 43-44: "Dios te salve, Anunciación. / Morena de maravilla". 26 Ibid.,[55][56] De los 70 versos que consta "San Gabriel. Sevilla", 28 están dedicados al diálogo (el 40 %) y 42 a la descripción (el 60 %). 29 En cuanto a la estilística, los diferentes tipos de endecasílabos y sus características, véase Tomás Navarro Tomás, Los poetas en sus versos: desde Jorge Manrique a García Lorca, Barcelona, Ariel, 1973, pp. 35 Los primeros veintiséis versos de "San Gabriel. Sevilla" están enteramente dedicados al relato, es decir, a la descripción del arcángel. Observamos, de manera general, que el octosílabo trocaico, ampliamente mayoritario (en 14 de los 26 versos, el 53,8 %), ritma el poema y sirve para subrayar la belleza física del ángel. El poeta usa el octosílabo dactílico (en 8 de los 26 versos, el 30,8 %) con otros fines. Este verso, más lento que el precedente (lleva como máximo tres acentos), siempre está asociado con la expresión de la sensualidad, a la que simboliza métricamente. En cuanto al octosílabo mixto (en 4 de los 26 versos, el 15,4 %), se reserva más bien al entorno del protagonista.

2-4-7 1-3-5-7 1-4-7 1-3-5-7 1-4-7 1-5-7 3-7 1-4-7 4-7 1-3-7 4-7 2-7 4-7 3-7 5-7 4-7 2-4-7 3-7 3-5-7 3-5-7 3-7 3-
Además del título del poema, el subtítulo que encabeza la primera parte, "San Gabriel", no deja ninguna duda en cuanto a lo que sigue: el arcángel anunciador, considerado como perteneciente al patrimonio del saber universal, es el personaje principal, si no el único, de esos versos. Su condición angélica, aunque supuestamente conocida y sugerida por el recuerdo de su santidad, no se menciona explícitamente.

   Un bello niño de junco,
Tras las precisiones brindadas por el título y el subtítulo, la indeterminación del artículo "un" es desviada en favor de otro significado. Presentado bajo los rasgos de "un […] niño", el ángel está considerado desde el principio como un hijo de hombre, su condición celeste permanece oculta. Sin embargo, antes que su naturaleza misma, el poeta quiere revelar su cualidad principal, que desarrollará a lo largo del poema: su belleza. A ello se añade la juventud del ángel, así como su aparente fragilidad y su origen andaluz, sugeridos por la metáfora final 30 . La aliteración en "o", sistemáticamente colocada después de la sílaba que lleva el acento, confiere a este octosílabo mixto redondez y armonía. Este primer verso prefigura el conjunto del poema y contiene en germen todos los temas presentes a continuación: humanidad, belleza plástica y gitanismo del ángel. 30 Véase también "Oración", Poesía varia, op. cit. supra, n. 5, p. 764, vv. 1-2: "San Rafael Arcángel, / muchachito de azafrán".

 

  anchos hombros, fino talle, Prosigue la descripción con la apreciación de la silueta del ángel, o mejor dicho de su tronco, cuyas proporciones obedecen a los cánones de la estatuaria griega clásica. Los anchos hombros, subrayados por las vocales abiertas y las nasales que aumentan su amplitud, simbolizan la robustez y la fuerza viril del personaje. La cintura fina, que el cierre repentino de la vocal "i" permite ver tipográficamente es, por el contrario, la señal de que el ángel, aunque de complexión potente, también es refinado. La impecable regularidad acentual de este octosílabo trocaico traduce por lo demás el equilibrio y la elegancia del ángel. Del octosílabo mixto al trocaico, Lorca ha pasado de la descripción de la belleza a su representación métrica, fenómeno que observaremos varias veces en el poema.

   piel de nocturna manzana,
Del busto del muchacho, el ojo del poeta se desliza luego sobre su piel. La mirada es casi impúdica, y la imagen poética elegida eminentemente sensual. El adjetivo "nocturna" evoca el cutis cobrizo del ángel y confiere a la escena una suave intimidad. En cuanto a la manzana, fruta supuesta del pecado original asimilada con la tentación de la carne, sólo puede figurar el atractivo físico que Lorca le presta a San Gabriel. Puestos en vilo por un juego de sinestesias, todos los sentidos están solicitados aquí. La vista y el tacto se funden en una caricia única, el olfato y el gusto encuentran en la manzana un perfume fresco y azucarado.

En cuanto al oído, éste aprecia las palabras de las que nacen estas imágenes ricamente labradas. La apertura progresiva de las vocales y la aliteración final en "a" traducen un abandono, al principio indeciso y luego total, hacia el placer. Los acentos, siempre tan regulares pero más espaciados de este octosílabo dactílico, son la representación métrica de la sensualidad del ángel lorquiano. Como si se arrepintiera de haber expresado con tanto ardor y lascivia la apariencia física de San Gabriel, el poeta vuelve con este octosílabo trocaico a una evocación más moderada de su belleza. No renuncia sin embargo a observar su boca, así como sus ojos. Los adjetivos que se aplican a estas partes del rostro del ángel podrían de hecho calificar su persona entera. La impresión de tristeza que se desprende del muchacho lo aureola de misterio y le añade atractivo. En cuanto a sus ojos, grandes pero inexpresivos, quizás sean el único indicio de su inaccesible naturaleza divina.

   nervio de plata caliente, Este octosílabo dactílico marca el retorno a la evocación de la sensualidad del ángel.

El poeta recorre con la mirada la musculatura evocadora del muchacho, púdicamente designada por la sinécdoque "nervio" 31 . Lorca subraya de nuevo el vigor del personaje, su fuerza típicamente varonil. La imagen poética utilizada para describir los músculos es asombrosamente rica. El carácter precioso del metal aumenta el valor y la robustez del ángel, mientras que su estado líquido, y por lo tanto el calor que se desprende de él, simbolizan su sangre pura y noble, generosa e impetuosa 32 . La impresión causada por este verso es la de un irresistible poder de seducción y de una suma voluptuosidad.

   ronda la desierta calle.
Por fin el arcángel está puesto en movimiento -ronda y recorre la calle-e integrado en un lugar que el artículo definido "la" sitúa en el ámbito de un pretendido saber común al poeta y al lector. El paraje es solitario, lo cual añade aún más a la misteriosa grandeza del personaje.

Busca o espera algo o a alguien, y su expectativa se apoya métricamente en la larga pausa acentual que caracteriza el verso. Este octosílabo trocaico marca una ruptura en el desarrollo del poema. Aísla los siete primeros versos, que sin duda son los más ricos y magistrales de "San Gabriel. Sevilla". De ahora en adelante, el ritmo de la composición cambia. Los versos están reunidos por cuatro y así es como vamos a seguir estudiándolos. 31 El singular cobra aquí el valor de un eufemismo. 32 Véase también "Reyerta", Primer romancero gitano, op. cit. supra, n. 5, pp. 418-419, vv. 15-16: "Ángeles con grandes alas / de navajas de Albacete".

breves lutos celestiales.

El arcángel es naturalmente hermoso, pero también coqueto, ya que sus zapatos de charol muestran lo preocupado que está por su apariencia. La criatura espiritual estaría totalmente humanizada si los versos siguientes no introdujeran un toque maravilloso. El poeta nos hace penetrar en el mundo de lo invisible que su imaginación crea por la poesía, mundo delicado e impalpable, formado de flores y música. Los octosílabos trocaicos y dactílicos se abrazan, como para recordar la estrecha unión entre amor casto a la belleza y amor carnal. El octosílabo dactílico, que Lorca ha convertido en el símbolo de la sensualidad, está reservado a la evocación de la dalia, flor exuberante por su forma y colores, y a la de los ritmos, principios armónicos esenciales para el nacimiento de la música.

  En la ribera del   no hay palma que se le iguale,     ni emperador coronado    ni lucero caminante.
La abundancia de imágenes lujuriantes inventadas por el poeta se enriquece aquí con una serie de hipérboles, introducidas por una anáfora que añade énfasis a estos cuatro versos.

Nada en la tierra ni en el cielo puede igualar en belleza al arcángel Gabriel. Su esplendor supera al de la naturaleza, cuyos componentes elige Lorca de modo que evoquen el mundo mediterráneo. La Andalucía del poeta se superpone a la Galilea de la Virgen, bordeada por el mar y plantada de palmeras. La magnificencia del ángel supera a la del hombre, aunque fuera el ser más potente del planeta, así como a la de los astros en el firmamento. A los tres elementos de comparación -naturaleza, hombre y astros-, corresponde un tipo de octosílabo particular. En efecto, se evoca el entorno inmediato del poeta mediante un octosílabo mixto, mientras que la infinita belleza del cielo sólo puede expresarse por un ritmo trocaico. El hombre en cambio, posible objeto de pasión, se merece la sensualidad del octosílabo dactílico. Estos cuatro octosílabos trocaicos siguen expresando la belleza del ángel y revelan un nuevo aspecto de ésta. Como Orfeo, San Gabriel es tan dueño de la música que las guitarras se estremecen para él y la naturaleza le obedece 34 . Su enemistad para con los sauces quizá encuentre su explicación en la continuación del mito. Abatido por la pérdida definitiva de Eurídice, Orfeo rehúsa todos los requiebros de las numerosas mujeres que suspiran por él. Se 33 Cantar de los Cantares 5 : 14b, Ezequiel 28 : 13 y Apocalipsis 4 : 3, op. cit. supra, n. 2, pp. 890, 1194 (A. T.) y p. 340 (N. T.). 34 Existe una duda en cuanto al sentido de "palomillas". Sin embargo, el sentido más corriente de este término, y sin duda el más adecuado aquí, es el de "mariposa pequeña". Véase el relato que Ovidio hace del mito de Orfeo en Les Métamorphoses, trad., introd. y notas de Joseph Chamonard, Paris, Garnier-Frères, 1966, pp. 253-255. hace, por el contrario, el intérprete de la belleza y la juventud masculinas 35 . Pero las mujeres tracias que le persiguen insistentemente un día se ensañan contra él hasta quitarle la vida.

