Stéphane Vaiedelich

Claudia Fritz

Perception of old musical instruments

Defining what is an old instruments is complex. In a general way, we can distinguish two categories of old instruments. On one hand, there are the instruments which are not or barely in use nowadays. Due to a period of abandonment, those instruments are representative of an epoch different from ours and can be relatively easily dated. On the other hand, there are the instruments which are still being played despite having been made a few centuries ago. Time and use have usually given them a patina, so they are perceived as old in terms of visual aspect, but they have usually been modified as well and so they can be perceived as contemporary in terms of sound. Do we understand an old instrument by hearing it? Do we actually even need to hear it to understand it? This article is meant to provide some thoughts on these questions, to highlight the links between sight and hearing in our perception of a musical instrument and to illustrate how documentation and scientific knowledge can influence this perception 1 .

Introduction

For most thinkers -from a Western society at least -the musical instrument seems to have to be defined, a priori, through its sound production function. In this way, it is defined as a device/an artificial machine/a human material product, allowing the supply of "sound objects", considered to be musical by the culture within which the instrument is used [START_REF] Sève | L'instrument de musique, une étude philosophique, l'ordre philosophique[END_REF]. Thus, whether the instrument is in a playing state or not and is submitted to scientific scrutiny as a whole or a fraction, the study and documentation of this musical instrument cannot leave out the analysis of its functionalities and their relationships with the material object itself.

Encountering a musical instrument never leaves one indifferent and usually arouses a feeling of curiosity close to admiration towards this alloy of matter, form and sound. Though it always seems to be conceived and produced in order to be listened to, the stories that it can tell are numerous.

An instrument can be a simple object in appearance or a clever assembly of tools and sound materials. It can be entirely conceived and designed as a new object or it can result from repurposing an artefact from its primary function (musical stone, washboard, musical glasses, . . . ).

The perception of an old musical instrument is complex. Its forms and plasticity as they appear to us nowadays are sometimes a very distant reflection of what they were originally. Its sound, when it still exists, has suffered as well due to ageing. Some instruments do not have a sound anymore because they cannot be played, or because the playing technique or the musical practice are lost. Therefore, the perception of this material and sound object relies on different senses, among which the sight is probably the most important. The involved senses provide the audience member the informations, on which they shape an identity to the instrument that is presented to them. Starting from a definition of the old musical instrument, we propose here, through chosen examples, to show how the input of sciences (both natural and human) allow the supply of factual elements which provide the instrument, beyond its sound, with a common sense based on "objective" documentation, that can be shared by a large audience.

The old musical instrument 2.1 Attempts towards a definition

Identified within a culture, a musical instrument is the product of an era to which it seems sometimes difficult to constrain it. It is indeed common that an instrument, like the guitar or the harpsichord for instance, that was thought to be forgotten was actually reused at a later time, in quite different musical ways, showing sometimes potentialities which were unexpected at the time of its original conception. The notion of old instrument is therefore difficult to define. Is it about an instrument newly made from an old template? Is it about an instrument that we can show, by an analysis of its material (dendrochronology if it happens to be in wood [START_REF] Kufar | A review on dendrochronology[END_REF]), that one part or the whole instrument dates from an old time relative to the observer and on which it would not be possible to play a contemporary repertoire? When a relatively long period of no use exists in the history of the instrument, the definition becomes easy. This period of abandonment, comparable to the period of burying of an archaeological object, is the guarantee of the testimony of another time. It places objectively the instrument in a time period, based on which the observer (scientist, musician, curator, general audience, . . . ) builds the mental image of a temporal distance. In this case, the evocative power of the instrument is not constrained to a "simple" auditory dimension, as the passage of time has stamped the whole object with various and complex significations and dimensions. Listening, seeing, touching, playing such a musical instrument is an open door to the time as highlighted by Robert Barclay: "Historic musical instruments have a very special place in modern society. They are touchstones to the past to an extent that other artefacts are. Playing music upon them allows us briefly to bridge the gap between the here and now and the over and done with" [START_REF] Barclay | The preservation and use of historic musical instruments : display case or concert hall?[END_REF].

On the other hand, when such abandonment period does not exist, defining the epoch of an instrument and thus defining what an old instrument is becomes very complex, as we will see in the particular case of the violin.

Particular case of the violin

Since its first appearance in the 16th century, the violin did not have to suffer a period of abandon and has been continuously played. It possesses, even now, a very privileged place in the European instrumentarium. Numerous are the musicians who are specialist in contemporary repertoire but play on instruments with a famous signature from the 18th century. The famous third partita for solo violin (BMW 1006) composed by Johan Sebastian Bach in 1720 was played on the violin made by Nicolas Lupot in 1803, kept in the collection of Musée de la Musique in Paris (figure 1), during the opening ceremony of the new museum in 1995. A few years later in 2002, the same violin allowed to hear the piece for solo violin written by Ianis Xenakis in 1950, in a tribute concert to this composer. From which period really is this instrument? Is it even from a precise period? What is the real signification ot the manufacture year still visible on the authentic maker's tag inside the instrument? What do we hear in each concert: a violin from 1995 or 2002, date of the concerts? An instrument from 1720 or 1950, dates of the creation of the pieces? Or an instrument from 1803 which has the incredible capability of being used for both music written way before and way after its fabrication? ). Sometimes called "The French Stradivari", Lupot was very active in Paris at the beginning of the 19th century. His instruments are largely inspired by Stradivarius models, which he was able to access thanks to his reputation as a repairer. The label inside the violin reads "Nicolas Lupot luthier rue de Grammont à Paris l'an 1803". Its neck was changed at least once and its set-up is typical of the first half of the 20th century, as indicated by its " Émile Français Paris" iron-stamped bridge. Copyright Cité de la musique -Philharmonie de Paris Defining an old violin is not easy. The transformations, that the instrument has necessarily undergone during successive maintenance to be kept in playing condition, make it change era.

During the 19th century, Parisian instrument makers were particularly active in the development of conservation techniques, some of which are still being in use. The study of technical treatises written by luthiers in the 19th and 20th centuries shows how the definition of an old violin can be ambiguous [START_REF] Marconi | Concevoir la restauration du violon au XIX e siècle[END_REF]. For them, it is an instrument which body (mainly the exterior and visible elements of it) has kept traces of its origin. These traces are magnified by the patina, a mark of regular and continuous use, representing the time passing but mainly the continuity of the instrument's existence, certainly old, but eternal as the instrument belongs to the past as well as to the present. Thus, the transformation of the neck and some other functional elements, which are well known to play an important role in the tone quality, are swept away like details as soon as they do not modify this immediate and mainly visual perception. "Le rebarrage des tables et le changement des poignées étant des réparations et non des modifications, ces instruments sont donc tels qu'ils ont été construits" (Changing the bracing of the plates and the neck being repairs but not modifications, these instruments are thus as they were made) wrote Laurent Grillet in his famous work Les ancêtres du violon et du violoncelle [START_REF] Grillet | Les ancêtres du violon et du violoncelle[END_REF], talking about some famous violins of his time. Beyond the fact that this argument can be seen as weak in the light of the so called Theseus paradox (a thought experiment that raises the question of whether an object that has had all of its components replaced remains fundamentally the same object), it still shows an interesting change in paradigm as braces of the plates and neck would nowadays be considered as constituant parts of the instrument (rather than secondary accessories in Grillet's time).

