Zhenyu Lin

William Ma

Ronald H Coase

In a recent manuscript, Nsoesie et al. analysed vehicle counts in hospital parking lots and internet search trends, and suggested that the COVID-19 outbreak might have started in Wuhan, China as early as August 2019. This claim received widespread media coverage despite the lack of validation from peers. This review serves as a prepublication evaluation of the study. We identify several problems, even questionable research practices, including but not limited to: inappropriate and insufficient data, misuse and misinterpretation of statistical methods, and cherry-picking internet search terms. We also reflect on scientific publishing in a time of public emergency.

Background

The manuscript under review [START_REF] Nsoesie | Analysis of hospital traffic and search engine data in Wuhan China indicates early disease activity in the Fall of 2019[END_REF] applies digital epidemiology to determine the time of the initial outbreak of the ongoing COVID-19 pandemic caused by a novel coronavirus, SARS-CoV-2. The authors obtained vehicle counts extracted from satellite images of hospital parking lots and Baidu search trends of two symptoms related to COVID-19. In both datasets, they observe an increase in August 2019, leading the authors to suggest an outbreak of COVID-19 much earlier than the first documented case on 1 December 2019 [START_REF] Huang | Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China[END_REF].

The study is posted to DASH, a repository for researchers related to Harvard University. 1 At the time of preparing this review, this manuscript is featured at the top of "trending works" of DASH. Indeed, the study attracted widespread media coverage (e.g. [START_REF] Folmer | Satellite data suggests coronavirus may have hit China earlier: Researchers[END_REF]), even political attention, partly thanks to the self-promotion of some of the authors through social media.

There is no doubt that the manuscript has gained its popularity, and its authority might never be questioned outside the academic community. However, the merit of an academic paper can only be established through peer reviews, that is, careful evaluations by people with related expertise.

With our expertise in mathematics, statistics, and medical sciences, we consider it our duty to prepare the following review for the methods and results of the manuscript. This would hardly affect the popularity of the study, but we seek to leave a record in the literature for future reference.

Our review may overlap with some remarks made in social media and mainstream media (e.g. [START_REF] Giles | Coronavirus: Fact-checking claims it might have started in August 2019[END_REF]). We did consult the media and participated in discussions on social media. But in this review, we only include arguments that can be solidified to meet the academic standard. On the other hand, our original analysis involves advanced statistics. We make effort to simplify our explanation to reach a broader audience.

Satellite imagery

The authors selected six hospitals for their study. They claim to have excluded "sub-specialty hospitals" such as Wuhan Asias Heart. But they curiously included "Hubei Women and Children's Hospital" (湖北省妇女儿童医院), better known as "Hubei Maternity and Child Health Care Hospital" (湖北省妇幼保健院). This hospital is specialized in gynecology, obstetrics, and pediatrics, but has no pneumology for adults. It is not a logical choice for patients who develop COVID-19 related symptoms. The traffic volume at its parking lot might be more sensitive to seasonal children diseases, such as children diarrhea caused by rotavirus in the peak seasons of autumn and winter [START_REF] Zeng | Epidemiological surveillance of norovirus and rotavirus diarrhea among outpatient children in five metropolitan cities[END_REF].

Satellite images of the hospitals were obtained from the company Remote Sensing Metrics (RS Metrics). The company also delineated parking lots and streets by automated feature extraction, then identified and counted the cars. The authors claim that (word by word) "Images with tree cover, building shadow, construction and other factors that present difficulties in defining the contours were excluded since this could lead to over-or under-counting of the number of vehicles."

The authors did not publish the original images and data, apart from five figures as examples. This makes it impossible to check their counts. But this is understandable, as the amount of data could be very large.

A problem arises, however, from the media coverage of the study. The news report of the American Broadcasting Company (ABC) [START_REF] Folmer | Satellite data suggests coronavirus may have hit China earlier: Researchers[END_REF] uses several images with tree covers, building shadows, and constructions; see Figure 1. These images, also bearing the logo of RS Metrics, should have been excluded by the authors. In fact, the flawed images only exemplify how these elements could lead to extremely inaccurate counts, as the authors warn in the manuscript. Even worse, the photos are clearly taken from different angles. Some invisible areas in the 2018 photos become visible in the 2019 photos, making the count even more misleading. The captions of the flawed images make it clear that they were "provided to ABC". It is not clear if they were provided by the authors or by RS Metric.

We suggest the authors clarify that the images were not provided by them and are not used in their analysis. In particular, they need to explain how they handled the parking lot of "Wuhan Women and Children's Hospital" that was under construction near the end of 2018, as annotated on the left of the top panel of Figure 1.

For the sake of this review, we assume no wrongdoing on the part of the authors.

Analysis of vehicle count

The dataset of vehicle counts is of very poor quality. In a period of more than two years, from January 2018 to early April 2020, we count 7, 8, 10, 23, 24, and 37 data points, respectively, for the six hospitals from Figure 2(a) of the paper. Note that "Hubei Women and Children's Hospital", a specialty hospital not quite related to COVID-19, contributes 24 data points. The data points are not uniformly distributed, not only among the hospitals, but also in time. About 30 data points accumulate after April 2020, as shown in Figure 2a.

Nevertheless, the authors decide to draw a smooth curve from the scatterplot using a local regression (LOESS), which usually requires a large, densely sampled dataset to produce good results. They then observe "a steep increase in volume starting in August 2019 and culminating with a peak in December 2019" from the LOESS curve.

Validation and tests are essential for statistical analysis, as should have been taught to everyone who deals with data. The authors claim to have used "high traffic areas including the Huanan Seafood Market and two railway stations (Wuchang and Hankou) for validation", but did not give any detail.

To illustrate the problem of the authors' practice, let us make the following observation on the LOESS curve: There is a very steep decrease in volume starting in December 2019 and culminating with a valley in February 2020. Following the analysis of the authors, we may suggest that there has been a trend of social distancing starting as early as 1st December 2019. See Figure 2a.

This, however, does not match reality. The outbreak went unnoticed until the end of December 2019 [START_REF]Pneumonia of unknown cause -China[END_REF], and a new coronavirus was identified as the cause only in early January 2020 [START_REF] Wu | Severe acute respiratory syndrome coronavirus 2 isolate Wuhan-Hu-1, complete genome[END_REF]. The local government did not implement the public health lockdown until 23 January 2020 [START_REF]武汉市新型冠状病毒感染的肺炎 疫情防控指挥部通告[END_REF]. In fact, before smoothing, the data points clearly show a sudden drop in traffic volume on this date; see Figure 2a.

The problem here is the use of the smoothing method and the interpretation of the smoothed curve.

LOESS [START_REF] Loader | Local regression and likelihood[END_REF] is one of many methods that generate a smooth curve from scattered data. It is a non-parametric regression, as one does not need to specify a function of any form to fit the data. This is both an advantage and a disadvantage, depending on the situation. On the bright side, LOESS is very flexible when there is no theoretical model. However, LOESS may distort the data in several ways. In the case of a peak or valley, for example, a LOESS smoothing will decrease its height or depth but increase its width. This phenomenon is The LOESS curve (α = 40%) given in the manuscript. We point out that the data is not uniform in time, that the LOESS curve suggests an earlier decrease of volume that does not match reality, and that there are outliers to which the LOESS method is sensible. b. Reproduction of the curve using extracted data points. c. LOESS curve with α = 30% reveals two more peaks. d. LOESS curve with α = 80%, default value for the geom smooth function in the ggplot2 package, has no peak at the end of 2019. demonstrated above by the lockdown valley. Moreover, the LOESS curve is very sensitive to outliers. This is the case in the dataset under discussion (see Figure 2a). Note that one outlier is contributed by "Hubei Women and Children's Hospital", possibly caused by a seasonal children disease. Thus, the LOESS smoothed curve is merely an indicator of a trend, but never a measurement of the starting point of a peak or valley, as the authors seem to believe.