Entonces la naturaleza entera viste de luto y Dionisos metamorfosea a las culpables en árboles 36 . ¿No sería irrazonable ver en este relato algunos aspectos de la personalidad de Federico García Lorca -su homosexualidad y su genio poético-y una de las fuentes de "San Gabriel. Sevilla"? La descripción del arcángel se acaba con una breve evocación de su misión: anunciar y por lo tanto permitir la encarnación de Jesús en el seno de la Virgen. El penúltimo endecasílabo -mixtositúa de nuevo al protagonista en un contexto humano definido y entrañable para Lorca. El ángel está presentado como el amigo de los gitanos, y recibe de ellos un regalo que nutre su inclinación hacia la coquetería, subrayada por la lentitud sensual del octosílabo dactílico: una prenda de vestir. El velo del misterio cae de nuevo en esta escena poético-maravillosa, dejando al lector hechizado. Los ángeles lorquianos son humanos, es cierto, pero poseen una belleza fuera de lo común, aureolada por un halo sobrenatural. *** Es interesante ver hasta qué punto esta criatura incorporal e inmaterial que es el ángel nutre la imaginación de Lorca. Sus representaciones poéticas son una revelación, consciente o inconsciente, de sus creencias, valores y preocupaciones. El poeta siente la necesidad de dar una apariencia física a los ángeles, conforme o no a los textos bíblicos y al imaginario colectivo. La mitad de los ángeles a los que Federico García Lorca atribuye forma y materia poseen características claramente humanas y masculinas, sin duda porque su homosexualidad 35 Ibid., le incita a elegir al hombre como referente para sus ángeles. Como verdadero esteta, el poeta aprecia la belleza, y la transcribe de manera apasionada en sus poemas, desplegando su estilo en imágenes suntuosas y ricamente labradas. ***

Verso « 10. San Gabriel. Sevilla » Acentos en las sílabas… Tipo de octosílabo

		Anunciación de los Reyes
		bien lunada y mal vestida,
		abre la puerta al lucero
		que por la calle venía 26 .
	multiplica para con él diminutivos cariñosos y exclamaciones admirativas 27 .
		Por razones similares, el mensaje divino se difumina en favor de la descripción del
	mensajero, por lo cual el poeta prefiere el relato al diálogo 28 . Sin embargo, lo más interesante
	del poema reside, a nuestro parecer, en los recursos retóricos y estilísticos de los que Federico
	García Lorca se vale para subrayar la belleza sensual del ángel, recursos que nos proponemos
	estudiar en la primera parte del poema 29 .
		SAN GABRIEL
	1	Un bello niño de junco,
	2	anchos hombros, fino talle,
	3	piel de nocturna manzana,
	4	boca triste y ojos grandes,
	5	nervio de plata caliente,
	6	ronda la desierta calle.
	7	Sus zapatos de charol
	8	rompen las dalias del aire,
	9	con los dos ritmos que cantan
	10	breves lutos celestiales.
	11	En la ribera del mar
	12	no hay palma que se le iguale,
	13	ni emperador coronado
	14	ni lucero caminante.
	15	Cuando la cabeza inclina
	16	sobre su pecho de jaspe,
	17	la noche busca llanuras
	18	porque quiere arrodillarse.
	19	Las guitarras suenan solas
	20	para San Gabriel Arcángel,
	21	domador de palomillas
	22	y enemigo de los sauces.

  La inversión del primer verso introduce en este endecasílabo trocaico una larga pausa acentual. La impresión de lentitud así creada subraya el ademán grave e infinitamente majestuoso del ángel. El segundo verso, un octosílabo dactílico, reanuda con el simbolismo de sensualidad que se le concede casi sistemáticamente desde el principio del poema. El poeta se detiene de nuevo en el torso de San Gabriel y se inspira de las Escrituras para describirlo. Los textos que parecen haberle servido de modelo a Lorca son tres: la evocación del amado en el Cantar de los Cantares, la del rey de Tiro -tradicionalmente considerado como una imagen de Luzbel-en Ezequiel, y la de Dios en el libro del Apocalipsis 33 . Sea como fuere, el resplandor del arcángel provoca una reacción sobrenatural del cosmos, expresada por una hipérbole, señal sin embargo de una humilde sumisión y de una gran reverencia de la creación para con la criatura excepcional.

			
	sobre su pecho de jaspe,
			
	la noche busca llanuras
			
	porque quiere arrodillarse.
		
			
	Las guitarras suenan solas
			
	para San Gabriel Arcángel,
		
	domador de palomillas
		
	y enemigo de los sauces.
	
			
	Cuando la cabeza inclina
		

"Nocturnos deMarzo", Poesía varia, ibid., 41. Véase también "Esta tarde en el río (verde pensamiento)…", ibid., t. IV (Primeros escritos), p. 532, vv. 3-6.

"La oración brota de la torre vieja…", ibid., t.IV, vv. 16, 18.

"San Miguel. Granada", Primer Romancero Gitano, ibid., t. I,[26][27][28][45][46][47][48]

Además de los poemas ya citados, véanse también "Reyerta" (Primer Romancero Gitano), "Demonio" (Odas) y "Oración"(Poesía varia), ibid., vv. 20,

25, 40, p. 764, vv. 1-2.

ángelbelleza -Gabriel significa "Fuerza de Dios" o "embarazada a pesar de sus muchos años y su apodo de "mujer estéril" 2 , y a María


João Cavalcanti Santos

Lenaïc Cuau

Philippe Poignet

Nabil Zemiti

Decoupled Model Predictive Control for Path Following on Complex Surfaces

Decoupled Model Predictive Control for Path Following on

A Convergence of Algorithm 1

The convergence of Algorithm 1 can be proved based on the assumption that each of the steps ∥p d,k+1 -p d,k ∥ are finite, which is the typical application of the proposed control scheme. A detailed discussion on this matter is presented in the following.

Theorem 1: Consider any given sequence of desired points P d = {p d,1 , p d,2 , . . . }, end-effector position p ∈ R 3 and positive integer k 0 . If there is a positive scalar ε such that

∆p d,k = ∥p d,k+1 -p d,k ∥ ⩾ ε > 0, ∀ k ∈ {k 0 , k 0 + 1, . . . } (23)
then the Algorithm 1 converges to a positive integer k ⩾ k 0 .

Proof of Theorem 1. Consider a given k ⩾ k 0 such that e T j (p -p d,j ) ⩾ ∆p d,j ∀ j ∈ {k 0 , k 0 + 1, . . . , k}.

with e j = (p d,j+1 -p d,j )/∥p d,j+1 -p d,j ∥. This represents a case in which a true value for the condition λ j = e T j (p -p d,j ) ⩾ ∆p d,j

was obtained in step 3 of Algorithm 1 for j ∈ {k 0 , k 0 + 1, . . . , k}. The solution of Algorithm 1 is obtained when j = k such that (25) is violated (false value in step 3).

Let V k+1 be the defined as

V k+1 = v ∈ R 3 : ∥v -p d,k ∥ ⩾ ε and (26a) v -p d,k ∥v -p d,k ∥ T (p -p d,k ) ⩾ ∥v -p d,k ∥ . (26b)
The set V k+1 is used to study the conditions on which v = p k+1 will lead to a false or true value in condition (25) for j = k + 1. Thanks to (26b),

p d,k+1 ∈ V k+1 ⇔ p d,k+1 -p d,k ∥p d,k+1 -p d,k ∥ T (p -p d,k ) = e T k (p -p d,k ) ⩾ ∥p d,k+1 -p d,k ∥ = ∆p d,k ,
leading to a true value of condition (25). Conversely, since (23) implies that v = p d,k+1 necessarily satisfies (26a), then

p d,k+1 / ∈ V k+1 ⇔ e T k (p -p d,k ) < ∥p d,k+1 -p d,k ∥ = ∆p d,k ,
which implies that a false value is obtained in step 3 of Algorithm 1 for k, thus converging to k = k. Note also that the combination of (26a) and (26b) leads to

d k = ∥p -p d,k ∥ < ε ⇒ V k+1 = {∅} ⇒ p d,k+1 / ∈ V k+1 . (27)
In short, (27) implies that d k < ε leads to a false value in condition (25).