Perception of old instruments induced by visual inspection

Before it even sounds, a musical instrument is seen, as a material object. Whether the observer is expert or novice, the visual examination which is conducted leads to a quick categorisation in function of his/her degree of expertise. "This instrument is obviously old, juts by looking at it!" "This instrument seems to come from Asia, its decorations are typical of Chinese lacquers from the 18th century." "The decorations of the papers show that it is a Flemish harpsichord". The documentary and material approaches to identify the origin of a musical instrument and thus certify its authenticity are mostly guided by sight (rather than by hearing) and often only question the initial hypotheses derived by an initial visual perception. When the instrument belongs to different epochs or different places because of different organological modifications (for instance, a Flemish harpsichord from the 17th century which underwent a large renovation in France in the middle of the 18th century), there is often a conflict between the visual perception and the multiple origin of the instrument. When these modifications are subtle, discrete and frequent and the instrument has always been used in the instrumentarium, then it belongs simultaneously to many epochs. This is the case of the violin, which has been shown above to represent a real archetype on this matter and therefore has appeared relevant to us to pursue further our reflection and our experimental approach.

Case of old violins kept in playing condition

Stradivari and Guarneri 'del Gesu' may well be the greatest violin makers ever, but it takes an expert opinion based on visual and historical (rather than tonal) evidence to say whether a particular example is genuine. Playing and listening tests never enter the authentication process. However, the violins of these two masters as well as other Italian makers of the 17th and 18th centuries are widely believed to possess playing and sound qualities that are both immediately discernible to experienced violinists and not found in new instruments2 .

Recent studies [START_REF] Fritz | Players preferences among new and old violins[END_REF][START_REF] Fritz | Soloist evaluations of six Old Italian and six new violins[END_REF] have shown that, when blind-folded, players prefer in average new instruments and cannot tell better than chance level whether a violin is old or new. This highly suggests that the origin of the tonal superiority of the Old Italian violins is not grounded in particular playing and sound properties of the violins but in the knowledge that the violinists have of their origin and age. Through top-down processes, this knowledge changes the perception at a neural level [START_REF] Levitin | Expert violinists cant tell old from new[END_REF]: simply knowing the history or pedigree of an instrument can activate expectations for its sound that cause neural circuits -even lower level sensory-perceptual ones -to behave differently than they would without that knowledge. Violinists may really believe that the instrument sounds better, even if there is no acoustic difference in the physical world. However, though these studies, along with many informal studies since the beginning of the 20th century, have shown that new violins are in general more efficient as musical tools for soloists nowadays, the interest for these violins as well as their price have not decreased. This shows that these instruments are now part of our cultural heritage like old paintings for instance.

Case of museum violins

Perception is the the complement of materiality and their relationship can be more or less direct. In the case of a musical instrument, this perception is at least double, auditory and visual. While the first one is probably the most important one during a concert, the second one prevails in a museum.

Sound, form and colour maintain complex relationships. If the material dimension of an instrument can be grasped even if in an incomplete conservation state, its tone, to be perceived, requires the instrument to be complete, and moreover, to be played by a musician. However, it can be present in the imagination of the audience, who will refer to their memory to "hear" the instrument that they see. Thus, similarly to the face that a listener imagines when hearing a voice on the radio, the sound imagined by a spectator is linked to the aesthetic and visual aspect of the instrument that he looks at. Has a dark sounding violin a dark colour? The question asked this way may appear simplistic.

However, the answers collected to a series of questions asked to 29 observers (10 violin makers and 19 musicians) about two violins, of trifling and recent make, one with a brown varnish, the other one with a golden yellow one, presented to the observers in the same conditions, allowing them to see the instruments from all perspectives but not to touch them, are disturbing. After having been asked to describe both violins, the participants were asked about how they imagined the sound of the two violins. While 23 participants described spontaneously only the physical objects in terms of shape, colour, ..., without any consideration on their sound, only 2 of them (bowed string players) could not answer the second question. Among the 27 who described how they imagined the sound, a clear difference could unsurprisingly be observed between the makers and the musicians. For the first ones, their imagination of the sound was based on "objective" parameters of the making and the wood. Thus, 8 out of 10 talked about the "voûtes, mailles du bois, assez plat sous le chevalet, travail fin, pores serrés" (arches, medullary rays, quite flat under the bridge, fine work, tight pores).

For the latter, their imagination of the sound was mainly (15 out of 19) induced from the colours (two players mentioned one the bridge, the other one the making, and two others could not answer the question).

What is more interesting is the relative consensus on the sonority of each violin and how the difference in imagined sonority is clearly linked to the difference in colours. The brown violin was imagined as richer, warmer, rounder, more flexible, more powerful than the yellow violin which was, conversely, considered as less complex, harsher and more nasal, stiffer and tighter, less powerful.

Beyond this perceptual assessment, the symbolic history of colours brings to these results an additional historical and social dimension. The red colour in a western society takes on a particular importance 3 . The analytical results obtained these past years [START_REF] Echard | The nature of the extraordinary finish of Stradivari's instruments[END_REF][START_REF] Échard | Complementary Spectroscopic Analyses of Varnishes of Historical Musical Instruments[END_REF] on a panel of more than hundred instruments of the violin family, mainly from the 17 and 18th centuries, show the presence of red pigments (vermilion, cochineal, madder), particularly in the Italian craftsmanship and more generally in the instruments made by makers whose reputation was already assured during their life. This is confirmed by the analysis of the 242 descriptive notices (in French) from the database of the Musée de la Musique's collection [13] which shows that 62% of the 18th century violins from Italian origin have a varnish described as red 4 while this is the case for only 25% of same period 3 "Le rouge vient en dernier dans l'oeuvre alchimique" (Red comes last in the alchemical work) writes the alchemist Norton de Bristol in the 15th century [START_REF] Ball | Histoire vivante des couleurs, 5000 ans de peinture racontée par les couleurs[END_REF]. 4 There is no standardized way of describing the colour in the notices and so the varnish description can depend strongly on the perception of the person who wrote the notice("jaune doré" (golden yellow), "brun orangé"(orange violins from French origin. However, there is an important change during the 19th century, as a red varnish appears in 78% of the notices of French origin violins from the 19th century and 91% from the 20th century.

Table 1 shows that the colour of the varnish is much less described for violins from the 19th and the 20th centuries (more than 20% of the notices do not contain information on the colour), than for violins from the 19th and the 20th (less than 10%). However, there are a relatively large number of "experimental" violins in the 19th and 20th centuries, for which the colour is never described.

Having experimental status, as the Savart violin has, is the only property perceived to have value; therefore, the color does not seem interesting enough for the curator to describe. Thus, if we remove the experimental violins from the database, the number of non described notices is only twice as large for the 19th century than for the other centuries. The predominance of the red varnish in the 19th and 20th centuries as well as the fact that 19th century violins are much less described can be explained by the fact that all along the 19th and then 20th centuries, the old instrument has been invented and become archetype. Development of conservation techniques, appearance of large national collections, presentation of copies of old instruments in universal exhibitions take part in its conquest by the opinion and results in a production mainly based on copies of the old masters work. Therefore, an old varnish is more highly considered as a recent varnish, and only the varnishes from the 17th and 18th can have this status of "old varnish". Thus, their description -in particular their colour -appears, to the curators eyes, much more important than for more recent varnishes. This shows that the museal look is not neutral but as well that the varnish is constituent of the work in the case of the violin, unlike for paintings, whose varnish is often removed to be changed. In addition, the making school of reference is the Italian school, Cremonese and Venitian in first place, in which the famous and influential masters -Stradivarius, Amati, Guarnerius, Gofriller -largely used a red varnish in their production. All these reasons thus explain the enthusiasm for red varnishes in the 19th and 20th centuries which is well illustrated by the following quote from the maker and musician Auguste Tolbecque: "C'est ainsi qu'un Stradivarius un peu plus rouge qu'un autre vaut de suite dix mille francs de plus" (This is how a Stradivarius a little more red than another one costs directly 10 000 francs more) [14].

brown), "rouge brun" (red brown), ...). In addition, there are some chemical processes which, with time, can affect the original colour. For example, yellow resins have a natural tendency to become brown due to oxidisation. It is therefore not very easy to know the original colour of the violins based on their descriptive notices. However, red pigments are easily recognizable and would appear in some ways in the description, so we have therefore decided to group varnish colours in two categories: 'red' and 'not red'.