The LOESS method has two parameters: The degree of the polynomial used to locally fit the data, and the span parameter α, which can be understood as the size of the window within which the curve is locally estimated. Tuning these parameters is crucial for the quality of analysis: If the degree is small or the span parameter is too big, the data will be over-smoothed and important features will be erased. If the degree is too big or the span parameter is too small, the curve will not be smooth enough to reveal any trend. The authors used a span parameter α = 40% and appear to have used degree 1.

To check the authors' analysis, we use WebPlotDigitizer [START_REF] Rohatgi | [END_REF], a tool also used by the authors, to extract data from Figure 2a of the manuscript. The extraction is successful for data until early April 2020; data points afterward are too dense to distinguish from each other. The extracted data allow us to reproduce Figure 2a of the manuscript using the geom smooth function in the ggplot2 package of R [START_REF] Wickham | ggplot2: Elegant Graphics for Data Analysis[END_REF]; see Figure 2b. We conjecture that the authors used the same function in the same package.

However, if we change the span parameter, the smoothing curve becomes very different. With α = 30% (see Figure 2c), we observe two new peaks, one at the end of 2018, the other in the middle of 2019. They make the peak at the end of 2019 much less significant. These two peaks were not present in the LOESS curve with α = 40%, as the window is just large enough to smooth them out. With α = 80% (see Figure 2d), apparently the default span value of geom smooth, the peak at the end of 2019 is also smoothed out. We thus think that the authors are obliged to justify their choice of 40% span over other values, if not for their convenience.

In conclusion, the authors chose a smoothing method without justification, selected smoothing parameters without justification, misused the method on data of insufficient quantity and quality, and then over-interpreted the smoothing curve without any validation.

Baidu search trends

The authors obtained search volumes of the terms "cough" and "diarrhea" from the database of Baidu (http://index.baidu.com), the primary search engine in China, and observed "elevated levels of Baidu search queries" for these keywords. The search trend for the keyword "cough" coincides with yearly influenza seasons; hence the authors emphasize the keyword "diarrhea" which, according to the authors, is "a more COVID-19 specific symptom". The plot shows that the search trend for "diarrhea" is elevated since August 2019.

We failed to reproduce these data from Baidu's database using the Chinese words 腹泻 or 拉肚子, both translate to "diarrhea". In response to our inquiry, the authors revealed that the actual Chinese term they used was 腹泻的症状, which translates to "symptom of diarrhea". This should raise an alarm for several reasons.

• The authors did not report the exact Chinese keywords they used for the search trend.

As a search term, "symptom of diarrhea" sounds strange to the ear of native Chinese speakers. Baidu's search trends also confirm that "diarrhea" is usually a much more used keyword; see Figure 3 (top).

• The most common symptoms of COVID-19 reported from China were fever (88.7% during hospitalization) and cough (67.8%), while diarrhea was uncommon (3.8%) [START_REF] Guan | Clinical characteristics of coronavirus disease 2019 in China[END_REF].

According to a large-scale study in the UK [START_REF] Docherty | Features of 16,749 hospitalised UK patients with COVID-19 using the ISARIC WHO Clinical Characterisation Protocol[END_REF], diarrhea is only the 7th most common symptom of COVID-19, behind cough, fever, shortness of breath, fatigue, confusion and cough (sputum).

• The observation of the authors cannot be reproduced by usual Chinese words for "diarrhea" (see Figure 3 top), nor by other terms such as "short of breath" or "difficulty breathing" (see Figure 3 middle, also [START_REF] Giles | Coronavirus: Fact-checking claims it might have started in August 2019[END_REF]).

• Unfortunately, even the search term "symptom of diarrhea" does not support the authors' claim. The elevation in August 2019 is not unique for Wuhan but nationwide observed in China. See Figure 3 (bottom).

In conclusion, the authors failed to justify their choice of the keyword "diarrhea", did not use the correct Chinese translation, and did not check more keywords, nor the nationwide trend, to validate their claim. This can only be described as cherry-picking.

Internet epidemiology done right

The idea of using internet search trends to monitor infectious diseases is certainly interesting and deserves more attention. In fact, there has been a successful application of digital epidemiology in the COVID-19 pandemic [START_REF] Wang | WeChat, a Chinese social media, may early detect the SARS-CoV-2 outbreak in 2019[END_REF]: A recent study by Wang et al. used WeChat, the largest social media in China, to detect and track the COVID-19 outbreak in China. Instead of a few cherry-picked terms, Wang et al. made an extensive list of keywords related to the COVID-19 disease, including "SARS", "Feidian" (非典, abbreviated Chinese word for SARS), "novel coronavirus", "coronavirus", "fever", "cough", "shortness of breath", "fatigue", "diarrhea", "runny nose", "infection". For each of these terms, the trends were obtained from an index of posts from the platform of WeChat, as well as an index of searches of Baidu. The data were then analyzed and reported.

Wang et al. crosschecked their findings with the earliest reported dates of symptom onset and confirmed cases of COVID-19 in December 2019. It was found that the spikes in WeChat Index closely followed the symptom onset dates for confirmed cases. In particular, the WeChat trend for "SARS", typically used by physicians more alerted to the situation, spiked in the first three days of December, in coincidence with the symptom onset of the first confirmed COVID-19 cases [START_REF] Huang | Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China[END_REF]. The trend of "Feidian", typically used by the public, began to rise on December 15 and stayed at a relatively high level until December 29, in coincidence with 33 cases confirmed from 15 to 27 December 2019. These terms are more sensitive than the symptoms of COVID-19 used in the Harvard study because, in the very early stage of the outbreak, people did not understand the disease beyond the fact that it is similar to SARS. There is no noticeable spike in the trend of COVID-19 related terms in November.

In the manuscript under review, the authors argue that "diarrhea" is a "more COVID-19 specific symptom" as its trend "was neither seen in previous flu seasons (n)or mirrored in the cough search data". This is a very weak justification for their choice of the search term.

We now hypothesize that "shortness of breath" is a much better predictor for COVID-19 deaths. First of all, "shortness of breath" is a much more common symptom than "diarrhea" [START_REF] Docherty | Features of 16,749 hospitalised UK patients with COVID-19 using the ISARIC WHO Clinical Characterisation Protocol[END_REF]. We obtain Google Trends for "shortness of breath" from 1 March to 15 May 2020, and observe a unique trend that was neither seen in previous flu seasons nor mirrored in the "cough" or "diarrhea" search data.

To further test our hypothesis, we obtain the smoothed COVID-19 deaths data from the Institute for Health Metrics and Evaluation (IHME), a research institute at the University of Washington working in the area of global health statistics and impact evaluation. It was found that Google Trends for "shortness of breath", when adjusted for the 28-day time lag, was highly correlated with the daily COVID-19 deaths in the United States, with Pearson's R=0.96; see Figure 4. This significant association may help guide clinical management decisions.