The Theorem 1 is proved showing that V k shrinks of a finite volume for each step of Algorithm 1. This is proved studying the upper bounds of the sequence

d k = ∥p -p d,k ∥. ( 28
)
To do so, consider the maximum value d k+1 that the expression d k+1 = ∥p -p d,k+1 ∥ can lead for any p d,k+1 ∈ V k+1 for given p and p d,k . The value of d k+1 can be computed according to

d k+1 = max v∈R 3 ∥v -p∥ (29a) s.t. ∥v -p d,k ∥ ⩾ ε and (29b) v -p d,k ∥v -p d,k ∥ T (p -p d,k ) ⩾ ∥v -p d,k ∥. ( 29c
)
After some straightforward manipulations, the first order Karush-Kuhn-Tucker optimality conditions [START_REF] Nocedal | Numerical optimization[END_REF]Section 12.3] for the problem (29) can be written as

(v -p) + λ 1 (v -p d,k ) + λ 2 (p -p d,k ) -2 (v -p d,k ) = 0 (30a) f 1 (v) = ∥v -p d,k ∥ -ε ⩾ 0 (30b) f 2 (v) = (v -p d,k ) T ∥v -p d,k ∥ (p -p d,k ) -∥v -p d,k ∥ ⩾ 0 (30c) λ 1 ⩾ 0 (30d) λ 2 ⩾ 0 (30e) λ 1 f 1 (v) = 0 (30f) λ 2 f 2 (v) = 0 (30g)
The solutions of (30) can be summarized into the following three sets:

(I) ⇒      v ∈ R 3 : (v -p d,k ) T (v -p d,k ) = ε 2 (v -p d,k ) T (p -p d,k ) = ε 2 λ 1 = λ 2 = 1 (31) (II) ⇒        v = p d,k + p-p d,k ∥p-p d,k ∥ ε λ 1 = 1 - p-p d,k ε λ 2 = 0 (32) (III) ⇒ v = p λ 1 = λ 2 = 0 ( 33
)
The square of the cost function (29a) for a solution v = v (I) of case (I) can be rewritten as

∥v (I) -p∥ 2 = (v (I) -p) T (v (I) -p) = (v (I) -p d,k + p d,k -p) T (v (I) -p d,k + p d,k -p) = (v (I) -p d,k ) T (v (I) -p d,k ) ε 2 -2 (v (I) -p d,k ) T (p -p d,k ) ε 2 + (p -p d,k ) T (p -p d,k ) d 2 k = ε 2 -2 ε 2 + d 2 k = d 2 k -ε 2
where the conditions of ( 31) and (28) were used. For a non-empty V k+1 , (27) implies that

∥v (I) -p∥ = d 2 k -ε 2 ⩾ d k -ε ⩾ 0.
Accordingly, the sets of solutions (I)-(III) can be sorted as in

∥v (I) -p∥ = d 2 k -ε 2 ⩾ ∥v (II) -p∥ = d k -ε > ∥v (III) -p∥ = 0 (34)
Therefore, the global solution of ( 29) is given by

d k+1 = d 2 k -ε 2 (35)
taking as optimal argument p * d,k+1 = v (I) any vector satisfying

(p * d,k+1 -p d,k ) T (p * d,k+1 -p d,k ) = ε 2 (p * d,k+1 -p d,k ) T (p -p d,k ) = ε 2 (36)
The geometrical interpretation of the solution of (29) is illustrated in Figure 9.

ε p constraint (30b) p d,k-1 p d,k p e k-1 v (I) v (II) d k d k + 1 d,k+1
constraint (30c)

Figure 9: Geometrical interpretation of the solutions of (29).

One may repeat the optimization (29) to obtain d k+2 substituting p * d,k+1 into p d,k for (29b)-(29c), which leads to

d k+2 = d 2 k+1 -ε 2 (37)
Repeating the procedure recursively, one can obtain for any i > k

d 2 i+1 -d 2 i = -ε 2 ⇒ (d i+1 -d i )(d i+1 + d i ) = -ε 2 ⇒ (38) d i+1 -d i = -ε 2 d i+1 + d i ⩽ -ε 2 2 d k . ( 39
)
Therefore, the sequence d i can be majorized by

d i ⩽ d k - ε 2 2 d k (i -k), (40)
such that, in accordance with (27

) i > 2 d k ε 2 (d k -ε) + k = k ⇒ d i ⩽ d i < ε ⇒ V i+1 = {∅}. (41)
Therefore, the solution k of Algorithm 1 satisfy

k ⩽ 2 d k ε 2 (d k -ε) + k, (42)
proving the assertion of the theorem.

B Self-intersecting Curves

The ability to track self-intersecting curves is a crucial characteristic for path following control schemes. Several strategies fail on this matter, e.g. [START_REF] Goncalves | Vector Fields for Robot Navigation Along Time-Varying Curves in n -Dimensions[END_REF][START_REF] Yao | Path following control in 3D using a vector field[END_REF]. In the case of vector field path following control, for instance, the vector field is singular in crossing points, which leads to a null guidance signal [START_REF] Yao | Singularity-Free Guiding Vector Field for Robot Navigation[END_REF]. Similarly to [START_REF] Gill | Spline path following for redundant mechanical systems[END_REF], the proposed control scheme is able to track self-intersecting reference curves thanks to the sequential search proposed in Algorithm 1. As a reminder, the applied rationale is illustrated in Figure 10. For k = {k 0 , k 0 + 1, . . . } evaluated sequentially in an increasing order, the algorithm stops whenever e T k (p -p d,k ) < ∆p d,k .

p d,k + 1 0 p d,k + 2 p d,k + 3 0 ∆ p d, k ∆p d,k k = k 0 k = k + 1 0 k = k + 2 0 k = k + 3 = k 0 ∆ p d ,k p d,k + 4 0 0 < 0
S k = v ∈ R 3 ∃ j ∈ {k 0 , k 0 + 1, . . . , k} : e T j (v -p d,j ) < ∆p d,j (43)
Note that Algorithm 1 converges when reaching a k such that p ∈ S k . Since this condition is tested in an increasing sequence k = {k 0 , k 0 + 1, . . . }, the algorithm is stopped for the first k satisfying p ∈ S k . Thanks to that, a reference point following the correct sequence in a selfintersecting path is obtained instead of missing a closed portion of the curve. This is shown in Figure 11 and in Video (i)1 . Taking a k 0 for which p d,k0 is close to a self-intersecting point, Figure 11 shows how the searched region S k is extended until p ∈ S k . Four different cases are considered. In particular, scenario (c) (Case (II) in the video) shows a worst case of the position p in which S k includes In order to complete the demonstration that the proposed algorithm is able to track a selfintersecting curve, a 3D printing was performed applying the reference path depicted in Figure 12. The printing procedure is available in the Video (ii)2 .

C External Controllers

An external controller should be integrated to the proposed PPFC in order to control the interaction between the tool and the surface. More precisely, the external controllers proposed as typical applications can be divided with respect to the controlled physical variable as follows:

(i) Distance between the tool position and the surface;

(ii) Applied force perpendicular to the surface.

The next paragraphs discuss more in detail how these types of external controllers can be integrated to the proposed control scheme.

C.1 Distance control

In case (i), the displacements along n are used to track a distance h d between the tool and the surface based on the measured distance h m . Reminding that n points towards the interior of the object of interest, a simple strategy able to perform the tracking of h d could apply an external controller position velocity ṗec as proposed in the letter:

ṗec = k lh (h m -h d ) n, (22)
with a positive scalar gain k lh . Although (22) applies velocities based uniquely on the measured error e h = h d -h m , it is possible to show that e h → 0 for t → ∞. A draft of the proof of this assertion is presented in the following.

Consider that the errors due to the tracking of the PPFC scheme are negligible. Figure 13 illustrates that the trajectory generated with null velocity along n leads to a constant distance between the tool and the surface. Since the orientation n is continuously updated, the translational velocity ṗP P F C generated by the PPFC controller is kept tangential to the surface. Therefore, the nominal variation of the measured distance is caused uniquely by the external controller velocity ṗec . More precisely, in accordance with Figure 14,

ḣm = -∥ ṗec ∥ = k lh (h d -h m ). ( 44
)
For a constant h d , the closed-loop system (44) leads to

ėh = ḣd -ḣm = -k lh (h d -h m ) = -k lh e h , (45)
which is a first order linear system with e h (t) = e h (0) e -k hl t ⇒ lim t→∞ e h (t) = 0 lim t→∞ ṗec (t) = 0 (46) In short, in spite of the simplicity of (22), this external controller is asymptotically stable if the path following control is able to track the desired positions p d,k and orientations n d,k with sufficient precision.

Clearly, this conclusion depends on the stability of the path following control. The experimental results presented in Section V indicate that the introduced strategy leads to satisfactory results in this matter. A formal stability analysis of the proposed predictive control should be addressed in future works.