How science can change or enrich this perception 4.1 The input of scientific documentation

Our perception of the instrument is global. How we imagine its musical possibilities and its sound is linked to the visual examination of the object as illustrated with the example described in the previous section showing a direct relationship between how observers imagined the sound of a violin and the colour of this violin. But mechanical and acoustical analyses can provide objective informations which can modify our a priori and enrich our understanding of the instrument. Two illustrative examples among many are given below.

The singular Stroh instruments are hard to classify. Instantaneously categorised as "violin-like instrument" in a listening test, these instruments seem however "half-string, half-wind" instrument (Figure 2). A scientific documentation resulting from a pluridisciplinary collaboration (acousticians, curators, musicians) shows that what could be considered as just curious instruments actually appear to be the work of a talented maker, resulting from the process of the engineer Augustus Stroh to meet his original need, i.e. solving part of the difficulties of the sound recording techniques in their infancy [START_REF] Gautier | Acoustic characteristics of the Stroh-violin[END_REF]. By shedding light on its genesis and history, and explaining their acoustical functioning, this scientific documentation give these instruments a new place in the instrumentarium and make them integral musical instruments. In this sense, this documentation can disrupt the values likely to be conveyed by the instruments and modify their perception by spectators. Another illustration can be found in the Selmer jazz guitar. In 1931, Mario Maccaferri filed a patent on a guitar with a double resonator. Produced by Selmer, a company renown for its saxophones, this guitar called "big mouth" will be adopted by the famous guitarist Django Reinhart to become, for some time, the emblem of the gypsy jazz guitar. Evicted in 1934, taking back his rights and patents, Maccaferri imposed the company to build a new model, to avoid a conflict of counterfeiting. A new guitar was born as soon as 1935. Its adoption by Django Reinhart would be immediate. The visual comparison between the two instruments is unambiguous (Figure 3): one has a wide oval hole, a neck with 12 frets and a system of simple internal bracing; the other one has a tiny circular hole, a neck with 14 frets and a system of double and reinforced bracing. These obvious organological differences would normally foreshadow a sound contrast between the two instruments. However, acoustical analyses contradict this assumption. The global responses of both instruments appear similar: the spectral analysis presented in Figure 4 shows a similar distribution of the radiated energy as a function of frequency, mainly spread between 200 and 400 Hz (band 2). The measured differences between the two guitars are of the same order of magnitude as the differences between guitars of the same model.

The complex relationship between science and perception

Will this contradiction between science and perception (induced by visual inspection) eventually modify the global perception? The answer is complex because the global perception and understanding of an instrument is linked to its cultural and symbolic status as well as its physical appearance.

As demonstrated by many historical examples, scientific investigations in the field of mechanics seem to coexist with the magic around musical instruments rather than suppress it. For instance, in 1761, based on the first ideas of the Irish Richard Puckeridge, Benjamin Franklin presented the glass harmonica 5 , whose sounds are produced by finger friction on tuned glass bowls. Accused of being responsible for premature deliveries or the weakening of the strongest man in less than an hour, this instrument was forbidden by a police decree in some German cities in 1835 and disappeared rapidly from the instrumentarium before being rediscovered later in the 20th century.

Thirty years earlier, in 1802, the physicist Ernst Florens Friedrich Chladni described very precisely, in his famous work Die Akustik [START_REF] Kufar | A review on dendrochronology[END_REF], the geometric figures which result experimentally from bowing glass or metal plates. Then, in 1822, Joseph Fourier laid out the mathematical foundations for the theory of decomposition in series of sinusoidal signals [START_REF] Barclay | The preservation and use of historic musical instruments : display case or concert hall?[END_REF]. As we can see, the scientific tools allowing an explicit formulation of the phenomena involved in the glass harmonica were already available at the time of its official ban, without having tarnished the magical dimension attributed to this musical invention.

Another example is the case of the violin. Since the end of the 19th century, many blind listening and playing tests [START_REF] Coggins | Blind faith[END_REF] have shown a preference towards new instruments compared to the old Italians as well as no systematic ability of players to recognise them. However, violinists (even some soloists who took part in the recent scientific studies [START_REF] Fritz | Players preferences among new and old violins[END_REF][START_REF] Fritz | Soloist evaluations of six Old Italian and six new violins[END_REF]!) continue to claim such statements as "the one thing that you cannot put into a new violin is that it's been played for 300 years-these instruments change and develop" [START_REF] Belluck | A Strad? Violinists can't tell[END_REF]. Therefore, although those studies could appear as "the final nail in the coffin for those who would believe that old musical instruments sound demonstrably better than new instruments," they are "unlikely to change many musicians' minds" [START_REF] Levitin | Expert violinists cant tell old from new[END_REF] and Stradivarius legend and the magic behind his instruments will remain mostly unaffected by science.

The example of the violin is actually not isolated and many instruments (at least in the Western society) carry in their history the construction of myths, regardless of scientific studies. The electric guitar, a more recent addition to the instrumentarium, is another example. Leo Fender's guitars have become emblematic instruments, similar to the Stradivarius, and the reputation of these instruments will likely remain unaffected by recent scientific work [START_REF] Navarret | Caractériser la guitare électrique : définitions, organologie et analyse de données verbales[END_REF][START_REF] Paté | About the electric guitar: a cross-disciplinary context for an acoustical study[END_REF].

Many musical instruments can acquire a leading role in the cultural identify of a population just as the electric guitar is easily associated with the 20th century North American culture. Human sciences, by shedding light on the cultural and symbolic dimensions of these instruments, enlarge their significance and enrich our perception of them. For example, various legends describe the origin within the Mongolian culture of the vielle morin huur (represented in Figure 5), which was added to the Unesco World Heritage in 2008. The generic Mongolian term huur refers to all cordophones and morin huur literally means a huur with a horse head. This vielle is closely associated with pastoral life as it punctuates the daily activities during the sedentary periods and the seasonal migrations 6 .

It generally accompanies tales and legends but possesses as well the power of preserving herds from diseases and evil spirits. This vielle matters as much to herd keepers as their own horses. The assembly of its eight components is achieved in a particular ceremony and the instrument cannot be played before the horse head has been coated with butter or cream. The intimate relationship with the horse is not just material (through the horse head) but is symbolic too. It is said that some monotonous chants played by morin huur artists reproduce the sound of the gallop in the steppe. Nowadays, the art and the technique of playing are changing. New musical approaches are leading to organological modifications. Tomorrow, the morin huur may be silent forever or newly played across the world. Either way, the horses' gallops will have moved away. But the scientific investigations of the instrument and its history, formalising and conceptualising the knowledge that has accompanied its origin, will offer audiences the keys for understanding this instrument. Beyond the existence of the object itself, these studies will allow this instrument to be the vehicle and the survival tool of an oral culture in movement.

Conclusion

As illustrated in this paper, an old musical instrument cannot just be perceived as a simple music tool as it happens to be the vector of cultural, esthetical, technical, historical values, which are invisible to visual inspection. A multidisciplinary scientific study and documentation is therefore necessary for us to perceive these multiple identities [START_REF] Lacombe | L'instrument de musique : identité et potentiel[END_REF] and modify and enrich our contemporary eye.

So is the case of the lute. As testified by iconography, it was a musical instrument which was really played and present in all social classes during the 17th century. Nowadays, it is reserved to interpretations called "historical" and occupies only an anecdotal place in the current musical landscape. De facto, it is generally perceived as an obsolete instrument. This however does not take into account the role, litlle known, that it seems to have played in the history of the western modern thought. The lute was indeed also used as an experimental scientific tool by Vincenzo Gallilei, to demonstrate that the frequency of a vibrating string changes as the square root of its tension.