We might have convinced many readers that our hypothesis sounds reasonable. However, careful statistical analysis and validation are necessary to ensure academic rigour. As we have no intention to extend our argument any longer in this review, we stop here and refrain from making any premature claim.

Reflections

The ongoing COVID-19 pandemic caused by the SARS-CoV-2 virus is presenting an unprecedented challenge to the scientific community. While researchers around the world strive to analyze and understand the disease, fast dissemination and circulation of research results become critical to an effective response to the public health emergency.

Many scientific journals have expedited publications of coronavirus-related submissions, but can hardly accommodate the demand. As a consequence, many scientific manuscripts are made public as "preprints" without formal peer reviews [START_REF] Kwon | How swamped preprint servers are blocking bad coronavirus research[END_REF]. This practice certainly accelerates the exchange of knowledge among researchers, as has been observed in disciplines that are long accustomed to preprints (e.g. mathematics and physics).

However, insufficient quality control facilitates studies of poor quality to sneak into the public sphere [START_REF] Dinis-Oliveira | COVID-19 research: pandemic versus "paperdemic", integrity, values and risks of the "speed science[END_REF]. At the time of preparing this review, Retraction Watch records more than 20 retracted papers about COVID-19 [START_REF]Retracted coronavirus (COVID-19) papers[END_REF]. These studies, intentionally or not, often become the source of misinformation and conspiracies that harm the global effort to battle the pandemic.

Authors of the manuscript include renowned pioneers in the new field of digital epidemiology, the use of "non-traditional" data streams such as satellite images and internet search trends for early detection of epidemics. They cited some previous works, including two perspectives [START_REF] Salathé | Influenza A (H7N9) and the importance of digital epidemiology[END_REF][START_REF] Brownstein | Digital disease detectionharnessing the Web for public health surveillance[END_REF] and a feasibility study [START_REF] Nsoesie | Monitoring disease trends using hospital traffic data from high resolution satellite imagery: a feasibility study[END_REF], to support the validity of their method.

The study under review is a bold move to use digital epidemiology to claim an event undetected by traditional surveillance. Unfortunately, it fails to maintain academic rigour nor academic integrity. The authors misused statistical tools on insufficient data and misinterpreted the result, leading to premature claims that only bring media popularity but harm academic reputation. It is a pity that a field at such an early stage already gets plagued by statistical fallacies. Moreover, the cherry-picking of search keywords for internet trends is a very questionable research practice [START_REF] Gewin | Uncovering misconduct[END_REF]. We understand that digital epidemiology suffers from insufficient data, as the authors admit in the manuscript. This does not justify the abuse of statistical methods and the suppression of evidence.

Before ending this review, we would like to express the following opinions in view of the current crisis of scientific publishing:

• Preprint repositories should all implement basic moderation.

Either before or in the early stage of the COVID-19 outbreak, many repositories (arXiv, medRxiv, ChemRxiv, etc.) implemented basic screening for plagiarisms, nonscientific contents, and papers that potentially endangers public health [START_REF] Kwon | How swamped preprint servers are blocking bad coronavirus research[END_REF]. ArXiv also removes "papers in need of significant review and revision". But this does not seem to be the case for DASH, an institutional repository of Harvard that hosts the manuscript under review.

• Peer-review should be possible to all manuscripts at every stage of publishing.

Traditionally, peer reviews are organized by journal editors after receiving the submission. This begins to change in recent years; services like PubPeer (https: //pubpeer.com/) enable peer reviews before submission or after publication. However, the manuscript under review does not have a DOI nor an arXiv ID, hence not indexed by PubPeer. This leaves us with no other option but to write a separate paper. It would be nice if PubPeer and similar services could cover more manuscripts, including those posted to institutional repositories or personal websites.

• News media should stop reporting preprints as established studies.

The purpose of preprint is to gather opinions from peers, not to draw attention from new media. Nevertheless, more and more studies are prematurely made public through media coverage. Many preprint repositories warn that preprints "should not be reported in news media as established information". We think that news media should follow this warning instead of ignore it. Moreover, media have the duty to pass on this warning to their readers when reporting preprints.

Figure 1 :

 1 Figure 1: Images used in the ABC coverage of the study [3]. One clearly sees tree covers, building shadows, and even a construction site (annotated on the left of the top panel). They should have been excluded from the study, as the authors claim. Many areas covered by obstacles in the 2018 photos become visible in the 2019 photos, leading to inaccurate and misleading counts.

Figure 2 :

 2 Figure 2: LOESS smoothing results. a.The LOESS curve (α = 40%) given in the manuscript. We point out that the data is not uniform in time, that the LOESS curve suggests an earlier decrease of volume that does not match reality, and that there are outliers to which the LOESS method is sensible. b. Reproduction of the curve using extracted data points. c. LOESS curve with α = 30% reveals two more peaks. d. LOESS curve with α = 80%, default value for the geom smooth function in the ggplot2 package, has no peak at the end of 2019.

Figure 3 :

 3 Figure 3: Baidu search trends. Top: The usual translation for "diarrhea", 腹泻, shows no elevation as claimed by the authors. It turns out that the authors used the keyword 腹泻的症状, which translates to "symptom of diarrhea". Middle: No elevation is observed before January 2020 in the trend of 呼吸困难 (difficulty breathing) or 气短 (shortness of breath). Bottom: The elevation in the trend of 腹泻的症状 (symptom of diarrhea) is more visible in the nationwide trend, showing that the trend is not Wuhan-specific.

Figure 4

 4 Figure 4

The full record once contained the entry "dc.description.version: Accepted Manuscript". It is later updated to "Author's original".

Acknowledgement

We are grateful to the following people for reading a preliminary version of our review and providing valuable feedback: Dr. Chenyu Jin from the University of Bayreuth, Dr. Housen Li from the University of Göttingen, Yan Wen from the University of Chicago, Prof. Feifang Hu from the George Washington University, Dr. Juan Qin from the University of California, Los Angeles, Dr. Qin Li and Dr. Xiao Tong from Harvard University, Dr. Tuotuo Yu from Central University of Finance and Economics, Dr. Qu Chen from Zhejiang University of Science and Technology, Sha Hua from the Wall Street Journal, Juan Chen from Yicai Global, Yanzi Deng from the South China Morning Post, Michael Anti from Caixin Globus, Muyi Xiao from the New York Times, Wanyuan Song from BBC News, and Jennifer Zhu Scott from Radian Partners.


Keywords: gravitation, elastic shocks, virtual acceleration

The ancient theory of Fatio de Duillier and Lesage on pushing gravity has been mainly criticized because of the extreme heating which would be produced in the case of inelastic shocks, supposed to be necessary to produce a gravity field. Here we investigate in an extremely simplified situation the possibility of creating a virtual acceleration with purely elastic repeated shocks and we derive some estimates on the mass and velocity of gravitons based on this model.