C.2 Force control

In case (ii), the displacements along n are used to control the contact force normal to the surface. A simple strategy able to perform the tracking of a desired force f d based on a measured force f could apply ṗec

= k f (f d -f ) n, (47)
with a positive scalar gain k f . One may apply a rationale similar to the one used in the last section. Nevertheless, the interaction between the tool and the surface leads to additional complexities in this case.

Firstly, one should verify that the precision of the path following control is robust with respect to the friction due to the contact between the tool and the surface. Secondly, the relation between the tool penetration s and the normal force f should be addressed, as illustrated in Figure 15. In order to apply the same rationale used in the last section, one could consider that the normal force can be written as a strictly increasing function of the penetration, i.e.

f = f (s) : df ds > 0 ∀ s ⩾ 0. ( 48
)
In words, (48) implies that f can be written as function of s such that increased penetrations lead to increased forces. The condition s ⩾ 0 is necessary to guarantee that the tool is in contact to the surface.

s(t) f(t)

Figure 10 :

 10 Figure 10: Illustration of a case in which the searched area is extended until e T k (p -p d,k ) < ∆p d,k and p ∈ S k for k = k 0 + 3.

Figure 11 :

 11 Figure 11: Four different cases indicating that the proposed algorithm is able to track the correct sequence of reference positions for self-intersecting paths. Cases (a), (b) and (c) show that the searched area S k is expanded starting from k 0 . Case (d) illustrates multiple positions of p that lead to e T k0 (p -p d,k0 ) < 0, such that p is within the first searched area and Algorithm 1 converges with k 0 = k.

Figure 12 :

 12 Figure 12: Experimental results for a self-intersecting path.

Figure 13 :

 13 Figure13: Illustration that if the tracking of n is able to keep this vector normal to the surface, the PPFC translational velocity is tangential and the h m is constant.

Figure 14 :

 14 Figure 14: Influence of ṗec on the distance h m .

Figure 15 :

 15 Figure 15: Notations used to correlate the penetration and the normal force.

https://seafile.lirmm.fr/f/a93502cf54ea454abd0b/

https://seafile.lirmm.fr/f/074035dfe33446c2b2f1/

Additionally, as in relation (44), one can conclude that

Considering the Lyapunov function V (f ) = (f d -f ) 2 the following conditions can be verified:

In addition, for s > 0, the time derivative of V can be written as

which, thanks to (48), leads to

Relations ( 50) and (51) prove that the closed-loop system obtained with (47) is asymptotically stable, such that lim t→∞ f (t) = f d lim t→∞ ṗec (t) = 0 (52)


Daniel Joly
email: daniel.joly@univ-fcomte.fr

Daniel Gilbert

Maria Diaz-De-Quijano

Mohamed Hilal

Mathieu Joly

Nadine Bernard

Enhancing air quality forecasts by geomatic downscaling: an application to daily PM10 concentrations in France

Keywords: particulate matter, land use regression methods, spatialization, public health

Public health institutions need high-resolution next-day forecasts so they can order appropriate measures when there is a risk of air pollution exceeding regulatory thresholds. The MOCAGE model, the chemistry transport model developed by Météo-France, forecasts hourly surface PM10 concentrations at a resolution of 0.1° throughout France (7.6 km). To obtain more efficient forecasts, a downscaling method is applied using topographic data (250 m resolution) and inventory data (2.2 km). All these disparate inputs are spatially standardized in a geographical information system to construct continuous daily fields at 250 m resolution. This method is suitable for large territories with widely varying environments (mountains, lowlands, coastlines, urban areas, etc.) and areas with a low density of monitoring stations. The parameters used to improve MOCAGE forecasts are derived from "global" and "local" regressions describing the links between the daily PM10 concentration averages collected at 325 monitoring stations and seven explanatory variables (three topographic and four emission-inventory variables). One of the main results shows that the topographic and emission variables respectively explain 6% and 13% of PM10 variance in France. Analysis by local regression accounts for 74% of the spatial variation of PM10 concentration while the global regression accounts for 49%. The results show above all that, if the authorities responsible for human health protection had used the downscaling method instead of MOCAGE raw forecasts in 2016, they would have informed or alerted ten times as many people about the information and recommendation threshold (50 μg.m -3 ) and alert threshold (80 μg.m -3 ) being exceeded.

Introduction

Due to their harmful effects on health [START_REF] Bentayeb | Association between long-term exposure to air pollution and mortality in France: A 25-year follow-up study[END_REF][START_REF] Riviere | Air pollution modeling and exposure assessment during pregnancy in the French Longitudinal Study of Children (ELFE)[END_REF][START_REF] Meier-Girard | Association of long-term exposure to traffic-related PM10 with heart rate variability and heart rate dynamics in healthy subjects[END_REF], it is especially important to monitor concentrations of air pollutants and to enhance air quality forecasts. Air pollutant effects on human health vary with the duration and level of exposure [START_REF] Barba-Vasseur | Does low to moderate environmental exposure to noise and air pollution influence preterm delivery in medium-sized cities?[END_REF][START_REF] Mariet | Multiple pregnancies and air pollution in moderately polluted cities: Is there an association between air pollution and fetal growth?[END_REF]). Among the major air pollutants, particulate matter and ozone have the largest impact on human health at present [START_REF] Pascal | Assessing the public health impacts of urban air pollution in 25 European cities: results of the Aphekom project[END_REF]. Exposure to particulate matter (PM) with an aerodynamic diameter of 10 μm or less (PM10) has notably been linked to respiratory and cardiovascular diseases and related deaths [START_REF] Analitis | Short-term effects of ambient particles on cardiovascular and respiratory mortality[END_REF][START_REF] Valavanidis | Airborne Particulate Matter and Human Health: Toxicological Assessment and Importance of Size and Composition of Particles for Oxidative Damage and Carcinogenic Mechanisms J Environ Sci Health Part C-Environ[END_REF][START_REF] Ayres-Sampaio | An investigation of the environmental determinants of asthma hospitalizations: An applied spatial approach[END_REF][START_REF] Fischer | Air Pollution and Mortality in Seven Million Adults: The Dutch Environmental Longitudinal Study (DUELS)[END_REF][START_REF] Dehghan | The relation between air pollution and respiratory deaths in Tehran, Iran-using generalized additive models[END_REF][START_REF] Riant | Associations between long-term exposure to air pollution, glycosylated hemoglobin, fasting blood glucose and diabetes mellitus in northern France[END_REF]. In this context, particle measurements and monitoring levels have featured prominently in European Legislation Directives that impose a duty on the authorities to inform the population when a certain threshold of mean annual or daily concentrations of pollutants is reached, especially in urban areas [START_REF] Marco | Air Quality Legislation and Standards in the European Union: Background, Status and Public Participation Advances[END_REF][START_REF] Kuklinska | Air quality policy in the US and the EUa review Atmospheric[END_REF]. The European and French air pollution networks result from European legislation (Directive 2008/50/EC on ambient air quality and cleaner air for Europe; French decree n°2010-1250 concerning air quality) making monitoring mandatory in densely populated areas. Moreover, Member States of the European Union are encouraged to predict air pollution events using chemistry transport models (CTMs) in order to warn the population, with a particular focus on at-risk populations. CTMs are numerical tools that simulate pollutant transport and dispersion, providing estimated concentrations for a particular area (El-Harbawi 2013) and time period (Zhang et al. 2017). Several air quality models such as HYSPLIT [START_REF] Makra | The effect of different transport modes on urban PM10 levels in two European cities[END_REF][START_REF] Waked | Investigation of the geographical origins of PM10 based on long, medium and short-range air mass back-trajectories impacting Northern France during the period 2009-2013[END_REF], Polyphemus/Polair3D [START_REF] Lecoeur | Dynamic evaluation of a multi-year model simulation of particulate matter concentrations over Europe Atmospheric[END_REF], CHIMERE [START_REF] Monteiro | Long-term assessment of particulate matter using CHIMERE model[END_REF][START_REF] Menut | CHIMERE 2013: a model for regional atmospheric composition modelling[END_REF][START_REF] Mailler | CHIMERE-2017: from urban to hemispheric chemistry-transport modeling[END_REF][START_REF] Potier | Characterizing the regional contribution to PM10 pollution over northern France using two complementary approaches: Chemistry transport and trajectory-based receptor models[END_REF], MOCAGE [START_REF] Sič | Modelling of primary aerosols in the chemical transport model MOCAGE: development and evaluation of aerosol physical parameterizations[END_REF][START_REF] Guth | First implementation of secondary inorganic aerosols in the MOCAGE version R2150 chemistry transport model[END_REF], CAMx [START_REF] Nopmongcol | Modeling Europe with CAMx for the Air Quality Model Evaluation International Initiative (AQMEII)[END_REF][START_REF] Milford | Measurements and simulation of speciated PM25 in south-west Europe[END_REF], MSC-W [START_REF] Simpson | The EMEP MSC-W chemical transport modeltechnical description[END_REF]CMAQ (Appel et al. 2012) have been used to simulate PM10 concentrations across France and Europe. These CTMs with coarse spatial resolutions (> 4 km) are typically used to analyze population exposure to pollutants in epidemiological studies, health risk assessments, and environmental damage studies [START_REF] Liu | Land use regression models coupled with meteorology to model spatial and temporal variability of NO2 and PM10 in Changsha, China[END_REF]. However, recent improvements have generated air quality models (for example HERMESv2.0) with spatial resolutions of up to 1-2 km [START_REF] Baldasano | Integrated assessment of air pollution using observations and modelling in Santa Cruz de Tenerife (Canary Islands)[END_REF][START_REF] Guevara | An improved system for modelling Spanish emissions: HERMESv20[END_REF]. These output results are still perfectible both in terms of resolution and quality of the estimation, for example for mountain areas, where the complexity of relief largely influences local air pollution (Gryning and Millán 2012) by causing local temperature inversions [START_REF] Chemel | Valley heat deficit as a bulk measure of wintertime particulate air pollution in the Arve River Valley[END_REF][START_REF] Largeron | The Atmospheric Boundary Layer during Wintertime Persistent Inversions in the Grenoble Valleys Frontiers[END_REF][START_REF] Paci | The Passy-2015 field experiment: atmospheric dynamics and air quality in the Arve River Valley[END_REF], or for high-density urban areas.