("Discorso interno alla diversita delle frome del diapason" probably written in 1589, rediscovered by Claude Palisca in the 1960s and translated in [START_REF] Baskevitch | Musique et sciences à Florence à la fin du XVI e siècle : la Camerata et les Galilei[END_REF]). This result contradicted the commonly accepted theory due to Pythagoras that the frequency should change linearly with the tension. As accepted theories of nature were tied to religious beliefs, contradicting such theory had implications far beyond the musical world [START_REF] Lindsay | The Story of Acoustics[END_REF].

Figure 1 :

 1 Figure 1: Violin made by Nicolas Lupot (E.996.10.1, Collection of Musée de la Musique -Paris [13]). Sometimes called "The French Stradivari", Lupot was very active in Paris at the beginning of the 19th century. His instruments are largely inspired by Stradivarius models, which he was able to access thanks to his reputation as a repairer. The label inside the violin reads "Nicolas Lupot luthier rue de Grammont à Paris l'an 1803". Its neck was changed at least once and its set-up is typical of the first half of the 20th century, as indicated by its " Émile Français Paris" iron-stamped bridge. Copyright Cité de la musique -Philharmonie de Paris

Figure 2 :

 2 Figure 2: A violin (E.996.10.1, Collection of Musée de la Musique -Paris [13]) and a cello (E.2010.4.3) made by Augustus Stroh, after his patent n • 9418 "Improvements in Violins and other Stringed Instrument", filed 4 May 1899 and registered 24 March 1900. Copyright Cité de la musique -Philharmonie de Paris.

Figure 3 :

 3 Figure 3: A "big mouth" Selmer-Maccaferri guitar at the top and a "small mouth" Selmer jazz guitar below, as presented during the Django Reinhart exhibition at Musée de la Musique in 2012. Copyright Cité de la musique -Philharmonie de Paris -photo Matthias Abhervé.

Figure 4 :

 4 Figure 4: Spectral analysis performed on a chromatic scale played in a similar way on the two guitars presented in Figure 3. The relative energy contained in six frequency bands -[50 200], [200 400], [400 800], [800 1200], [1200 1800] and [1800 2500]-is expressed in percentage of the total energy. Figure by Sandie Le Conte.

Figure 5 :

 5 Figure 5: Viele morin huur (E.2009 2.1, Collection of Musée de la Musique -Paris [13]). Copyright Cité de la musique -Philharmonie de Paris

  Old musical instruments represent thus an exceptional material archive containing information about humankind's technical knowledge, sensibilities, beliefs and dreams. A full understanding of an old musical instrument requires a pluridisciplinary study which addresses the technical, artistic, religious, cultural and historical dimensions of this archive. Such study enriches our global perception of what the instruments were or are. In fact, an old musical instrument may not always be

Table 1 :

 1 Percentage of violins of Musée de la Musique's collection not described in terms of colours for each century, when considering the full database and when removing the violins considered as experimental.

		full database database without the the experimental violins
	17th	8	8
	18th	7	7
	19th	24	19
	20th	21	10

In this paper, perception refers not only to the processing of sensory information but also to the understanding of the instrument in societal, historical and cultural terms

Testimonies are numerous. For example, we can refer to Simone. F. Sacconi, famous maker of the 20th century, author of "Il secreti di Stradivari"[START_REF] Sacconi | secrets" de Stradivarius[END_REF] in which he reported the testing of a Stradivarius cello at Carnegie Hall, during which a famous player, deliberately sitting in the gallery, turned around instinctively from the first few notes, with the impression that the instrument was behind him.

The famous composer Gaetano Donizetti used it in its opera Lucia di Lammermoor, created in Napoli in 1835.

See the documentary film[START_REF]L'Histoire du chameau qui pleure, film documentaire réalisé par Byambasuren Davaa[END_REF] which presents a poetic and well documented description of the vital importance that the instrument can represent.

Acknowledgements

The authors would like to thank Sandie Le Conte and Timothy Wofford for fruitful discussions as well as CNRS and Cité de la Musique -Philharmonie for supporting this research.

what we think it is.The question of hearing it can then become completely irrelevant ...


T H Kauffmann
email: thomas.kauffmann@univ-lorraine.fr

M D Fontana

Inorganic salts diluted in water probed by Raman spectrometry: Data processing and performance evaluation

Keywords: Raman spectrometry, Salt solutions, Inorganic salts, Calibration, Normalization, Uncertainty

The paper reports on a methodology for the determination, by means of Raman spectrometry, of the concentration of inorganic salts dissolved in aqueous solutions. This procedure is then applied to nitrate solutions with a varying content from 0 to 100 mM (mmol/l). For this we exploit the concentration dependence of the Raman peak lying at 1047 cm -1 specific to the NO 3 -anion. Different signal processing and normalization methods are used and compared to deduce the most reliable and robust calibration. Then cross-validations are done using the "leave-one-out" method to validate the prediction models. Several figures of merit such as sensitivity, signal to noise ratio, limits of detection and quantification are calculated to estimate the efficiency of our methodology for nitrate solutions. Various sources of uncertainties are considered and evaluated according to the ISO GUM and the standard uncertainty of the concentration is then calculated.

Introduction

The detection and identification of chemicals in water by Raman spectroscopy (RS) has been demonstrated since several decades [START_REF] Nakamoto | Infrared and Raman Spectra of Inorganic and Coordination Compounds[END_REF][START_REF] Irish | Interactions in aqueous alkali metal solutions[END_REF]. The advantages of RS compared with other techniques to detect anionic species are well known [START_REF] Nakamoto | Infrared and Raman Spectra of Inorganic and Coordination Compounds[END_REF]3]. Indeed chromatography, mass spectrometry, colorimetry, ion selective electrodes, even if these techniques are sensitive, require sample preparation, or use of reagent, and/or frequent calibration [START_REF] Giambra | Application of ion chromatography to qualitative and quantitative determination of the main inorganic ionic components of samples from a production process of potassium sulphate[END_REF]. The vibrational spectrum provides something like a fingerprint of one substance. However vibrational spectra of solute species by IR absorption is impossible owing to water absorption in IR range, so that only RS can be efficiently used. Furthermore Raman lines corresponding to substances are relatively narrow and generally distinct from the OH bands. In addition to remote sensing substances in tanks, lakes or rivers, portable Raman spectrometers are now inexpensive and robust [START_REF] Lewis | Handbook of Raman Spectroscopy[END_REF].

The paper is devoted to the method allowing a quantitative and reliable exploitation of Raman spectra. It is well known that the Raman intensity is proportional to the species content [START_REF] Loudon | The Raman effect in crystals[END_REF][START_REF] Fontana | Raman spectroscopic sensors for inorganic salts[END_REF], so that it can be used for the calibration of a substance although it is recognized that the main drawback of RS is the weakness of Raman intensity so that it is usually applied to large concentration. Nevertheless it is possible to measure concentration below 1 mM (mmol/L) using high performance laboratory spectrometers which are cumbersome and very expensive [START_REF] Rudolph | Quantitative analysis in alkaline aluminate solutions by Raman spectroscopy[END_REF]. Furthermore, a long acquisition time of several minutes is generally needed and it is frequently necessary to apply several heavy treatments (pure water spectrum subtraction or deconvolution for example) in order to reach the micro molar level [START_REF] Rudolph | Analysis of ionic species in natural mineral waters studied by Raman spectroscopy[END_REF]. UV Raman resonance and SERS are also proposed to overcome the difficulties of normal RS. However, despite advantages, compared with normal RS, these techniques use more complex instrumentation, are more expensive, and specific to one or two species only [START_REF] Ling | Reflectance based optical fibre sensor for ammonium ion using solid-state Riegler's reagent[END_REF]. The linearity of the response versus concentration could also be affected in resonance conditions.

We show here that classical RS through portable spectrometer can provide reliable measurements of dissolved substances even for high dilutions. We consider as an example for all measurements and treatments the aqueous solutions of sodium nitrates. As they are used in fertilizers, aerosols and explosives a detection of nitrates and a reliable determination of their concentration is an important challenge [START_REF] Camargo | Nitrate toxicity to aquatic animals: a review with new data for freshwater invertebrates[END_REF][START_REF] Monteagudo | River eutrophication: Irrigated vs. non-irrigated agriculture through different spatial scales[END_REF][START_REF] Moorcroft | Detection and determination of nitrate and nitrite: a review[END_REF].