Introduction

The cause of gravitational force has become a basic question right after the discovery by Newton of the gravitational field produced by matter. Around that time, Fatio de Duillier, then Lesage, cf. e.g. [START_REF] Chabot | a theoretician of gravitation in search of legitimacy[END_REF][START_REF] Chabot | Nombres et approximations dans la théorie de la gravitation de Lesage[END_REF][START_REF] Edwards | Pushing gravity: New perspectives on Lesage's theory of gravitation Apeiron[END_REF][START_REF] Edwards | Photon-graviton recycling as cause of gravitation[END_REF]) formulated the idea that gravity might be the result of interaction of matter with tiny unseen particles, qualified as "ultra-mundane", pushing, as a consequence of a mutual 3D shield effect, any pair of massive objects towards each other. At that time the atomic theory of matter was not found yet, but today one might think, if we follow this theory, that the "gravitational mass" of a material object is determined by the number of nucleons (protons+ neutrons), since it is not clear at all how the electronic cloud might interact with the particles. Fortunately, the "inertial mass" of an electron is inferior to that of a nucleon by 3 orders of magnitude.

It was explained in [START_REF] Haraux | About dark matter and gravitation[END_REF] how the pushing gravity might provide a solution to the missing mass enigma. It is therefore of interest to try to understand the mechanism leading to the emergence of gravitational acceleration in the case of a single nucleon bombed by ultra-mundane corpuscles in the way assumed by pushing gravity. If no object is interposed in the trajectory of corpuscules, the nucleon will not undergo any notable force, or more precisely the force will change sign at a very small time scale, so that no motion of the nucleon will be detectable. On the other hand it is quite interesting to examine what happens when a single direction is blocked (for instance by another nucleon) on one side only. This short note is devoted to that purely mathematical, physically unrealistic toy model that might however give some relevant quantitative information about the possibility of pushing gravity in a purely elastic framework, a choice which has been rejected until now by all specialists with the argument that no gravitational field would emerge in this context.

Elastic shock of two particles with colinear velocities.

Although this is a simple and rare case, it is interesting, for the very simplified model that we shall consider, to recall the following well known result concerning the purely elastic shock of two point masses. Consider two particles with respective masses m 1 , m 2 , and velocities u 1 , u 2 before collision, v 1 , v 2 after collision. The equality of the total linear momentum before and after the collision is expressed by:

m 1 u 1 + m 2 u 2 = m 1 v 1 + m 2 v 2 .
Because the shock is perfectly elastic, the conservation of the total kinetic energy gives :

1 2 m 1 u 2 1 + 1 2 m 2 u 2 2 = 1 2 m 1 v 2 1 + 1 2 m 2 v 2 2 .
These equations may be solved directly to find v 1 , v 2 when u 1 , u 2 are known:

v 1 = m 1 -m 2 m 1 + m 2 u 1 + 2m 2 m 1 + m 2 u 2 (2.1) v 2 = 2m 1 m 1 + m 2 u 1 + m 2 -m 1 m 1 + m 2 u 2 (2.2)
3 A very simple toy model.

We now consider a nucleon, assimilated to a punctual mass M, and a punctual "graviton" of much smaller mass m, both particles being on the real line, with for instance the nucleon placed at the origin. The nucleon has an initial speed v 0 and the graviton meets the nucleon with initial speed u before the shock. For the moment we do not make any hypothesis on the signs and we are only interested in the velocity v 1 of the nucleon after the shock. In the previous set of formulas we therefore take m 1 = M , m 2 = m, u 1 = v 0 and u 2 = u. From the first formula (2.1) we find, mutatis mutandis

v 1 = M -m M + m v 0 + 2m M + m u (3.3)
3.1 Analysis of the first shock.

Since we want to analyse the effect of successive shocks with the gravitons which are supposed to be high speed particles, we may assume that v 0 = 0 . In this case we find

v 1 = 2m M + m u (3.4)
showing that the velocity of the incoming graviton is transferred to the nucleon with a reduction factor 2m M +m . The incoming graviton "rebounds" with a velocity

v 2 = m -M m + M u
in the opposite direction, the absolute velocity undergoing a relative loss of 2m M +m . So the gravitons lose a fixed proportion of their energy each time they collide with a motionless nucleon. But this does not mean anything from a global point of view since they might regain energy when they bump on another nucleon with a velocity opposite to theirs. This is consistent with the conservation of total energy .

Analysis of the completely isolated case.

One important objection against the Duillier-Lesage pushing gravity theory is the fact that in the "free directions", without obstacles for gravitons, the absence of motion is the result of an equilibrium between forward and backward shocks, those shocks giving rise to inefficient forces which cancel each other and may nevertheless produce a very high temperature. In the case of perfectly elastic shocks this objection vanishes. A simple way to understand that is to consider the simplest case where back and forth shocks are systematically alternated. Of course the alternance of forward and backward shocks can be different with an arbitrary level of complexity in the successions, but intuitively the result will be similar in the long run if the signs are reasonably distributed as expected from the random character of incoming particles. In the simplest case, assuming that all gravitons have speeds ±v, we are led to study the inductive sequence

∀k ≥ 0, v k+1 = M -m M + m v k + (-1) k 2m M + m v (3.5)
with v 0 = 0. Setting

v k = 2m M + m vw k ; ρ = M -m M + m we are reduced to the simple relation ∀k ≥ 0, w k+1 = ρw k + (-1) k ; w 0 = 0 (3.6)
whose solution is immediate

w 1 = 1, w 2 = -1 + ρ, w 3 = 1 -ρ + ρ 2 , ...w k = (-1) k+1 k-1 j=0 (-ρ) j = (-1) k+1 1 -(-ρ) k 1 + ρ
As a consequence we obtain

v k = (-1) k+1 1 -(-ρ) k 1 + ρ 2m M + m v
showing that the nucleon is bound to perform small oscillations of size less than 2m M +m |v|. In practice such oscillations will be indiscernable from rest. We can say that the gravitons have no noticeable effect on an isolated nucleon, or more generally produce no significant motion of the nucleon in the directions where no obstacle comes to perturb the incoming particles.

The case of a unilateral obstacle.

Let us now consider the case where an obstacle (for instance another nucleon) is placed on the right of our original nucleon, prohibiting shocks with gravitons coming from the right. Now incoming gravitons come from the left as a flux of particles with identical speeds u = V > 0 which we can, for simplicity, assume separated by a fixed (small) interval of time h > 0. The gravitons reach the origin at time kh and they rebound against the nucleon at some time t k > kh (to be computed later) since this particle has been pushed forward by the previous shocks. Then if no other force comes to, constrain the nucleon, the succesive velocities

v k = v(t k ) are given by ∀k ≥ 0, v k+1 = M -m M + m v k + 2m M + m V (3.7)
Now the same reduction as previously gives ∀k ≥ 0, w k+1 = ρw k + 1; w 0 = 0 (3.8) which provides

w k = k-1 j=0 ρ j = 1 -ρ k 1 -ρ and finally v k = (1 -ρ k )V (3.9)
This looks at first sight contrary to our ambition of justifying the emergence of a gravitational acceleration. Indeed from (3.9) it follows at once that

lim k→∞ v k = V (3.10) and lim k→∞ (v k+1 -v k ) = 0 (3.11)
precluding once and for all the existence of a fixed induced acceleration for very large times.

A fundamental remark.