The satellite based data used to spatialize PM provide greater spatial-temporal coverage compared to sparse ground monitoring stations. Indeed, the PM estimations are mostly at 1 × 1 km spatial resolution and can thus meet the demand expressed by the health services for PM10 estimates of good quality at high spatial resolution [START_REF] Stafoggia | Estimation of daily PM10 concentrations in Italy (2006-2012) using finely resolved satellite data, land use variables and meteorology[END_REF][START_REF] Shtein | Estimating daily and intra-daily PM10 and PM25 in Israel using a spatio-temporal hybrid modeling approach[END_REF]. However, the problem with satellite data is that they do not provide ground pollution data when the cloud cover is compact. Under these circumstances, the use of geomatic techniques applied to low-resolution CTM model outputs is another solution for downscaling CTM that has proven its effectiveness [START_REF] Denby | Sub-grid variability and its impact on European wide air quality exposure assessment[END_REF][START_REF] Theobald | Improving the spatial resolution of air-quality modelling at a European scaledevelopment and evaluation of the Air Quality Re-gridder Model (AQR v11)[END_REF]. They are based on regressions between numerical models of PM10 concentrations and meteorological predictors such as temperature, wind speed, boundary layer height, etc. [START_REF] Stadlober | Quality and performance of a PM10 daily forecasting model[END_REF], whose resolution is finer than that of the numerical models. The land use regression (LUR) models use air pollution measurements and land use predictor variables to estimate pollutant concentrations at unsampled locations at different temporal (day, month, year) resolutions [START_REF] Hoek | A review of landuse regression models to assess spatial variation of outdoor air pollution[END_REF][START_REF] Ryan | Review of Land-use Regression Models for Characterizing Intraurban Air Pollution Exposure[END_REF]. LUR models are usually used to estimate pollutant variations at the within-city scale or at local scale [START_REF] Hoek | and Use Regression Model for Ultrafine Particles in Amsterdam[END_REF][START_REF] Eeftens | Development of Land Use Regression Models for PM25, PM25 Absorbance, PM10 and PMcoarse in 20 European Study Areas, Results of the ESCAPE Project[END_REF][START_REF] De Hoogh | Development of Land Use Regression Models for Particle Composition in Twenty Study Areas in Europe[END_REF][START_REF] De Hoogh | Development of West-European PM25 and NO2 land use regression models incorporating satellite-derived and chemical transport modelling data[END_REF][START_REF] Shahraiyni | Statistical Modeling Approaches for PM10 Prediction in Urban Areas[END_REF][START_REF] Miri | Estimate annual and seasonal PM1, PM25 and PM10 concentrations using land use regression model[END_REF] or for isolated and sparsely populated areas (Diaz- [START_REF] Diaz-De-Quijano | A more cost-effective geomatic approach to modelling PM10 dispersion across Europe[END_REF]. However, few studies have modeled broad geographic areas (European subdomains) at high resolutions (< 0.5 km) [START_REF] Vienneau | Comparison of land-use regression models between Great Britain and the Netherlands[END_REF]2013).

The main objective of this study is to enhance PM10 forecasts at country scale using LUR methods in order to obtain the best possible statistical modeling of the spatial variation of PM10 at regional scale over large areas characterized by regional and local spatial heterogeneity. Another aim is to improve the information given to populations about air quality in accordance with recommended thresholds and to better predict risks for the population. Here, a LUR geomatic method based on a set of explanatory variables has been used to estimate spatial variations of PM10 concentrations through regressions. The findings are then applied to downscale MOCAGE, a chemistry-transport model (CTM) that forecasts mean daily PM10 concentrations. The case study is applied to the whole of continental France at regional scale with a 250 m spatial resolution.

Material and methods

Data input

Monitored PM10 concentration

In France, the PM10 observation network measurements are managed by the air quality monitoring association network. The PM10 data used consist of daily averages calculated from data monitored at 325 stations in 2016 (366 days), using hourly averages. When there are fewer than 20 hourly measurements for a day for a given station, data for that day are considered to be missing; otherwise the daily average is calculated from the available hourly values. The stations sample urban (50%), periurban (10%), and rural (8%) areas; 10% and 22% are industrial and traffic stations respectively. All monitored data have been validated by the AASQA (Associations Agréées pour la Surveillance de la Qualité de l'Air -Authorized Associations for Air Quality Monitoring) and the LCSQA (Laboratoire Central de Surveillance de la Qualité de l'Air -Central Air Quality Monitoring Laboratory). Validated data are then collected by the European Environment Agency (EEA) as part of its Air Quality e-Reporting. PM10 monitoring stations are unevenly distributed (Fig. 1). Clusters appear around the main cities (Paris, Lyon, Marseille, Lille, Toulouse), while rural and mountain areas (Alps, Pyrenees, Massif Central, Jura, etc.) are not very densely sampled. For this reason, only a comprehensive study of France as a whole will be carried out. The model has been used for a wide range of scientific studies on tropospheric and stratospheric chemistry at various spatial and temporal scales; it simulates a lot of chemical species and, among them, PM10 concentration. It has been applied for example for studying the impact of climate on chemistry [START_REF] Teyssèdre | New tropospheric and stratospheric Chemistry and Transport Model MOCAGE-Climat for multi-year studies: evaluation of the present-day climatology and sensitivity to surface processes[END_REF][START_REF] Lacressonnière | Watson, L How realistic are air quality hindcasts driven by forcings from climate model simulations?[END_REF][START_REF] Lamarque | The Atmospheric Chemistry and Climate Model Intercomparison Project (ACCMIP): overview and description of models, simulations and climate diagnostics[END_REF] or tropospheric-stratospheric exchanges using data assimilation [START_REF] Amraoui | Mid latitude stratosphere-troposphere exchange as diagnosed by MLS O3 and MOPITT CO assimilated fields[END_REF][START_REF] Barré | Diagnosing the transition layer at extratropical latitudes using MLS O 3 and MOPITT CO analyses[END_REF]. Météo-France also contributes to the Copernicus Atmosphere Monitoring Service (http://www.regional.atmosphere.copernicus .eu/). MOCAGE is one of the seven models providing input for the regional overall forecasting system for Europe [START_REF] Marécal | A regional air quality forecasting system over Europe: the MACC-II daily ensemble production[END_REF].

The configuration of MOCAGE used in the present study has been run daily since 2005 for air quality operational 4-day forecasts in the framework of the PREV'AIR consortium [START_REF] Honoré | Predictability of European air quality: Assessment of 3 years of operational forecasts and analyses by the PREV'AIR system[END_REF][START_REF] Rouïl | PREV'AIR: an operational forecasting and mapping system for air quality in Europe[END_REF], http://www2.prevair.org/). The main characteristics of the configuration are the following. Three nested domains are used, with decreasing resolutions (globe at 1°, Europe at 0.5°, and France at 0.1°). For the global and the European domain, the hourly input meteorological forcing fields are Météo-France ARPEGE forecasts. For France, and for the first day of forecast, MOCAGE uses the operational outputs of Météo-France's non-hydrostatic AROME model. Emissions are prescribed at the surface, based on state-of-the-art inventories (MACCity over the globe and TNO-MACC-III over Europe), or dynamically depending on the local meteorology (e.g., for sea salts and desert dusts). MOCAGE simulates the concentration of gases [START_REF] Josse | Radon global simulations with the multiscale chemistry and transport model MOCAGE[END_REF][START_REF] Dufour | Observed and modelled "chemical weather" during ESCOMPTE[END_REF]), primary aerosols [START_REF] Martet | Evaluation of long-range transport and deposition of desert dust with the CTM MOCAGE[END_REF][START_REF] Sič | Modelling of primary aerosols in the chemical transport model MOCAGE: development and evaluation of aerosol physical parameterizations[END_REF], and secondary inorganic aerosols [START_REF] Guth | First implementation of secondary inorganic aerosols in the MOCAGE version R2150 chemistry transport model[END_REF].