This paper proposes in a first part a general method to achieve a reliable calibration of a substance diluted in water from a set of Raman spectra. At first, Raman signature specific to a substance and the concentration indicator are needed. Different data treatments are then described to evaluate the calibration methods. Several criteria revealing the quality of the quantitative analysis are presented. At last the methodology used to calculate the uncertainties coming from several error sources is described according to the Guide of Uncertainty and Measurement [START_REF]Guide to the Expression of Uncertainty in Measurement[END_REF]. In a second part the suggested procedure with its different steps is applied to the determination of the concentration in sodium nitrate solutions. The results are presented and the figures of merit are calculated and discussed underlining the power of the method.

Methods for a quantitative exploitation of Raman data

Species Raman signature and concentration indicators

RS is well known analytical tool to detect inorganic and organic substances as well [START_REF] Nakamoto | Infrared and Raman Spectra of Inorganic and Coordination Compounds[END_REF]. Thus database systems can be employed for the identification of species in the mixture. By contrast, the quantitative exploitation of Raman spectra is less frequent. It is mainly due to the difficulty to extract the absolute Raman intensity which is a complicated function of the set up characteristics (spectrometer, laser, detector…) and experimental conditions (temperature...). In fact it is possible to use the relative intensity to determine the species concentration C with a good reliability, if using appropriate procedure of data treatment. Different processes are presented and compared below.

Before any treatment, one (or two) Raman lines which are unambiguously attributed to a molecule or a substance have to be chosen [START_REF] Fontana | Raman spectroscopic sensors for inorganic salts[END_REF]. These lines are generally intense and well defined. The position of this line (i.e. the frequency of one vibrational mode assigned to the line) and its width depend on the state of the substance and environment conditions [START_REF] Mabrouk | Raman study of cation effect on sulfate vibration modes in solid state and in aqueous solutions[END_REF]. Moreover the width is varying with spectrometer resolution. This line is the own signature of the substance to be analyzed.

Intensity at the maximum of this peak I p is proportional to the substance content C, and therefore can be used to calibrate, and then to determine the content. Obviously the integrated intensity J p (peak area) is proportional to C as well, so that both quantities I p and J p can be chosen as indicator parameters to derive the concentration C. As they depend on several internal (linked to the set up) and external (experimental conditions) parameters, I p or J p has to be related to C univocally. For this, I p and J p are required to be independent of these parameters. This is provided by the normalization of spectra, and thus transformation of I p and J p , so that only relative intensities are really exploited. Each use of I p or J p as concentration indicator has some advantages and drawbacks. The line position can be shifted by temperature change, or spectrometer drift causing a possible change in I p value. The integrated intensity J p is nearly unaffected by these small changes if the line width is not too small (several cm -1 ) but requires additional calculation compared with I p .

Data processing

Pre-treatments

To improve the quality of the Raman signal, some data treatments are often applied on the spectra [START_REF] Lasch | Spectral pre-processing for biomedical vibrational spectroscopy and microspectroscopic imaging[END_REF]. The first process is generally a smoothing using several methods (moving average, Savitsky-Golay etc.) in order to diminish the spectral noise [START_REF] Palacky | SVD-based method for intensity normalization, background correction and solvent subtraction in Raman spectroscopy exploiting the properties of water stretching vibrations[END_REF]. Then a baseline correction is usually done to overcome the background noise which may be due to the absorption of the solution for example.

Normalization

In order to achieve a calibration, a classical method in spectrometry consists in choosing a specific signature of the species, the intensity of which is concentration-dependent and to study its behavior. Thus, the intensity of a specific peak usually evolves with the species concentration and is therefore used for the calibration. The problem is that the intensity of a spectrum depends of several external parameters like the laser power, the integration time, the temperature or the optical probe, which give significant differences between series of spectra, independently of the sample [START_REF] Fontana | Raman spectroscopic sensors for inorganic salts[END_REF]. Thereby if the spectra are not recorded under the same conditions they have to be normalized by reducing them on the same scale for comparison. For this, different methods of normalization exist which have all theirs advantages and drawbacks [START_REF] Palacky | SVD-based method for intensity normalization, background correction and solvent subtraction in Raman spectroscopy exploiting the properties of water stretching vibrations[END_REF].

Within our method procedure we propose three kinds of normalization. One can consider the normalization to a reference peak the intensity I ref which should be independent of the concentration and the indicator is therefore I p ref = I p / I ref after the normalization process. In the second method the intensity can be compared to the intensity maximum I max taken over the whole spectrum. We therefore have in this case I p max = I p / I max . The third method is the Standard Normal Variate (SNV) transformation which consists in subtracting the mean value of the spectrum I and dividing by the standard deviation s which leads to I p SNV = (I p -I) / s. Each method can be applied to the indicator J p as well. We have respectively J p ref = J p / J ref where J ref is the area of the chosen reference, J p max = J p / J max where J max is the whole area under the spectrum and J p SNV which is the area J p after the SNV transformation.

Criteria for quantitative analysis 2.3.1. Figures of merit

Several figures of merit (FOM) can be defined in order to estimate the performance of the methodology for the determination of the species concentration [START_REF] Taverniers | Trends in quality in the analytical laboratory. II. Analytical method validation and quality assurance[END_REF]. Some of FOM are linked to the signal to noise ratio SNR and thus to the instrument characteristics and the Raman efficiency of the substance. Related to SNR the limits of detection LOD and quantification LOQ are chosen and evaluated for a solution under study with a concentration C.

The signal to noise ratio is estimated by SNR = (I p -N) / σ N where the signal is calculated as the difference between the intensity of the peak I p and the baseline of the spectrum while the noise σ N is given by the standard deviation of the intensity in a signal-free range.

LOD and LOQ are respectively estimated for a confidence level of 99% by LOD = (3×C) / SNR and LOQ = (10×C) / SNR. LOD is the minimum detectable concentration which corresponds to a signal just above the noise level and LOQ is the limit from which we have a reliable concentration value. In practice LOQ is given by LOQ = 3.3×LOD.

We consider also the sensibility S defined as the relative change in intensity (I p or J p ) with the concentration and provided by the slope of the calibration.

Calibration model evaluation

The efficiency of a regression within a calibration model can be evaluated by different methods and criteria. Here, as usually in many studies [START_REF] Kang | Quantitative analysis of mixed hydrofluoric and nitric acids using Raman spectroscopy with partial least squares regression[END_REF][START_REF] Estienne | Multivariate calibration with Raman spectroscopic data: a case study[END_REF], three parameters are calculated in order to estimate the power of the regression: the regression coefficient (R 2 ), the root mean square error (RMSE) and the bias (BIAS) which are detailed hereafter.

The regression coefficient corresponds to the spreading predictions parallel to the ideal line. R 2 is usually given for a regression but this parameter although necessary is not sufficient. Thus, the RMSE evaluates the differences between values ŷ i predicted by the model and the observed value y i and has the same unit as the predicted value. The BIAS provides the offset on the prediction. The true values are underestimated or overestimated if the BIAS is negative or positive respectively. These three criteria can be calculated by the following expressions:

R 2 = 1- ∑ i= 1 n ( y i -ŷ i ) 2 ∑ i= 1 n ( y i -ȳi ) 2 RMSE= √ ∑ i= 1 n ( y i -ŷ i ) 2 n-2 BIAS = ∑ i = 1 n ( ŷ i -y i )
n where n is the number of points used for the regression and y i the mean of the y i values.

Repeatability

The repeatability of the method is determined using several replicates made on one solution under the same experimental conditions. It is given here through the coefficient of variation CV = σ / µ where µ and σ are respectively the mean value and the standard deviation from a statistical study made on the predicted concentrations given by the model.