But this is not the end of the story! First we observe that (3.10) was predictable. Indeed the velocity is increasing after each shock, and if it falls above V, no shock with a graviton coming from the left is possible any longer. Easy to understand ... afterwards. Then we observe that from the equation

v k+1 -v k = 2m M + m ρ k+1 V
Between the sucessive times of impact t k and t k+1 the derivative suddenly jumps from v k to v k+1 , displaying a "virtual acceleration "

v k+1 -v k t k+1 -t k = 2mV M + m × ρ k+1 t k+1 -t k .
As mentioned previously the time interval t k+1 -t k is not exactly equal to kh. In order to compute the times t k we observe that at time t k+1 the abscissa of the nucleon has become

x k+1 = t 1 v 1 + (t 2 -t 1 )v 2 + ...(t k+1 -t k )v k+1 so that [t k+1 -(k + 1)h]V = t 1 v 1 + (t 2 -t 1 )v 2 + ...(t k+1 -t k )v k+1 Similarly we have [t k -kh]V = t 1 v 1 + (t 2 -t 1 )v 2 + ...(t k -t k-1 )v k
and by subtracting both sides of these inequalities we find

[t k+1 -t k -h]V = (t k+1 -t k )v k+1 = (t k+1 -t k )(1 -ρ k+1 )V
and finally

t k+1 -t k = h ρ k+1
Incidentally this yields the formula

t k = h k 1 1 ρ j = 1 -ρ k ρ k -ρ k+1 h
which tends to kh as ρ → 1, but the main point here is the formula

v k+1 -v k t k+1 -t k = 2mV M + m × ρ 2(k+1) h .
displaying a "virtual acceleration "

γ ∼ 2m M V h as long as k is not too large because ρ is very close to 1 provided m/M is sufficiently small. This suggest that me may recover a gravitational force if the gravitons have sufficiently small masses and V is sufficiently large. Quantifying this idea with real data is the object of next section.

Some estimates

We have

ρ = 1 - 2m M + m > 1 - 2m M
If we wish for instance to secure the inequality ρ k > e -1/1000 ∼ 0, 999 which seems to be comfortable, we require

kln(1 - 2m M ) > -1/1000
which amounts essentially to 2m M < 10 -3 1 k We would like to reach for k, to fix the ideas, the total number of possible shocks spaced in time by the interval h during a time corresponding to the currently estimated age of the universe. Therefore we choose, just to see what it will imply

k = A/h
where h is the fraction of a second separating two shocks and A is the age of universe in seconds. this gives the sufficient condition

m M < 10 -21 h
For instance if we assume a flux of 1000 gravitons per second, we must assume that the gravitons are 10 -24 times lighter than a nucleon. This is very small , much smaller than the present estimate for the mass of neutrinos. But why not?

Now what about the velocity V ? The virtual acceleration undergone by the nucleon as a consequence of shocks is

γ ∼ 2m M h V
It seems reasonable to compare this value of γ with the acceleration that would be produced, via Newton, by the interposition of a nucleon on the right of the given one, at the minimal possible distance, corresponding to the radius of an average atom, about one angstrom or 10 -10 m. That acceleration is about 10 -17 m/s 2 . Assuming for instance that m/M has the highest permitted value 10 -21 h we find 2 × 10 -21 V > 10 -17 m/s, thus V > 10 4 m/s is enough.

Even classical sub-luminic particles can fulfill the conditions. On the other hand, there are motions apparently driven by gravitation which are much faster than 10 4 m/s, so logically the velocity of gravitons, if they exist, is much higher than our lower estimate.

Concluding remarks

If our simplified punctual 1D model has some relevance, for macroscopic objects, the "virtual acceleration" will become an acceleration in the classical sense as a consequence of averaging effects. In addition, in reality, even for one nucleon, the "jump" from the velocity v k to v k+1 is certainly not instantaneous. Now the most important, and maybe surprising, information obtained in the above calculations is that, in order for a pushing gravity to function, the gravitons must have a mass smaller than 10 -20 times the mass of a nucleon. In addition, the model suggests an exponential exhaustion of the gravitational forces for very large time, even if the decrement is extremely weak. At this point, we must insist on the fact that the necessity of a very small mass for gravitons results, in our calculations, from the hypothesis that the gravitational affects remained essentially constant in the past 10 10 years, which is not at all a certitude. Moreover, in space, the massive objects tend to move, which might have the effect of "reinitializing " their position with respect to the flux of gravitons. As a conclusion, the gravitons might be heavier than what was computed, but in such a case one should be able to detect an important decay of the gravitational constant over very long periods of time. This is, for instance, not contradictory with the evolution scheme based on the Big Bang.

Now we must admit that this very simple toy model is, of course, far from reality, since we condensed all directions into one to see what happens. In many cases of gravitational attraction, the obstacle is viewed from the attracted object under a very small angle, but it is not always the case and in order to rehabilitate the Fatio-Lesage pushing gravity theory in a purely elastic framework, much more has to be done. It seems to be quite difficult to build a continuous model that would allow macroscopic calculations giving more credit to the hypothesis of pushing gravity. And recovering rigorously Newton's formulas on a discrete model is probably even more difficult, with the possible need of powerful computers as a guide to a full understanding of the situation. Finally it is clear that in order to appraise the gravitational constant as a function of the flux of gravitons, the mass and the velocity, a three dimensional model has to be devised.


N B Roozen

Q Leclère

C Glorieux

M Rychtáriková

A global error estimator for the uncertainty of a multi-channel spectral analysis

Keywords: uncertainty, multi-channel, spectral analysis PACS: 43.20.-f, 43.20.+g, 43.40

A global estimator of the uncertainty of the average frequency response function in multi-channel spectral analysis measurements is proposed. The proposed global estimator is a generalization of the random error estimator of the frequency response function magnitude of a single-input-single-output system. In principle, the signal-to-noise ratio (and thus the quality of the frequency response function estimation) is increasing with increasing number of averages M , according to √ M . However, in the situation that, for practical reasons, there is a maximum imposed upon the total measurement time T max , it is clear that there is a trade-off between the number of averages M and the record length T [s] that is used to obtain an estimate of a single-averagefrequency-response-function. There is a choice between a few long records or many short records, with the requirement that, assuming zero overlap, the number of averages M times the record length T may not exceed the total available measurement time, i.e. M × T ≤ T max . In addition to the existence of such an optimum, a minimum record length is required as well which is related to the reverberation time of the system. The newly proposed global estimator is used to determine the optimal record length of a multichannel system, such that a minimum error of the average frequency response function is obtained. It is also shown by experimental results that indeed the

Introduction

In this article a global estimator is proposed. For demonstration purposes the estimator is applied to the multi-channel spectral analysis of a vibrating building element. The measurements were carried out by means of a scanning laser Doppler vibrometer and were intended to determine the radiated sound power at low frequencies (as described in more detail in [START_REF] Roozen | Determination of radiated sound power of building structures by means of scanning laser Doppler vibrometry[END_REF]). An interesting complicating factor in this experiment is that the building element is excited by a reverberant acoustic field, which exhibits a random characteristic in time. By looking for a minimum of the proposed global estimator, an optimum record length is sought, for a given reverberation time of the room. This is a classical problem, discussed by Piersol in 1978 [START_REF] Piersol | Use of coherence and phase data between two receivers in evaluation of noise environments[END_REF], and later also by Jacobsen and Nielsen in 1987 [START_REF] Jacobsen | Spatial correlation and coherence in a reverberant sound field[END_REF], where it was concluded that the analysis record lengths should be at least as long as the reverberation time to avoid serious errors in the coherence estimate. The measurements also indeed demonstrate nicely that the minimum record length is required to be longer than the reverberation time of the system.

