N B O U K H D O U D

R G R O S A

Effect of agricultural practices and coastal constraints on soil microbial functional properties in Mediterranean olive orchards

This research examines how agricultural practices commonly used in Lebanese olive tree orchards (no-tillage, co-culture with Vicia sativa L. var. sativa and conventional tillage) affect soil microbial and chemical characteristics, and whether this depends on geographical context (coastal or inland areas). Four coastal and four inland sampling sites were selected, and at each site three practices were considered (one practice per plot). For each plot (200 m 2 ), 20 soil samples were taken and homogenized to obtain a composite sample from which soil was characterized chemically and microbiologically. Differences in chemical properties (smaller N content and alkyl C fraction in the coastal area) depended on geographical location rather than on agricultural practices. For microbial properties, catabolic structure varied with geographical location; the index of catabolic diversity was larger in the coastal area. Importantly, the basal respiration was similar in both areas although the soil contained less nitrogen (N) in the coastal area. The negative effect of conventional tillage on microbial functioning of the soil was observed only in relation to geographical location. We found that this effect was reduced by co-culture, for example with Vicia sativa L. var. sativa. Our study reveals that geographical location affects how agricultural practices affect soil properties (in the coastal area, enhanced mineralization of labile C and change in microbial catabolic profile). This suggests that soil management should take into account the environmental conditions specific to coastal areas, which enhance the adverse effects of conventional tillage. Consequently, this type of management should not be implemented in coastal areas of the Mediterranean.

Highlights

• What are the appropriate practices for olive trees under coastal constraints? • Coastal-specific constraints were the main factors controlling microbial communities.

• Agricultural practices affected enzyme activities depending on distance from the sea.

Introduction

Olive trees (Olea europaea L.) are among the longest cultivated fruit trees in the Mediterranean basin [START_REF] Ferrise | Climate change impacts on typical Mediterranean crops and evaluation of adaptation strategies to cope with[END_REF], where olive oil has been produced for thousands of years. Their cultivation is an important socio-economic feature and a Mediterranean cultural symbol. Mediterranean soil is known to be poor in organic matter and subject to erosion [START_REF] Kosmas | The effect of land use on runoff and soil erosion rates under Mediterranean conditions[END_REF]. In this context, olive orchards contribute to ecosystem services by improving soil conservation. The main constraint is summer drought [START_REF] Castro | Effects of different olive-grove management systems on the organic carbon and nitrogen content of the soil in Jaén (Spain)[END_REF][START_REF] Sofo | The olive tree: a paradigm for drought tolerance in Mediterranean climates[END_REF], but coastal zones also experience windy conditions and higher temperatures, in addition to factors such as sea spray, all of which induce osmotic stress, increase desiccation and affect microbial functioning of the soil. Climatic projections for the coming decades suggest an increase in the number of heat waves and drought periods in the Mediterranean region, which is considered to be particularly sensitive to these expected extreme conditions [START_REF] Giorgi | Climate change projections for the Mediterranean region[END_REF]. Appropriate management is clearly required to maintain soil quality for this typical Mediterranean crop.

Inappropriate agricultural practices can have a substantial effect on chemical properties of soil (i.e. salinity, pH, nutrient content and balance, and C/N ratio) [START_REF] Nieto | Uso sostenible de suelos de olivar. Efecto del manejo del suelo en sus propiedades[END_REF] and can affect the C and N cycles governed by soil microbial communities. Previous studies have shown that certain agricultural practices favour carbon sequestration in the soil of olive tree orchards [START_REF] Sofo | Net CO 2 storage in mediterranean olive and peach orchards[END_REF] and that olive tree biomass has a long-term carbon storage capacity [START_REF] Nieto | Uso sostenible de suelos de olivar. Efecto del manejo del suelo en sus propiedades[END_REF]. Weed control in olive tree orchards is, for instance, traditionally based on tillage. Tillage is also considered necessary in the topsoil to reduce compaction and improve water infiltration temporarily. Chemical treatments are commonly used to limit olive grove pests such as Bactrocera oleae (Rossi) (olive fruit fly), especially in the coastal area where more pronounced heat favours the development of this pest. Recent studies have focused on improving soil management, with cover crops under olive trees, for instance, to increase soil organic matter and improve the structure and fertility of soil [START_REF] Castro | Effects of different olive-grove management systems on the organic carbon and nitrogen content of the soil in Jaén (Spain)[END_REF]. In Lebanon, some farmers have opted for conservation agriculture by planting Vicia sativa L. var. sativa (Fabaceae family) under olive trees to improve nitrogen fixation and soil fertility to maintain or even increase crop yields [START_REF] Chalak | Current status of agro-biodiversity in Lebanon and future challenges[END_REF].

Our study aims to understand how common agricultural practices in Lebanese olive tree orchards, such as no-tillage, co-culture with Vicia sativa L. var. sativa and conventional tillage, affect functioning of the soil. In addition, we investigated how responses of both soil chemical and microbial properties depend on the interaction between geographical location and practice. To this end, we characterized both the chemical (total C and N, CaCO 3 and organic matter quality by solid-state NMR of 13 C) and the microbial (catabolic structure, extracellular enzyme activities, basal respiration and biomass) properties of the soil according to the geographical location (i.e. coastal or inland area). We hypothesized that coastal environments, where hydric, thermic and osmotic stresses are amplified (mainly because of wind conditions and exposure to sea spray), might affect the structure of microbial communities differently and lead to differences in both C and N dynamics and in the soil's functional responses to management.