Uncertainty evaluation

According to the ISO GUM [START_REF]Guide to the Expression of Uncertainty in Measurement[END_REF], the standard measurement uncertainty can be evaluated via two kinds of methods. The type A evaluation consists into a statistical study from several measurements of the same quantity obtained in assumed constant experimental conditions. The result is chosen as the average of the data and the standard uncertainty is given by the standard deviation of the mean. Uncertainty evaluation by means other than statistical analysis provides the type B contributions.

In the calculation of the whole uncertainty on the concentration, the squared of standard uncertainties obtained on the various sources of error have to be added. To deduce the standard uncertainty from the experimental uncertainty of type B, a rectangular distribution is conventionally assumed since no indication of the confidence, with which the error is estimated, is supplied by the manufacturer from the different used instruments [START_REF] Kauffmann | Optical sensor of salt concentration: uncertainty evaluation[END_REF].

The different sources of uncertainty are as follows. Firstly the uncertainty coming from the preparation of solutions used for the calibration is considered. Then the linear regression applied on the calibration points gives rise to an uncertainty evaluated by a type A method. Finally the uncertainty coming from the intensity measurement related to the instruments is taking into account.

Salt solutions uncertainty

Salt solutions used to achieve the calibration were prepared by dilution of a standard solution. The standard solution of a concentration C 0 was done according to C 0 = m / V 0 by weighing and adding a mass m of the solid substance into a flask containing a volume V 0 before filling with distilled water. The relative standard uncertainty of the standard solution concentration is therefore:

( u(C 0 ) C 0 ) 2 = ( u (m) m ) 2 +( u (V 0 ) V 0 ) 2
Instrumental uncertainties ∆m and ∆V 0 are given by the manufacturer so that the corresponding standard uncertainties are u(m) = ∆m / √3 and u(V 0 ) = ∆V 0 / √3 (type B evaluations).

Then the set of solutions were prepared by dilution of the standard solution and several concentrations are calculated according to C = C 0 .V / (V+V water ) where V is the pipetted volume of the standard solution and V water the volume of the added distilled water. Same pipettes of 5 mL were used for the standard solution and the water so that the uncertainty of the two volumes are the same and equal to ∆V = ∆V water i.e. a standard uncertainty u(V) = ∆V / √3. Final volume V+V water is equal to 40 mL so that 8 pipetting of 5 mL were required to prepare a solution. Relative concentration uncertainties of the diluted solutions u sol (C) are then given by:

( u sol (C) C ) 2 = ( u (C 0 ) C 0 ) 2 +8 ( u(V ) V ) 2 2.4.2

. Calibration uncertainty

The confidence on the calibration depends on the reliability of the results derived from the fit of the plot I = f(C) with a regression linear law I i = a.C i + b. For this we apply the type A method and calculate the standard deviation s(I) between experimental data of I and the values as predicted by the law [START_REF] Rosengard | Probabilités et statistiques en recherche scientifique[END_REF][START_REF] Feinberg | Validation interne des méthodes d'analyse[END_REF]:

s( I )= 1 √ n-2 √ ∑ i= 1 n ( I i -Î i ) 2
where I i and Î i are the experimental and predicted values respectively and n is the number of data used for the calibration. Then we calculate the standard uncertainty on the concentration C coming from the linear regression by the inverse calibration method with the formula [START_REF] Feinberg | Validation interne des méthodes d'analyse[END_REF][START_REF] Lu | Uncertainty evaluation of humidity sensors calibrated by saturated salt solutions[END_REF]:

u calib (C )= s( I ) a √ 1 N + 1 n + ( I -Ī ) 2 a 2 ∑ i= 1 n (C i -C )
2
where I is the value coming from N replicates, I and C are the averages of the calibration values, C i the different concentration values used in the calibration and a the slope of the calibration line. It can be noted that a corresponds to the sensitivity S of the probe to the sought substance. Moreover, one notes that the expression of s(I) / a is the same as the RMSE used to evaluate the calibration regression.

Instrumental uncertainty

Here are concerned the uncertainty coming from the instruments and the chosen concentration indicator. Our indicator can be the intensity I p or the area J p of the signature peak after signal processing and normalization. If the area J p is taken instead of the intensity of the peak I p , the peak area is approximated by a rectangle and calculated by J p ≈ I p × dw p where dw p is the full width at half maximum (FWHM). The relative uncertainty on J p is given by the calculation:

( u( J p ) J p ) 2 = ( u ( I p ) I p ) 2 +( u (dw p ) dw p ) 2
The uncertainty on the intensity value u(I) is given by the spectral noise estimated by the standard deviation of the intensity σ n while the uncertainty u(dw) of the width is estimated from the uncertainty of the line position by u(dw) = ∆dw / √3 considering a type B evaluation. ∆dw is provided by the CCD manufacturer as equal to ±1/2 pixel.

When normalization is applied on the indicator using a reference (I ref or

I p ref ) 2 = ( u ( I p ) I p ) 2 +( u ( I ref ) I ref ) 2 ( u( J p ref ) J p ref ) 2 = ( u ( J p ) J p ) 2 +( u( J ref ) J ref ) 2 = ( u( I p ) I p ) 2 +( u( dw p ) dw p ) 2 +( u( I ref ) I ref ) 2 +( u (dw ref ) dw ref ) 2
The uncertainty u instr (C) of the concentration is then obtained by dividing u(I p ref ) or u(J p ref ) by the sensibility S (i.e. the slope a of the calibration).

Results and discussion

The method described above with its different stages is applied to aqueous solution of nitrate.

Experimental

Salt solutions preparation

11 solutions of sodium nitrate NaNO 3 (purity >99%) were made by a dilution series of a standard solution preparing by weight. The concentration range was 0-100 mM (mmol/l) with a step of 10 mM.

Instrumentation

Raman spectra were recorded at room temperature with a RXN-1 Raman spectrometer (Kaiser Optical Systems) with an excitation laser line at 532 nm. The laser light was focused through a 50� objective (NA = 0.5) at a distance above 1 cm from the liquid sample contained in a quartz cell (contactless measurement) with a power of 70 mW. The Raman light was collected in backscattering configuration by the same optics and is diffracted through a grating on a CCD camera (Andor) using two lines of 1024 pixels and cooled at -40 °C. Acquisition was done by the iCRaman software (Mettler Toledo). The recording time was 2×30 sec and the spectral range was between 100 cm -1 and 4350 cm -1 within a spectral resolution about 1 cm -1 .

Data treatments such as baseline correction and normalization processes were performed by a self made program using the R language [START_REF] Core | R : A language and environment for statistical computing[END_REF].

Nitrate signature

In the first step one chooses the specific signature of the nitrate in solution allowing its identification. Figure 1 shows the Raman spectrum of a sodium nitrate solution. The large peak located at 1047 cm -1 is present in all nitrate solutions and corresponds to the main vibrational mode of the nitrate anion NO 3 -. Its position was found to be independent of the monoatomic cation in aqueous solutions [START_REF] Mabrouk | Raman study of cation effect on sulfate vibration modes in solid state and in aqueous solutions[END_REF]. Besides the nitrate signature the spectrum exhibits the wide OH-stretching band (OH-SB) of the water molecule extending from 2800 to 3900 cm -1 and a weak band lying around 1635 cm -1 and corresponding to the OH-bending band (OH-BB). As shown in Figure 1 the intensity of the NO 3 -peak evolves monotonously with the nitrate concentration so that this peak can be considered for the quantitative analysis. A calibration can therefore be achieved using the intensity I p of the NO 3 -peak lying at 1047 cm -1 or the area J p p from 1000 to 1100 cm -1 after a linear correction of the baseline. -specific peak at 1047 cm -1 . The inset shows the peak evolution with the nitrate concentration in mM (mmol/l).