Theory

A multiple-channel-output system is considered as illustrated in Figure 1. A building element is excited by an acoustic field, as shown, resulting in a response being measured at different positions, i = 1...N . An additional position r is being probed simultaneously with each of the other positions, thus serving as a reference, allowing the responses at positions i = 1...N to be measured sequentially in time, retaining the phase information of the response at the individual positions (assuming the building element to respond stationary and time-independent). As we assume the exciting acoustic field to be caused by a single sound source (loudspeaker), we are dealing with a single-input-multiple-output system, for which a single reference point is sufficient. In case of a multiple-input-multiple-output system more reference points will be necessary. In the following the auto-spectra of the response of the structure at different positions is denoted by S ii , with i = 1...N , the auto-spectrum of the reference signal is denoted by S rr , and the cross-spectra between the response of the structure at different positions and the reference signal are denoted by S ir , with i = 1...N . The frequency response function between the response of the structure at different positions and the one of some reference signal can then be estimated using the H 1 -optimal estimate or the H 2 -optimal estimate, respectively [START_REF] Bendat | Engineering Applications of Correlation and Spectral Analysis[END_REF]:

H 1 ir = S ri S rr (1)
H 2 ir = S ii S ir (2)
It is well known that measurement noise is better suppressed when averaging cross-spectra, compared to averaging auto-spectra [START_REF] Bendat | Engineering Applications of Correlation and Spectral Analysis[END_REF]5,[START_REF] Randall | Frequency analysis[END_REF]. Taking a cross-spectrum enhances common frequency components in both signals that have a consistent phase relationship. Uncorrelated signal components average to zero. Since auto-spectra are positive valued, summing them accumulates not only correlated signal components, but also uncorrelated ones.

In view of this, as the estimate of frequency response functions is based upon both cross-spectra and auto-spectra, it is best to choose the H-estimate with the least noisy autocorrelation factor, i.e. use H 1 in case less noise is expected on the reference auto-spectrum S rr and use H 2 if less noise is expected on the response auto-spectra S ii .

Global coherence and global ǫ

A quantity that plays an important role in the statistical uncertainty of the averaged frequency response function of a single-input-single-output system is the coherence of the frequency response function, which is defined as [START_REF] Roozen | Determination of radiated sound power of building structures by means of scanning laser Doppler vibrometry[END_REF]

γ ri = |S ri | 2 S rr S ii (3)
Given the coherences γ ri related to each frequency response function H ri , the normalized standard deviations due to random errors in single-inputsingle-output systems are given by [START_REF] Roozen | Determination of radiated sound power of building structures by means of scanning laser Doppler vibrometry[END_REF]:

ǫ H ri = 1 √ 2M 1 -γ ri γ ri ( 4
)
where M is the number of averages.

As mentioned above, the goal of this paper is to introduce a global coherence function that considers the quality of all measurements simultaneously, as well as a global statistical uncertainty on the average frequency response function. As the coherence function γ ri as defined above can be seen as the ratio between the energy in the auto-spectrum of signal i that is coherent with the reference signal, γ ri S ii , and the total energy in the the auto-spectrum of signal i, S ii , we propose the following expression for the global coherence:

γ global = γ ri S ii N S ii N (5)
where .. N denotes the average over all N measurement points. The thus defined global coherence can be viewed as the ratio between the mean coherent auto-spectrum by the mean raw auto-spectrum.

In addition, we propose an estimate for the statistical uncertainty of the average frequency response function as follows:

ǫ global H ri = ǫ 2 H ri |H 2 ri | N N |H ri | 2 N = ǫ 2 H ri |H ri | 2 N N |H ri | N (6)
Indeed, the variance of the modulus of the average frequency response function is given by

V{ |H ri | N } = 1 N V{|H ri |} N (7)
while

V{|H ri |} = |H ri | 2 ǫ 2 H ri (8)
Hence, assuming a gaussian distribution, it can be stated with 95% certainty that the actual value of

|H ri | N lies between |H ri | N -2ǫ global H ri |H ri | N and |H ri | N + 2ǫ global H ri |H ri | N .
Note that in [START_REF] Roozen | Determination of radiated sound power of building structures by means of scanning laser Doppler vibrometry[END_REF] (section 8.2) a multiple coherence function was defined for a system with multiple inputs and a single output, which provides a measure of the linear dependence between a collection of inputs and an output, independent of the correlation among the inputs. However, to the best of our knowledge, a global coherence as defined above for a single input multiple output system, has not been proposed before. The same is true for the global error of the average of the frequency response function.

The multi-pass estimator

In the situation that the system being investigated responds stationary and time-independent, the frequency response functions H ri , i = 1...N , can be measured sequentially in time using a multi-pass measurement scheme, which puts less heavy demands on the measurement and data acquisition system in terms of the required number of channels. Assuming that a H 1 -type of estimate of the frequency response functions is required, a fixed reference signal, here denoted as the signal measured at point r, should be used in order to retain the phase information at each point i = 1...N , using Equation (1) for the estimation of the frequency response function:

H 1 ri = S ri S rr (9)
By estimating the frequency response function between this reference signal and the response of the structure at different positions, being measured sequentially in time, the phase between the response at the different measurement points and the reference can be determined, and thus also the phase between the individual responses.

A particularity of such a multi-pass measurement scheme is that the reference signal is acquired at each measurement pass, whilst all other signals are recorded during one pass only. Having N measurement passes to determine the N components of the frequency response functions H ri , i = 1...N ,we thus have N times an estimate of the reference auto-spectrum S rr(i) , with i = 1...N .

In this particular case, it is advantageous to estimate the auto-spectrum of the reference signal S rr as the averaged auto-spectrum of all sequentially in time determined auto-spectra S rr(i) , with i = 1...N , i.e.:

S rr N = 1 N N i=1 S rr(i) (10)
Taking the average auto-spectum of the input increases the signal-tonoise-ratio of the reference auto-spectrum with a factor √ N . In combination with the frequency response functions for different locations i calculated from respective auto-spectra and cross-spectra on the basis of records of limited duration, the averaged structural response function X i at each individual point i, can be estimated as:

X i = H 1 ri S rr N (11)
This method is called multi-pass acquisition [START_REF] Leclere | Multi-channel spectral analysis of multi-pass acquisition measurements[END_REF].

Provided the noise contributions are uncorrelated, the error in the estimate of S rr N is a factor √ N smaller than the error in the estimate of S rr , with obvious advantages for the accuracy of the X i estimate.

Practical application

In this section the multi-pass acquisition is used in a practical application, with the aim of illustrating its adequacy. In addition, the proposed global estimator ǫ global H ri is used to determine the optimal record length for the measurement, assuming a maximum allowable measurement time T max of 100 [s], thus finding the best compromise between the number of averages M and the record length T [s]. In addition, the minimum record length is being studied by the same estimator ǫ global H ri . In the considered experiment, we determine the response of a light-weight double wall panel that is subjected to a random, diffuse acoustic field at one side. The building element is mounted in a transmission test facility as illustrated in Figure 1.

The transmission rooms are of identical shape, with a slightly inclined test partition (see Figure 1), and with a volume V = 88.78 m 3 each and a total interior surface S = 121.15 m 2 each. The Schroeder frequency f s of the receiving room is approximately f s = c 3/2 T 60 log 10 (e)/4V = 350 Hz, assuming a reverberation time of 2.5 [s] (using the averaged T 20 , see also Figure 8(a)) and a propagation speed of sound c of 340 [m/s]. A light-weight double wall panel was mounted in the opening between both rooms, with dimensions 3.27 x 2.98 m 2 . The sound field in the sending room is generated by two sound sources emitting random noise. Its response was measured at N =145 different measurement positions (using two interleaved meshes of 8x8 and 9x9), as shown in Figure 2. 1). Strips of retro-reflecting material (145 in total) are visible to enhance the optical reflection on a grid of locations on the wall, as required for the scanning laser vibrometry measurements.