MOCAGE outputs are expressed in the WGS 84 geodetic system at a resolution of 0.1° in X and Y, with an average distance of 7.6 km from one point to another. The domain covered by MOCAGE extends over 10.9° in Y and 14.9° in X (West = 9.95°; East = -4.95°; North = 51.95°; South = 41.05°). Preliminary filtering has extracted the points belonging to continental France from the databases. Data refer to the 366 days of monitored data.

Explanatory variables

PM10 emissions register. The inventory of PM10 emissions, at the resolution of 0.025°, concerns the whole area covered by the monitored PM10 concentration and MOCAGE. Data derived from data collected by the National Spatial Inventory (INS, http://emissions-air.developpement-durable.gouv.fr/) were made available to the CNRM by INERIS (Institut national de l'Environnement et des Risques -National Institute for Environment and Risk Management) as part of the PREV'AIR consortium. They consist of 101 696 points whose values are daily averages expressed in μg.km² that concern the 10 sectors of SNAP (Selected Nomenclature for Air Pollution) emissions (Sup. material 1). Because some sources of pollution in France (i) are related to similar activities and (ii) some of them are too scarce, some SNAPs have been grouped together to form (i) the "industry + waste treatment" type (SNAPs 1+3+4+5+6+9) and (ii) the "road" type (SNAPs 7+8). SNAPs 2 (non industrial combustion plant) and 10 (agriculture) are left unchanged (Sup. material 2). Inventory points are also projected in RGF-L93. The value of each point resulting from the inventory (2.2 km resolution) is transferred to the 81 pixels of the resulting 9 x 9 pixels window (250 m resolution). The values are then smoothed by reference to a moving average (radius of 8 pixels).

Topographic variables.

The 250 m resolution Digital Elevation Model (DEM) provided by the French National Geographic Institute (IGN) is used to construct a large number of topographic variables [START_REF] Joly | Rank-ordering of topographic variables correlated with temperature[END_REF]. Preliminary Pearson correlation processing was done to select the variables that most frequently explain the spatial distribution of PM10 concentrations at the p<0.05 threshold. There are three variables: elevation [elev], magnitude of the positive landforms (hump amplitude [hump]), and valley depth [valley]. The hump value corresponds to the deviation in elevation between ground elevation and the virtual surface connecting all the thalwegs. Conversely, the valley value corresponds to the deviation in elevation between the ground and the virtual surface connecting all the ridges [START_REF] Joly | Rank-ordering of topographic variables correlated with temperature[END_REF]. These variables are intended to describe the capacity of topographic settings to impede or facilitate particle dispersion.

Methods

Regressions

The available information and the processing applied to obtain data for the regressions and downscaling are presented in Figure 2. The information provided in the form of dots is transformed into continuous spatial fields by kriging or geomatic processing. Values corresponding to monitored PM10 concentration (1.1.1 in Figure 2), simulated PM10 concentration (1.1.2 in Figure 2), PM10 emissions register and topographical variables (1.1.3 in Figure 2) for each of the 325 pixels where monitoring stations are located are extracted from the raster layers made up of 86 366 691 pixels. The result is three tables, all comprising 325 rows (number of monitoring stations) and either 366 days (in the case of monitored and simulated PM10 concentration) or seven columns (explanatory variables, four for the different types of PM10 emissions and three for the topographical variables).

To control the degree of collinearity between variables, the variance inflation factor (VIF, an index quantifying the level of collinearity of a variable when it is added to the others in a multiple regression) is calculated. A maximum VIF value of 5 [START_REF] Hair | Multivariate Data Analysis[END_REF]) has been recommended. Collinearities between the seven explanatory variables are very low (<1.5), except for elevation and hump, which have VIF values of 4.1 and 3.3. These values are high, but below 5 and so remain acceptable.

Figure 2. Available data and their transformation for further statistical processing

Based standard ordinary least squares (OLS) regression models are then used to establish the statistical links between the explanatory variables and the monitored PM10 concentration [START_REF] Arslan | Spatial Modelling of Air Pollution from PM10 and SO2 concentrations during Winter Season in Marmara Region: 2013-2014[END_REF]. A simple linear regression is performed for each of the 366 days between the 325 PM10 concentrations (response variables) and each of the seven explanatory variables. Only significant variables (p < 0.05) are selected and included in a multiple regression (final daily model). The "global" regressions referred to above provide poor results when applied to climatologically heterogeneous areas such as France. The solution chosen to overcome this difficulty is to carry out "local" regressions [START_REF] Joly | Temperature Interpolation by local information, the example of France[END_REF][START_REF] Diaz-De-Quijano | Modelling and mapping trace element accumulation in Sphagnum peatlands at the European scale using a geomatic model of pollutant emissions dispersion[END_REF]. This method is based primarily on the recognition of the n nearest monitored stations to each pixel according to a neighborhood criterion. With n low (e.g. n = 20), the catchment area of the stations being limited, the stations are climatologically coherent. On the other hand, the statistics are not very accurate [START_REF] Joly | Temperature Interpolation by local information, the example of France[END_REF]. With high n (e.g. n = 100), the statistics are strengthened but the catchment area of the stations becomes very large and the probability of integrating different climatic zones increases. This poses a problem for the consistency of the models especially in rural areas where the density is 1 station per 1700 km². In this study, parameter n was set to 40, to provide the best empirical trade-off between statistical and geographical characteristics. Pixels characterized by the same corpus of n stations are grouped together in the same polygon. France is thus segmented into a total of 13,434 polygons.

Regressions provide the parameters that describe these relationships. There are two types of parameters: the intercept and the slope of each explanatory variable included in the regression. The PM10 concentration at any point in the study area can be estimated from these two coefficients. Then, the analysis by multiple regression and the application of coefficients to the pixels in order to estimate PM10 concentrations is performed once for the global regression and as many times as there are polygons for the local regressions. The parameters obtained by multiple regression are used to compute the estimated PM10 concentrations and their corresponding residuals are calculated by cross validation [START_REF] Plutowski | Cross-validation estimates IMSE[END_REF][START_REF] Stone | Cross-validatory choice and assessment of statistical predictions[END_REF]). Two goodness-of-fit measures (RMSE and R² calculated using the Bravais-Pearson R) are computed to assess the performance of each model.

Upon completion of the regression analysis, 325 residuals are obtained for each of the 366 days. [START_REF] Joly | Improving spatial temperature estimates by resort to time autoregressive processes[END_REF] have shown that the application of an autoregressive process can reduce these residues. In time series such as daily PM10 concentrations, [START_REF] Neal | Application of a statistical post-processing technique to a gridded, operational, air quality forecast[END_REF] have demonstrated that the value recorded on any day at each station is dependent on the values recorded the day before. Physical mechanisms such as inertia account for this property of bodies that tend to stabilize or to repeat the same patterns over the course of time. This general principle holds for the climatic characteristics of a location which are, in part, dependent upon those of the previous day. The autoregressive process aims to estimate the residuals of the day d (response variables) by those of day d-1 (explanatory variable).

Downscaling

The objective of downscaling is to improve the resolution and quality of the PM10 fields simulated by MOCAGE. The downscaling of PM10 concentration is based, for each of the days analyzed, on the resolution of equation ( 1) whose regression parameters are adopted as such, while the value of the intercept returned by the regression is replaced by the value of PM10 from MOCAGE:

10 = + + ⋯ + + 10 (2)
in which the variables X and a are identical to those of the regression (1) while PM10MOCAGE replaces b.

The R² and RMSE (Root Mean Square Error calculated by cross validation) are used to measure the quality of fit.

Results

Monitored PM10 concentration overview

The national-averaged annual mean concentration measured in France in 2016 is 18.2 μg.m -3 and the three quartiles (Q1 to Q3) are 13.1, 16.9, and 22.1 μg.m -3 . Among the 366 daily averages, the lowest value was 8.0 μg.m -3 , on January 31 and the highest was 47.1 μg.m -3 (December 1). High daily values (median > 40 μg.m -3 ) occurred three times, especially in December. On the other hand, values are low from April to November with a quartile 3 rarely exceeding 30 μg.m -3 . The average daily amplitude (calculated from data from the 325 monitoring stations) is 46.6 μg.m -3 . The lowest amplitude (22.1 μg.m -3 on May 1) reflects high spatial homogeneity of PM10 concentrations throughout the country; however, the maximum amplitude (144.8 μg.m -3 on December 1) reflects very high spatial heterogeneity.

The minimum and maximum of the 325*366 daily monitored values are 0.4 and 194.2 μg.m -3 found in a rural station and in the Paris region, respectively. In 2016, 1.85% and 0.11% of the monitored data exceeded the information (50 μg.m -3 ) and alert (80 μg.m -3 ) thresholds for human health protection, respectively.

PM10 estimation by global regression

The three most frequently selected variables in the 366 global daily multiple regressions are roadtraffic emissions (94%), plant combustion emissions (59%), and elevation (57%). Then come valley depth, agricultural emissions, and magnitude of positive landforms with a frequency close to 30%. Industrial emissions are rarely significant (9%).