SNR, LOD and LOQ

The SNR is estimated for a nitrate solution of 10 mM as shown in Figure 2. The signal value is calculated as the difference between the NO 3 -peak intensity and the baseline and is equal to 1004 a.u. The spectral noise is given by the standard deviation of the intensity between 1200 and 1300 cm -1 and is equal to 30 a.u. We found therefore a SNR value of 33.5.

Finally the limit of detection is estimated by LOD = 3×10 / 33.5 and found to be equal to 0.9 mM. The limit of quantification is then calculated by LOQ = 3.3×0.9 and is therefore equal to 3 mM.

Fig2. Signal to noise ratio SNR evaluation considering the signal as the intensity of the NO 3 -peak at 1047 cm -1 and the noise as the standard deviation of the intensity between 1200 and 1300 cm -1 .

Comparison of normalization methods

The intensity of the signature peak is used for the concentration calibration but as it depends as well on many experimental conditions (recording time, laser power, CCD responses etc.) it must be normalized to be exploitable for the concentration determination. The different methods of normalization mentioned above are used and applied to both indicators I p and J p and then compared each to the others in order to obtain the best concentration calibration.

The first method is the normalization to the reference peak and instead the OH-SB often taken as reference for normalization in the literature [START_REF] Li | Hydrogen bonded structure of water and aqueous solutions of sodium halides: a Raman spectroscopic study[END_REF]27] we prefer to consider I OH-BB of the OH-BB of the water spectrum. This band is lying about 1635 cm -1 and even if it is less intense than the OH-SB it has as advantage to be not (or very slightly) affected by the nitrate introduction as shown in Figure 3. The second method takes the intensity as reference as the maximum of the spectrum, here I OH-SB of the OH-SB. The last method uses the intensity of the NO 3 -peak after a SNV transformation applied on the whole spectrum.

If the area J p under the NO 3 -peak is chosen as indicator, the first method takes the area under the OH-BB from 1500 to 1800 cm -1 and the second normalization method is to divide J p by the whole area of spectrum. As the OH-SB can be also used for comparison with the different normalization processes the area from 2600 to 3900 cm -1 is calculated as well. Finally within the third method the SNV transformation is applied to the spectrum before calculating J p .

Fig3. Raman spectrum in NaNO 3 solution of 0, 30, 50 and 100 mM showing the NO 3

-specific peak at 1047 cm -1 evolving with the nitrate concentration and the unaffected OH bending band at 1635 cm -1 .

In order to improve the quality of the calibration a linear baseline correction is applied after the SNV transformation. Figure 4 shows clearly the contribution of a baseline correction after SNV normalization. Here the regression coefficient R 2 found for the calibration using the area of the NO 3 peak increases from 0.812 to 0.899 after a SNV transformation and reaches 0.992 when a baseline correction is applied in addition. Baseline corrections are also applied on the NO 3 -peak and on the OH bending and stretching bands for the other normalization methods before the ratio calculation. After normalization and linear baseline correction, the different intensity or area values I p and J p of the NO 3 -peak are plotted in function of the concentration in order to obtain the calibration lines shown in Figure 5. Then a linear regression is applied to evaluate the regression coefficient R 2 and the sensitivity S. Results are summarized in Table 1. Firstly one can see that applying normalization significantly improves the regression coefficient in each case whatever are the indicator and the normalization method. Thus R 2 increases from 0.95 to over 0.99 within all normalization methods.

All normalization methods give rise to R 2 values very close each to the other. By contrast they yield very different values of sensitivity S. As a consequence S can be considered as the criterion in the comparison between the different normalization methods.

In the first case when I p is the indicator, the normalization to the reference peak i.e. the OH-BB gives the highest sensitivity equal to 3.1×10 -2 mM -1 more than one order of magnitude larger than S provided by the SNV transformation. The normalization by the spectrum maximum i.e. the intense OH-SB gives the worst sensitivity (52 times lower).

If the area J p is taken as indicator, the sensitivity is clearly better when the SNV normalization is applied and has the same value (3.1×10 -2 mM -1 ) as in the best situation of the first case. The normalization method with the OH-BB taken as reference gives a sensitivity 8 times lower. Other methods yield a bad sensitivity, nearly 3 orders of magnitude less than the SNV transformation. The SNV transformation provides a good calibration (R 2 and sensitivity) when the area J p is taking as indicator and is a simple and easy method to be applied. The choice of a reference band gives also good results in calibration (R 2 ) but the sensitivity is smaller particularly if the chosen band is wide. Moreover this method can be less convenient because it requires searching for a band reference which is independent of concentration and requiring several processing (smoothing, baseline etc.).

The normalization by reference to the spectrum maximum, here the intense OH-SB has to completely excluded, not only because it gives a very poor sensitivity (or very small calibration slope) but also since it can lead to an erroneous determination of C. Indeed there are no a priori reasons for which the maximum could be independent of C. In our case OH-SB is affected by the introduction of nitrate, and other substances as well.

Prediction model

The methodologies implemented to establish the calibration laws according to the indicator I p or J p can be summarized as follows:

(a) the successive treatments as follows are to applied to the indicator I p : Raw spectra ; Smoothing (optional) ; Baseline corrections on the NO 3 -spectral range (1000-1100 cm -1 ) and on the OH-BB (1500-1800 cm -1 ) ; Normalization I p OH-BB = I p / I OH-BB ; Plot I p OH-BB vs C ; Linear fit ; Calibration law.

(b) the successive treatments as follows are to applied to the indicator J p : Raw spectra ; Smoothing (optional) ; SNV transformation ; Baseline correction on the NO 3 -spectral range (1000-1100 cm -1 ) ; Area calculus J p SNV ; Plot J p SNV vs C ; Linear fit ; Calibration law.

The calibration models C pred vs C are then achieved by reversing the calibration laws in order to predict the concentration values. Results of calibration give regression coefficients R 2 equal to 0.999 and 0.992 for I and J respectively, root mean square errors RMSEC of 1.1 mM and 3.1 mM and zero BIAS.

The models are then checked by cross-validation using the "leave-one-out" method. One removes the first of the n samples and performs the calibration model with the n-1 remaining samples to predict the concentration of the removed sample. This sample is then set back in the calibration set while the second one is removed for prediction. The same operation is repeated for all samples. The cross-validation gives a good coefficient of regression R 2 equal to 0.998, a RMSEV of 1.4 mM and a zero BIAS using the intensity I p OH-BB . Results are worse using the area J p SNV with R 2 equal to 0.984, a RMSEV of 4.1 mM and a weak BIAS of 0.5 mM. Calibration and validation models are given in Figure 6. The repeatability is finally determined using 30 replicates of a solution of NaNO 3 . We found a coefficient of variation equal to 4.5 % with I p OH-BB as indicator and 3.7 % if J p SNV is chosen. Comparison results of the two best normalization methods are summarized in Table 2.

Uncertainty calculation

Here the uncertainties of the concentration are calculated for a solution of C = 10 mM for the two possible choices of indicator.

Solution preparation

Salt solutions used to achieve the calibration were prepared by dilution of a standard solution of C 0 = 100 mM of sodium nitrate (molar mass of 85 g/mol). The standard solution was done according to C 0 = m / V 0 by weighing and adding m = 4.25 g of NaNO 3 into a flask containing a volume V 0 = 500 mL before filling with distilled water. Uncertainties due to the weighing machine and the flask are given by the manufacturer and equal respectively to ∆m = ± 0.01 g and ∆V 0 = ± 0.5 mL so that the corresponding standard uncertainties are u(m) = 5.77×10 -3 g and u(V 0 ) = 2.89×10 -1 mL and the uncertainty of the pipetted volumes are equal to ∆V = ±0.03 mL which gives a standard uncertainty u(V) = 1.73×10 -2 mL.

The standard uncertainty of the standard solution is found to be equal to u(C 0 ) = 0.15 mM which gives for a solution of 10 mM an uncertainty u(C sol ) = 0.1 mM.