Given our ambition to measure the complex (amplitude and phase) vibrational behavior of a structure subjected to a reverberant acoustic field, rather than just estimating rough amplitude levels, special care needed to be taken to acquire and process the signals. Given the diffuse character of the sound fields in both rooms, the vibration of the building element has very complex, random looking characteristics in time. Nevertheless, as the response can be assumed to be linear, each acquired response constitutes of the same modal responses. As a result, whilst the temporal responses have a random nature, the spatial response should be coherent. In order to obtain a reasonable estimate of the spatially coherent response of the building element, the required measurement record lengths need to be longer than the (mechanical or acoustical) reverberation time of the acousto-mechanical system involved (in this case: the transmission chambers, as these reverberation times are much longer as compared to the mechanical reverberation time of the panel).

Indeed, as mentioned by Piersol [START_REF] Piersol | Use of coherence and phase data between two receivers in evaluation of noise environments[END_REF] in 1978, the analysis record lengths should be at least as long as the reverberation time to avoid serious errors in coherence measurements in a reverberant acoustic field. Obviously this also applies to the measurement of the vibrational response of a building element subjected to a reverberant acoustic field. In a publication by Jacobsen [START_REF] Jacobsen | Spatial correlation and coherence in a reverberant sound field[END_REF], it is mentioned that the sound field in a reverberant room can be considered fully coherent if the spectral bandwidth ∆f is less than 2.2/T 60 , and in a later publication by Jacobsen [START_REF] Jacobsen | The coherence of reverberant sound fields[END_REF] if the spectral spectral bandwidth ∆f is less than 1/T 60 , where T 60 is the reverberation time of the room. This requirement corresponds to a record length T = 1/∆f of at least equal to or larger than T 60 .

The reason for this requirement is that, following Piersol [START_REF] Piersol | Use of coherence and phase data between two receivers in evaluation of noise environments[END_REF], if there is a finite propagation time between two receivers, the resulting coherence measurement will include a bias error of the form

γ2 xy (f ) = γ 2 xy (f ) [1 -τ /T ] 2 (12)
where γ2 xy (f ) is the measured coherence between the two signals x and y, γ 2 xy (f ) is the true coherence, τ is the propagation time between the receivers and T is the record length. Measurements in a reverberant acoustic field contain multiple reflections, with different propagation times, τ i , i=1,2,3,. . . . The maximum τ i that delivers a significant contribution to the measured signal is a function of the reverberation time of the room T 60 . The same statement holds for the response of a mechanical structure that is excited by a reverberant field. Thus, in order to be able to estimate the true value of the coherence the record length T should at least be as long as the reverberation time of the room T 60 . Jacobsen [START_REF] Jacobsen | Spatial correlation and coherence in a reverberant sound field[END_REF][START_REF] Jacobsen | The coherence of reverberant sound fields[END_REF] draws the same conclusion in his papers. Another way of explaining the requirement that the record length T should at least be as long as the reverberation time of the room T 60 when dealing with a structure that is excited by a reverberant field is in terms of frequency response function and transmissibility measurements. In the present situation we measure the response y i at point i whilst using a reference signal y r at a fixed position r. Both signals are the response due to a reverberant acoustic field. If we denote the acoustic excitation by x we can write

y i = H i x y r = H r x (13)
where H i and H r are the transfer functions between an acoustic source at a specific position in the (reverberant) sending room and the vibration of the plate at points i and r, respectively. We are interested in the estimation of the transmissibility y i /y r , which is equal to H i /H r (in practice being estimated by means of auto-and cross-spectra, Eq. 9). The impulse responses H i and H r have a length that is approximately equal to T 60 , which can be truncated at t = T 60 without important loss of signal quality. This again results in the requirement that T needs to be larger than T 60 in order to be able to obtain a correct estimate of the transmissibility y i /y r , and to obtain an unbiased estimate of the coherence. For the experimental determination of the coherent spatial response of the building element, we made use of a laser Doppler vibrometer (LDV), a Polytec OFV-505 sensor head and a Polytec OFV-5000 controller. The phase information in the signals was retained by synchronously acquiring the reference signal of an accelerometer (Bruel&Kjaer type 4343 amplified with a Bruel&Kjaer charge amplifier type 2635) placed on the building element approximately 20 cm from the bottom edge of the send-side-panel.

The coherent, structural response of the building element, including the phase relations between the measurement points, was estimated by means of the multi-pass estimator [START_REF] Bendat | Engineering Applications of Correlation and Spectral Analysis[END_REF].

The quality of the determined low-frequency response of the building element to the reverberant sound field was assessed for different record lengths. In practice this was done by recording, for each measurement position, the two channels (the scanning laser vibrometer signal and fixed acceleration signal) in time-domain during 100 seconds using a sampling frequency of 20000 Hz, which was downsampled to 1000 Hz. Thus the practically imposed maximum total measurement time T max equals 100 seconds. The effect of varying record length T was assessed by considering parts (sub-records) of that total measurement record.

As a first step, the auto-and cross-spectra for all 145 measurement positions were estimated using a considered record length of 20 seconds (resulting in a frequency resolution of 0.05 Hz), taking 8 averages with 50% overlap (thus exploiting the full data record of 100 [s]). The auto-spectra are shown in Figure 3. The auto-spectra of the (fixed) reference signal, as shown in Figure 3(a), is rather noisy, as expected. Using the multi-pass approach the auto-spectrum of the reference signal is determined by averaging the N estimates of the auto-spectrum of the reference signal, yielding S rr N , with N =145. This led to a substantial quality improvement (in theory a factor √ N , [START_REF] Bendat | Engineering Applications of Correlation and Spectral Analysis[END_REF]) of S rr and consequently also of H ri . The final step in the data processing was to determine the individual frequency response functions for each measurement point relative to the reference signal, and calculate from these the respective response functions, by scaling with the square root estimate of S rr N , using Equation (11). The resulting velocity spectrum X i at an example point and the corresponding ordinary coherence (as defined in Equation ( 3)) are shown in Figure 4. The ordinary coherence at this example point, as well as at other points, is relatively high in the frequency range between 40 and 115 Hz. Below 40 Hz, the response of the used loudspeakers was weak. Above 115 Hz, the double wall insulated rather well, which, combined with the fact that the laser vibrometer measures velocity (and not acceleration), led to lower signal levels resulting in lower coherences. We refer to Figure 5 for the relationship between the record length T and the number of averages M using different overlap ratios. From this figure it can be seen that the number of averages M increase significantly with increasing overlap ratio's, which would appear to be interesting from a data processing point of view to increase the quality of the estimators in a cheap manner. However, it is shown in Appendix A that for randomly (Gausian) generated signals, using a Hanning window, an overlap greater than 50% does not decrease the variance of spectral estimators further, whilst Bendat's 1/ √ M -rule would suggest a further reduction. For instance, in the given situation, increasing the overlap from 50% to 99% would increase the number of averages from 8 to 400 (using a record length of 20 [s]; see Figure 5) and thus would the variance in theory be reduced by a factor 400/8 = 7. However, this predicted reduction of the variance is artificial. For that reason, in the remainder of this paper an overlap ratio of 50% is used, to ensure that the estimates for ǫ global H ri as given in Eq. ( 6) will be valid. The global coherence, as defined by Eq. 5, is depicted in Figure 6 as function of both frequency and record length. It can be seen that the global coherence is significantly lower than 1 for record lengths shorter than 1 or 2 seconds, which is logical as for these short acquisition times the signals are non-correlated; the measurement time is shorter as compared to the reverberation time of the rooms. For large sub-record lengths the coherence is close to 1. Also notice that for record lengths longer than about 70 [s], the coherence and the global coherence is 1 for all frequencies. This is because in that case (with 50% overlap) only one averages is taken (see also Figure 6), which yields a coherence of 1 by definition. Another issue that should be mentioned here is that for small values of the number of averages M the coherence value will be overestimated because not having converged to its actual value. This point is especially important because Eq. 4 is based on the actual value of the coherence, that in practice needs to be replaced by it's estimation. When the number of averages are high enough, this is allowed. But if this estimation has not converged, the coherence will be overestimated. Concluding, one should be cautious when interpreting coherence estimates and consequently also the proposed global coherence when the number of averages are low. The estimator will be reliable for values of approximately M ≥ 10, which corresponds to a record length T < 18 [s] for the present case (T max = 100 [s], 50% overlap, see Figure 5). Fortunately, the global coherence is already rather high for record lengths smaller than 10 [s].