Estimating PM10 concentrations by global regression provides poor results. The topographic variables and the emission variables provide low explanatory power: the R² is on average 0.17 and 0.13 respectively (Tab. 1). The multiple regressions grouping the seven variables (3 topographic + 4 emission inventory) slightly improve the scores which remain modest (R² = 0.27). Note that this value is not the sum of the previous two because of the collinearities between variables. The situation changes after the autoregressive process: the annual average of R² is 0.49. The RMSE of the global regressions is between 8.3 and 7.7 μg.m -3 depending on the model (tab. 1).

Topography (T)

Emissions

PM10 estimation by local regressions

The hierarchy that has been described for global regressions is roughly respected with the exception of industrial emissions, which are much more frequently significant (from 9% to 24%). Conversely, topographic variables are significant in only 13% of cases. The local regressions produce excellent results: R² values reach 0.74 as an annual average when the regressions relate to the seven explanatory variables and when the autoregressive process is used. The residues are low on average (RMSE = 4.7 μg. m-3 ), but, on certain days and at certain stations, they can be high: the minimum is -32.3 μg.m -3 while the maximum reaches 67.6 μg.m -3 . The highest errors (<-10 and >10 μg.m -3 ) have respective frequencies of 0.9 and 1.6%. However, 87% of the errors are between -4.9 and +4.9 μg.m -3 .

MOCAGE outputs and downscaling

In 2016, the correlation coefficient (R²) between the monitored PM10 concentration and the PM10 concentrations simulated by MOCAGE is 0.15. In 81% of the cases, daily R² values are less than 0.2. They are higher than 0.5 in 1.4% of cases. Downscaling using the topographic and emission variables associated with the autoregressive process improves the estimate since the R² annual mean is 0.23. If 5% of the daily R² are less than 0.2, 38% are now greater than 0.5, the maximum being 0.81.

Residuals (1.6 μg.m -3 as an annual average) indicate that the downscaling process does not produce statistical bias. They can be occasionally high: the greatest errors (< -10 μg.m -3 and +10 μg.m -3 ) have a frequency of 9.2 and 16% respectively. However, 60% of errors range from -4.9 to + 4.9 μg.m -3 . The RMSE of the downscaling estimates is 9.2 μg.m -3 .

Mapping of PM10 concentration in the most polluted situation

The case study selected for PM10 mapping is that of December 1, the most polluted day of the year 2016: the minimum, Q1, median, Q3, and maximum PM10 concentrations observed among the 325 stations are 3.2, 31.0, 40.4, 55.0, and 148.0 μg.m -3 respectively. The R² between the 325 monitored PM10 concentration and the values simulated by MOCAGE is 0.46 (Fig. 3); but MOCAGE, which does not currently represent all aerosol species, returns much lower values than the monitored ones: the average is only 24.3 μg.m -3 (as against 47.1) and the maximum is just over 56.0 μg.m -3 (as against 148.0). The southeast quarter of France is used to illustrate the results of the models. This regional zoom shows the value of the method in providing regional estimates of PM10 based on national information. Indeed, health agencies have a regional scope of intervention. The PM10 simulated by MOCAGE (Fig. 4A) shows that the highest values (≥ 30 μg.m -3 ) are located around Lyon with a clear extension to the south, along the Rhône valley and eastward to Marseille. Areas with lower pollution levels are located in the mountainous and rural areas.

After downscaling, the average downscaled values are 56.3 μg.m -3 and the R² is 0.80 (Fig. 3). The spatialization of PM10 concentration remains virtually unchanged in its main features (Fig. 4B). In detail, it should be noted that pollution in the main cities (Lyon, Marseille, Nice, etc.) and the Arve valley, one of the most polluted valleys in France, is prominent. The map shows more spotted areas, reflecting the contrast in emissions. The Rhône-Saône river valley (north and south of Lyon) stands out because of the road network which is taken into account in the multiple regression. Table 2. Percentage of PM10 exceeding the information and recommendation (50 μg.m-3) and alert (80 μg.m-3) thresholds on December 1, 2016 and number of inhabitants affected by these exceedances.

The proportion of pixels with a value above the information and alert threshold (50 and 80 μg.m-3, respectively) on December 2, 2016 is 0.5% and 0.07% with the MOCAGE raw forecast (tab. 2). It rises to 8.3% and 0.7% after downscaling. Using French 200 m gridded population data (https://www.insee.fr/fr/statistiques/2520034), the number of people to be alerted after exceeding the information and recommendation threshold of 50 μg.m-3 also depends on the model used: fewer than 5.9 million people with the MOCAGE raw forecast versus more than 10 million with the downscaling forecast. Alerts would be triggered for fewer than 2.5 million people with MOCAGE, but for almost 4.5 million with the downscaling technique. These estimates at d+1 allow decision-makers to take appropriate measures to prevent adverse health effects from exposure to PM10 [START_REF] Beloconi | Bayesian geostatistical modelling of PM10 and PM25 surface level concentrations in Europe using high-resolution satellitederived products[END_REF].

Discussion

The LUR geomatic method used, and based on a set of explanatory variables, provides the best estimate spatial variations of PM10 concentrations at country scale with a 250 m of resolution by using the so-called local regression with autoregression. The combination of methods (global and local regressions, autoregressions, alone or associated) and explanatory variables (PM10 emissions register, topographic variables) makes it possible to forecast PM10 concentrations (explained variables) with different qualities of estimates.

Quality of regression estimates

The global regression estimates provide low results: topography (R² = 0.17) and emission sources (R² = 0.13) account for little of the spatial variation in PM10 concentrations across France. The global approach cannot describe phenomena that depend on local conditions [START_REF] Stadlober | Quality and performance of a PM10 daily forecasting model[END_REF][START_REF] Czarnecka | The impact of thermal inversion on the variability of PM10 concentration in winter seasons in Tricity[END_REF]. After autocorrelation, the estimates result in an R² and RMSE of 0.49 and 7.7 μg.m -3 . These two indicators are similar to those derived from statistical adjustments intended to correct bias in the output of numerical models. But the estimates of the latter are based on annual or seasonal averages with resolutions of several kilometers while ours are daily with a resolution of 250 m. As an example, let us mention the estimate of PM10 concentrations in Great Britain and the Netherlands which provide R² values between 0.3 and 0.4 [START_REF] Vienneau | Comparison of land-use regression models between Great Britain and the Netherlands[END_REF]. In a similar experiment concerning variation in PM10 concentrations throughout the EU in 2001, the model explains 46% of the variation in PM10 with RMSE =5.19 μg.m -3 [START_REF] Beelen | Mapping of background air pollution at a fine spatial scale across the European Union[END_REF]. At a European scale and with a resolution of 20 km, the mean daily RMSE for April and June is around 13 and 7 μg.m -3 with values as low as 8 and as high as 20 μg.m -3 [START_REF] Hamm | A spatially varying coefficient model for mapping PM10 air quality at the European scale[END_REF]. In similar work, [START_REF] Denby | Interpolation and Assimilation Methods for European Scale Air Quality Assessment and Mapping Part I: Review and Recommendations ETC/ACC Technical Paper[END_REF] provide an optimal estimate of the PM10 spatial distribution over Europe at 35x25 km resolution using a statistical interpolation method; the total RMSE of the mean daily concentrations of PM10 at the validation stations is 9.2 μg.m -3 .

The situation is much better with local regressions which take into account the topographic and PM10 emission factors at a fine scale. The latter are not stationary but vary from one place to another across France. Thus, the influence of topography is more pronounced in mountainous areas [START_REF] Largeron | The Atmospheric Boundary Layer during Wintertime Persistent Inversions in the Grenoble Valleys Frontiers[END_REF] and that of emissions in areas where dynamic or thermal turbulence is lowest. There is thus a spatial differentiation that is well described by local regressions, but poorly captured by global regressions. There is a second problem with global regressions. Because the explanatory variables fail to describe the spatial variation of PM10 concentrations, the (single) intercept calculated by regression is close to the mean. However, depending on atmospheric conditions, there are significant variations in particulate matter from one point to another. Another merit of the local regression is that it suitably fits the intercept to the local PM10 concentration.

The downscaling carried out concerns the MOCAGE raw forecasts at day+1 and is part of a time scale analogous to high-resolution urban studies. It is also similar to the adaptations of the output of numerical models expressed as annual averages, which concern very large areas and most often entire countries. Downscaling improves the simulation given by CTM models. This has been reported many times in the literature [START_REF] Konovalov | Combining deterministic and statistical approaches for PM10 forecasting in Europe[END_REF][START_REF] Denby | Comparison of two data assimilation methods for assessing PM10 exceedances on the European scale[END_REF][START_REF] Hamm | A spatially varying coefficient model for mapping PM10 air quality at the European scale[END_REF][START_REF] Mok | Selection of bias correction models for improving the daily PM10 forecasts of WRF-EURAD in Porto, Portugal[END_REF]. It can also be seen in our experiment. The R² in downscaling (0.23) improves the score obtained from the MOCAGE raw forecasts (0.15). However, another benefit of downscaling is the increase in the MOCAGE raw forecast values, which are generally quite low, in part because organic secondary aerosols are not yet implemented in that CTM. The annual average of the MOCAGE values is 10.7 μg.m -3 compared to 19.8 μg.m -3 with downscaling (a value very close to the annual average of monitored data (18.4 μg.m -3 ).