Calibration

For the method using the intensity I p OH-BB , the standard deviation of the calibration between experimental and predicted data is found to be s(I p OH-BB ) =3.42×10 -2 . With a calibration slope a = 3.11×10 -2 , N = 2 replicates, n = 11 calibration values, I p OH-BB = 1.63, C = 50 mM and with an intensity value I p OH-BB = 4.14×10 -1 for C = 10 mM, we find a concentration uncertainty coming from the calibration of u calib (C) = 0.9 mM.

If J p SNV is chosen as indicator, we find a standard deviation of the calibration s(J p SNV ) = 9.62×10 - 2 and with a calibration slope a = 3.14×10 -2 , the same numbers of replicates and calibration values (N = 2 and n = 11), J p SNV = 1.78, C = 50 mM and with an area value J p SNV = 5.84×10 -1 for C = 10 mM, we find a concentration uncertainty coming from the calibration of u calib (C) = 2.6 mM.

Instrumentation

To calculate the uncertainty coming from the instruments, we assume first that the uncertainty on the intensity is given by the spectral noise and can be estimated by the standard deviation σ n used previously in the SNR calculation so that u(I p ) = u (I OH-SB ) = 30 a.u. Then for a solution of 10 mM we find a peak intensity I p = 1124 a.u. and an intensity of I OH-SB = 2715 a.u. for the OH-BB used here as reference giving I p OH-BB = 4.14×10 -1 . The calculation gives rise to an instrumental uncertainty of u instr (C) = 0.4 mM.

For the area method, the spectral noise is estimated by the standard deviation σ n of the intensity between 1200 and 1300 cm -1 but here after the SNV normalization and the uncertainty is found to be u(J p SNV ) = 8.9×10 -4 . The CCD manufacturer provides an uncertainty of ±1/2 pixel which represents for our spectrometer an uncertainty of ∆dw = ±1 cm -1 (2048 pixels for a spectral range of 4250 cm -1 ) and assuming a rectangular distribution for the uncertainty on the line position we have u(dw p ) = 1/√3. Therefore with the values of I p SNV = 3.29×10 -2 , with dw p = 12 cm -1 for the peak width and with J p SNV = 5.84×10 -1 we find in this case an instrumental uncertainty of u instr (C) = 1 mM.

Combined standard uncertainty

Finally we calculate a combined standard uncertainty of the concentration u c (C) = 1 mM with I p OH-BB as indicator (i.e. an expanded uncertainty U c (C) = 2mM) and u c (C) = 2.8 mM if J p SNV is chosen (i.e. U c (C) = 5.6 mM). Table 3 summarizes the various sources of uncertainty in both methods.

Conclusion

We propose a general procedure for the quick determination, by means of a portable Raman probe, of the concentration of an inorganic salt diluted in aqueous solution. We underline the effects of the normalization method and the indicator on the abilities and performances of the Raman sensor. Sensibility and accuracy are shown to be particularly dependent on these choices. The described method may is here applied on nitrates but may be used for other inorganic salts (sulfates, phosphates, carbonates, ...) giving a specific line, signature of the sought anion. Nitrate solutions from 0 to 100 mM were chosen as example and investigated through the Raman peak located at 1047 cm -1 as the specific signature of the NO 3 -anion. Several spectral processing and normalization methods were compared and discussed in order to get the best concentration calibration. Two methods give highest sensitivity equal to 3.1×10 -2 mM -1 in both cases and good calibration parameters (R 2 , RMSE and BIAS) according to the indicator chosen to calibrate the concentration variations. Thus SNV (Standard Normal Variate) transformation is better when the area under the peak is considered while a normalization by the OH bending band intensity taking as reference is preferable if the intensity at the maximum of the signature peak is used as indicator. In each case a cross-validation using the "leave-one-method" was done to validate the models.

Repeatability of the methods and some figures of merit are calculated for a nitrate solution of 10 mM. Limits of detection and quantification were therefore found to be 0.9 mM and 3 mM respectively and the repeatability of the both methods is less than 5%. The efficiency of our probe and technique can be pointed out if comparing the LOD value with the required water drinkability limit (50 mg/l or 0.8 mM).

Finally the uncertainty of the concentration is estimated according to the ISO GUM taking into account the various sources of uncertainties (solution preparation, calibration, instruments) and are found to be equal to 2 mM and 5.6 mM with the peak intensity and the area respectively taken as indicators.

  J ref ) relative uncertainties are convenient as well in the calculations. In this case we have I p ref = I p / I ref or J p ref = J p / J ref which give respectively: ( u( I p ref )

2. 4 . 4 . 2 (C )+u instr 2 (

 4422 Combined standard uncertainty Finally the combined standard uncertainty of the concentration u c (C) is obtained by adding the squared value of each uncertainty coming from the different error sources according to u c (C )= √ u sol 2 (C )+u calib C ) and the expanded uncertainty is calculated by U c (C) = k.u c (C) with a coverage factor k = 2 corresponding to a confidence level of 95%.

Fig1.

  Fig1. Raman spectrum of a NaNO 3 solution of 100 mM showing the NO 3-specific peak at 1047 cm -1 . The inset shows the peak evolution with the nitrate concentration in mM (mmol/l).

Fig4. NO 3 -

 3 Fig4. NO 3 -peak for several nitrate solutions from 0 to 100 mM: (a) Raw data, (b) after SNV normalization applied on the whole spectral range and (c) after baseline correction from 1000 to 1100 cm -1 .

Fig5.

  Fig5. Calibration lines using (a) I p and (b) J p as indicator for different normalization methods.

Fig6.

  Fig6. (a) Calibration and (b) cross-validation lines and results using the OH bending band normalization for I p and the SNV transformation for J p . R 2 , RMSE and BIAS are given in both cases.

Figure1

  Figure1Click here to download high resolution image

Table 1 .

 1 Calibration results compared for different normalization methods. Results without normalization process (raw spectra) are also given for comparison.

	Normalization methods	R 2	Slope S
	Indicator I p
	Raw spectra I p				0.9537	1.2 × 10 2
	OH stretching band I p	OH-SB (i.e. I p	max )	0.9984	5.9 × 10 -4
	OH bending band I p	OH-BB	0.9990	3.1 × 10 -2
	SNV I p	SNV				0.9986	2.3 × 10 -3
	Indicator J p
	Raw spectra J p			0.9571	1.6 × 10 3
	Whole Area J p	max			0.9951	1.3 × 10 -5
	OH stretching band J p	OH-SB	0.9926	1.8 × 10 -5
	OH bending band J p	OH-BB	0.9980	3.8 × 10 -3
	SNV J p SNV				0.9924	3.1 × 10 -2

Table 2 .

 2 Calibration and validation results, sensitivity and repeatability for NaNO 3 solutions according the best normalization methods.

	Accuracy	Calibration		I p	OH-BB	J p SNV
		R 2	unitless	0.999	0.998
		RMSEC	mM	1.1	3.1
		BIAS	mM	0		0
		Validation		I p	OH-BB	J p SNV
		R 2	unitless	0.998	0.984
		RMSEV	mM	1.4	4.1
		BIAS	mM	0		0.5
	Sensitivity	S	10 -2 /mM	3.1	3.1
	Repeatability	CV for 30 replicates of NaNO 3 %	4.5	3.7

Table 3 .

 3 Uncertainties calculated for a nitrate solution of 10 mM according the two possible indicators

	Source of uncertainty	Standard uncertainty
		I p	OH-BB	J p SNV
	Solution preparation u sol (C)	0.1 mM	0.1 mM
	Calibration u calib (C)	0.9 mM	2.6 mM
	Instrumentation u instr (C)	0.4 mM	1.0 mM
	Combined standard uncertainty u c (C)	1 mM	2.8 mM
	Expanded uncertainty U c (C) (k = 2)	2 mM	5.6 mM