The global error of the average of the response functions ǫ global H ri as de-fined by Eq. 6 is depicted in Figure 7 as function of both frequency and record length. However, the same aspect as mentioned above also applies here. When the number of averages M is small the coherence values will be overestimated. The overestimation of the (global) coherence will lead to an underestimation of the uncertainty parameter as proposed by Eq. 6. Thus, the results shown in Figure 7 should be interpreted with some caution for record lengths larger than T = 18 [s] for the present case, assuming that at least M ≥ 10 averages are required. In order to enable a more global view on these results for all frequencies simultaneously, the ǫ global was normalized to 1 for each frequency line individually (Figure 8(b)). The reverberation time (T 10 , T 20 , T 30 and EDT ) of the receiving room (Figure 8(a)) was determined from the measured room impulse responses (ISO 3282) using Dirac R software with exponential sweeps as testing signal.

The optimum record length, as indicated by the minimum of the normalized ǫ global , is of the same order of magnitude as the reverberation time of the transmission room (Figure 8(b)). The reverberation time of the transmission room is in this case determining the minimum record length, as the mechanical reverberation time of the building element is much lower.

Conclusions

A global coherence parameter and a global estimate parameter for the statistical uncertainty of the average frequency response function of a multi- channel structural-acoustic system is proposed. The proposed parameters are applied in an experimental test to assess the global uncertainty of the measured frequency response functions, and it is shown that the minimum record length needs to be larger than the reverberation time of the structuralacoustic system to avoid serious errors in the frequency response function estimate. In addition an optimal choice between the record length and the number of averages, for a fixed maximum measurement time, is determined based upon the suggested global parameters.

Figure 1 :

 1 Figure 1: Test facility measurement set-up.

Figure 2 :

 2 Figure 2: Double wall mounted in transmission test facility (as illustrated in Figure1). Strips of retro-reflecting material (145 in total) are visible to enhance the optical reflection on a grid of locations on the wall, as required for the scanning laser vibrometry measurements.

  Reference accelerometer autospectra S rr (in grey) and auto-spectra S rr N , with N =145 (in blue). Laser vibrometer auto-spectra S ii .

Figure 3 :

 3 Figure 3: Auto-spectra of reference accelerometer (fixed position) and laser vibrometer at 145 different measurement positions at receiving side, using a record length T =20[s].

Figure 4 :

 4 Figure 4: Example of the multi-pass estimator for the velocity at a single point (point number 1, at the left bottom of the building element), using a record length T =20[s].

Figure 5 :

 5 Figure5: Dependence of the number of averages and overlap ratio, for a fixed total measurement time T max of 100 seconds. Solid line: 0% overlap, dashed line: 50% overlap, dash-dotted line:75% overlap, dotted line: 90% overlap, diamonds: 99% overlap.

  3D view. (b) Color coded top view.

Figure 6 :

 6 Figure 6: Global coherence γ global as function of frequency and as function of the record length, using 50% overlap.

Figure 7 :

 7 Figure 7: Global ǫ global Hri as function of frequency and as function of the record length, using 50% overlap.

  Reverberation time of the receiving room, EDT (green dashed curve), T 10 (blue full curve), T 20 (red dash-dot curve), T 30 (light green dotted curve).

Frequency

Figure 8 :

 8 Figure 8: Measured reverberation time of the transmission test facility room and global ǫ global Hri normalized for each frequency line separately, using 50% overlap, as function of frequency and as function of the record length.

  .u.](a) Variance versus overlap ratio, using a Hanning window.

  .u.] (b) Variance versus overlap ratio, using a uniform window.

  Variance versus number of averages, using a Hanning window.

  Variance versus number of averages, using a uniform window.

Figure A. 9 :

 9 Figure A.9: Actual variance (solid blue curve) and theoretically predicted variance (dashed red curve) for a numerically generated signal with a normal distribution, total signal length 600000, FFT record length n = 100000.

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http://www.sciencedirect.com/science/article/pii/S0003682X14001583 Applied Acoustics 87 (2015) 57-63

Appendix A.

The aim of this appendix is to show that the usable amount of overlap is limited in practice. Exceeding a certain overlap ratio will not give better estimates.

It is known that the variance of the auto-spectrum of a gaussian signal averaged over M time windows is proportional to 1/M [START_REF] Bendat | Engineering Applications of Correlation and Spectral Analysis[END_REF]. Considering a signal x with a rms value equal to x, the variance of the double sided autospectrum S xx is given by (cf. [START_REF] Leclère | On the use of the H s estimator for the experimental assessment of transmissibility matrices[END_REF]) :

where n is the number of points of the FFT.

To illustrate the effect of overlap on the variance of the auto-spectrum of a signal, we consider a numerically generated signal with a normal distribution. The digital signal has a length of 600000. The Fourier transform is determined for this signal with a length n = 100000. Without overlap M = 6 averages could be taken, whilst with 50% overlap M = 11 averages could be take and with 99% overlap M = 500 averages.

According to the theory above, the variance should be inversely proportional to the number of averages taken. If we estimate the variance of the auto-spectrum, which resulted from the averaging process by calculating the variance over the n points of the averaged FFT-result, it corresponds well to the theoretical curve up to a certain level of the overlap ratio. But for the higher overlap ratios the theoretical curve does not correspond to the actual variance, the latter exhibiting a kind of saturation, as shown in Figure A.9. When using a Hanning window, the theoretical curve corresponds to the actual variance curve reasonably well up to about 50% (see Figure 9(a)). Plotting the variance as function of the number of averages, obviously gives a straight line on a log-log scale for the theoretical curve (see Figure 9(c)).

It's interesting to see that the cross-over depends upon the window type being used. For instance, although a rectangular window is not recommended for stochastic signals, the theoretical curve follows the actual variance curve up to about 20% when such a window is used (see Figure 9(b) and 9(d)).