Monitored PM10 versus simulated PM10 (MOCAGE)

There is a large difference in the quality of the estimates obtained from local regressions applied to monitored PM10 concentration and MOCAGE. The values provided by the numerical forecasts of the MOCAGE model are not of the same nature as the values of monitored PM10 concentrations. The CTM predicts the average concentrations in a cubic grid of several kilometers in the plane and tens of meters in height, while the monitored data concern a point concentration, very close to the ground where the boundary layer is often established (Zhang et al. 2017), in a much more complex environment. Part of the difference between models and monitored data is therefore due to monitored data being much more precise than models in terms of their spatial representativeness. In addition, the MOCAGE data used in this study forecast for day+1, with several sources of uncertainty, such as meteorological forcing (Gillian et al. 2015), the emissions inventory [START_REF] Frey | Quantification of Variability and Uncertainty for Air Toxic Emission Inventories With Censored Emission Factor Data[END_REF], and the representation of the physicochemical processes. In addition, in the current parameterization, pollutants are injected vertically in a decreasing manner into the first five layers of the model regardless of the weather situation. In December 2016, however, since the boundary layer was very low, the injection was partially above the boundary layer, resulting in an increased negative bias.

Accuracy of inputs

Insufficient knowledge of pollutant sources and emission inventories can lead to significant bias and error in air quality estimations of mechanistic models (Chang and Anna, 2004;[START_REF] Borrego | Procedures for estimation of modelling uncertainty in air quality assessment[END_REF]. Each monitored station is influenced by many sources (agricultural, industrial, road traffic, etc.), at varying distances, which contribute, in the form of different species (minerals, nitrates, organic carbon, etc.) to measured concentrations of PM10. It may be considered futile to attempt to precisely locate the sources influencing each station using a specific emission inventory. Depending on the processes involved in aerosol evolution (transport, sedimentation, deposition, scavenging, transformation, etc.), the radius of influence of the sources varies greatly with the type of emission, the species emitted, the weather conditions, and the local geography.

But one can also consider that a source impacts the measured concentration as a direct function of its proximity. The use of a more detailed resolution issuing register than the one used (2 km resolution) would substantially improve the estimates. In geomatics, the location of data is a crucial point. Its quality determines the quality of the results. The 366 values of 3rd quartile of daily monitored PM10 concentration (and the R² and RMSE that characterize them) were arranged into five classes: < 15 μg.m -3 , ..., ≥ 50 μg.m -3 . It appears that the R² values are perfectly well-ordered: they increase as the PM10 concentration increases (tab. 3). Statistical results are therefore better when quartile 3 values exceed 40 μg.m -3 than when they are lower than 15 μg.m -3 , whatever the variables included in the regressions. RMSEs are even more orderly.

Variation

Normally, RMSE values decrease when R² values increase. This is true for estimates relating to situations where the distribution of monitored data values is similar (mean, range of series). This is not the case here where the R² and RMSE characterize six ranges of non-homogeneous values. The estimates of low monitored data values (<15 μg.m -3 ) have a relatively low RMSE despite their relatively low R² because the range of the series concerned is also low: 36 μg.m -3 . By comparison, the mean range of the series with the highest observed values (> 40, > 50 μg.m -3 ) is much larger (83, 90 μg.m -3 ), resulting in a potential (and actual) higher error as well.

Conclusion

The LUR geomatic method used allows us to estimate PM10 concentrations in a large country, the whole of continental France (550 000 km²), at a fine scale (with 250 m resolution). This method also allows us to respond to another problem: France is made up of very dissimilar areas: (1) rural zones where there are few if any air quality monitoring stations, (2) mountain areas where current spatial resolutions of 7 km are inadequate for displaying variations in pollution distribution due to the influence of topography, and (3) urban areas where the local emission sources are predominant in terms of PM10 emissions. To achieve our objectives, disparate data were collected and homogenized in the form of tables and grids in a standardized format.

This study indicates that the explanatory variables we used (topography and emission inventories) provide very poor results, explaining respectively merely 6% and 12% of PM10 distribution. In contrast, the use of local regressions increased these values significantly to 46% and 57%, respectively. This gain stems from the dispersion of PM10 concentrations, which is primarily a local phenomenon. But this phenomenon is also of a regional nature, as indicated by the strong link between monitored PM10 concentrations and the intercept of local regressions. Finally, application of the autoregressive process shows that spatial patterns very frequently persist from one day to the next, which greatly improves the estimates. These estimates at d+1 show that it is possible to significantly improve the quality of forecasts as to when the information and recommendation threshold (50 μg.m -3 ) and the alert threshold (80 μg.m - 3 ) will be exceeded and at no additional costs for the public authorities responsible for pollution and health.

This study provides several useful lessons for the future. In particular, the initial emission inventory values at medium resolution (approximately 2 km) were transferred to high resolution matrices (250 m) by dilatation smoothing. However, it would be advisable to use emission inventories on a spatially finer scale than is used here. The use of regional inventories and their spatial compilation would be a preferable solution.

The same method could be applied to deeper, more detailed analysis on a smaller scale. For example, it would be very useful to explore the effect of local topography and built environment on the downscaling method. Improving aerosol modeling in numerical atmospheric chemistry models is a long-term task. Nevertheless, in the short term, downscaling PM10 concentration forecasts can be improved by identifying (i) finer-scale emission variables and (ii) other explanatory variables such as atmospheric variables, which are widely used to reduce the biases introduced by numerical models.

Figure 1 .

 1 Figure 1. Location of the 325 PM10 monitoring stations in continental France 2.1.2 Simulated PM10 concentration MOCAGE (MOdèle de Chimie Atmosphérique à Grande Echelle -Large Scale Atmospheric Chemistry Model) is the chemistry transport model developed by Météo-France.The model has been used for a wide range of scientific studies on tropospheric and stratospheric chemistry at various spatial and temporal scales; it simulates a lot of chemical species and, among them, PM10 concentration. It has been applied for example for studying the impact of climate on chemistry[START_REF] Teyssèdre | New tropospheric and stratospheric Chemistry and Transport Model MOCAGE-Climat for multi-year studies: evaluation of the present-day climatology and sensitivity to surface processes[END_REF][START_REF] Lacressonnière | Watson, L How realistic are air quality hindcasts driven by forcings from climate model simulations?[END_REF][START_REF] Lamarque | The Atmospheric Chemistry and Climate Model Intercomparison Project (ACCMIP): overview and description of models, simulations and climate diagnostics[END_REF] or tropospheric-stratospheric exchanges using data assimilation[START_REF] Amraoui | Mid latitude stratosphere-troposphere exchange as diagnosed by MLS O3 and MOPITT CO assimilated fields[END_REF][START_REF] Barré | Diagnosing the transition layer at extratropical latitudes using MLS O 3 and MOPITT CO analyses[END_REF]. Météo-France also contributes to the Copernicus Atmosphere Monitoring Service (http://www.regional.atmosphere.copernicus .eu/). MOCAGE is one of the seven models providing input for the regional overall forecasting system for Europe[START_REF] Marécal | A regional air quality forecasting system over Europe: the MACC-II daily ensemble production[END_REF].

  the monitored PM10 concentration at point ij, X 1, X2, Xn = variables included in the multiple regression, a 1, a2, …, an = slope of variables X1, X2, …, Xn, b = intercept.

Figure 3 .

 3 Figure 3. Scatter plot between the 325 monitored PM10 versus the values simulated by Mocage and the values predicted by downscaling; case of December 1, 2016

Figure 4 :

 4 Figure 4: Restitution of PM10 simulated by MOCAGE for December 1, 2016 by (A) kriging and (B) downscaling. 3.6 Population at risk of exceeding the 50 and 80 μg.m -3 threshold in France Mocage kriged Mocage downscaled LUR geomatic method pixels > 50 μg.m -3 (%) 0.5 8.3 13.7 pixels > 80 μg.m -3 (%) 0.07 0.7 3.0 Population exposed > 50 μg.m -3 5 880 300 10 434 500 13 901 000 Population exposed > 80 μg.m -3 2 484 400 4 467 200 7 833 400

Table 1 .

 1 Mean of the R² depending on the four local regressions steps

			(E)	T + E	T+E + autoregressive
					term
	R² average -	0.06	0.13	0.17	0.49
	GLOB R² average -LOC	0.46	0.53	0.57	0.74

of R² and RMSE according to PM10 concentration

		< 15	15 to 19.9	20 to 29.9	30 to 39.9	40 to 49.9	≥50
	Nobs	75	123	109	36	17	6
	R²	0.71	0.73	0.75	0.76	0.80	0.81
	RMSE	2.5	2.8	3.6	4.9	6.1	6.5
	Amplitude	35.9	39.8	49.2	59.6	83.5	89.7

Table 3 .

 3 Number of data in each class (Nobs), daily amplitude of monitored PM10, R² and RMSE for six classes of the 366 daily PM10 values (μg.m-3) of quartile 3; local regressions.

Acknowledgements

The authors would like to thank INERIS for making the 0.025° resolution emission files available and the LCSQA for providing validated PM10 monitored data for the year 2016.