Materials and methods

Sampling sites and experimental design

The study was carried out in El Koura (north Lebanon), a region with homogeneous geomorphological and pedological properties. Its climate is typically Mediterranean and characterized by an average annual rainfall of 900 mm year -1 and mean temperature of 16 ∘ C. The region has a long tradition of olive farming. Two areas located 200 m (coastal, C) and 6 km (inland, I) from the coastline were chosen. Average altitudes of the coastal and inland areas are 75 and 285 m above sea level, respectively. In each area (coastal and inland), four independent sites (coastal area, 34 ∘ 21 ′ 38 ′′ N, 35 ∘ 44 ′ 31 ′′ E; inland area, 34 ∘ 18 ′ 35.2 ′′ N, 35 ∘ 48 ′ 27.8 ′′ E) from 1 to 3 km apart were chosen for their homogeneous pedoclimate using the soil database and map from the Center for Remote Sensing (CNRS-Lebanon) [START_REF] Darwish | Digital Soil Map of Lebanon 1/50, 000[END_REF]. The pedoclimatic characteristics were mean annual temperature (10-15 ∘ C and above 15 ∘ C in inland and coastal areas, respectively) and mean annual precipitation (930 mm per year for the inland area and 900 mm per year for the coastal area). Other characteristics included aspect (W), slope (0-5%), soil type (calcic Luvisol in the coastal area or Luvisol in the inland area according to the IUSS Working Group WRB ( 2006)) and main soil properties (depth ranged from 70 to 100 cm, pH from 7 to 7.5, clay content from 200 to 300 g kg -1 , CaCO 3 concentration from 120 to 410 g CaCO 3 kg -1 and sandy loam texture).

The four sampling sites were selected at random in each area (coastal and inland). For each site, three management practices in 20-year-old olive tree orchards were considered: (i) no-tillage, where a cover of natural grass was maintained for 4-5 years, (ii) co-culture for 4-5 years at least with Vicia sativa L. var. sativa sown after conventional tillage and (iii) conventional tillage annually over the previous 4-5 years. For each site in each area, four plots per practice were selected at random to give a total of 24 samples: 4 plots × 3 practices × 2 areas. For each practice, sampling plots with similar tree density were selected and delimited at each site to reduce the effect of borders.

We sampled the soil in the first week of April 2013. For each plot (200 m 2 ), we took 20 soil samples at randomly selected sites by shovel at 0-10-cm depth. The samples were mixed thoroughly to homogenize them to obtain a composite sample. Soil samples were taken from rows and inter-rows (ten each), excluding the area beneath the canopy. The soil was sieved (< 2 mm) to remove plant roots, fauna and debris and hand mixed. Soil samples were stored at 4 ∘ C before the determination of microbial activities.

Chemical and microbial characterization of the soil

Chemical characterization. To measure pH and electrical con- ductivity, 4 g of soil was mixed with 40 ml of deionized water (30 minutes) and measured with a Metrohm 744 pH meter for pH and a DIST three electrode (HANNA Instruments, Limena, Italy) for electrical conductivity (EC). Soil samples were oven-dried for 48 hours at 90 ∘ C until the weight remained constant; water content (g kg -1 ) was then deduced from the samples' weights before and after drying. To measure total organic carbon (TOC) and total nitrogen concentration (TN), air-dried soil was crushed in a blender and analysed by combustion in an Elemental Analyzer, FlashEA 1112 (Thermofisher, Villebon-sur-Yvette, France). Determination of calcium carbonate (CaCO 3 ) in soil was based on the release of CO 2 after the addition of HCl 4 N using the calcimeter Bernard method. Organic carbon was calculated as the difference between total C content and carbonate C content. The quality of organic matter was determined by solid-state nuclear magnetic resonance (NMR) of 13 C, and spectra were obtained with a Bruker Avance (Rheinstetten, Germany) 400 MHz NMR spectrometer that operates at a 13 C resonance frequency of 106 MHz and uses a Bruker double-bearing probe. The 13 C chemical shifts were referenced to tetramethylsilane (at 0 ppm) and calibrated with a glycine carbonyl signal set at 176.5 ppm. Following [START_REF] Baldock | Aspects of the chemical structure of soil organic materials as revealed by solid-state 13 C NMR spectroscopy[END_REF], the 13 C NMR spectra, obtained from samples of 80 mg of dry ground soil spun at a magic angle spinning (MAS) rate of 10 kHz, were divided into four chemical shift regions: alkyl C (0-45 ppm), O-alkyl C (45-112 ppm), aromatic C (112-160 ppm) and carboxyl C (160-185 ppm). Two ratios were calculated from the NMR of 13 C data: the aromaticity ratio (the sum of aromatic-C to the sum of all regions except carboxyl C) and the humification ratio (alkyl C to O-alkyl C).

Microbial properties. The microbial catabolic structures were determined with BIOLOG ® EcoPlates (BIOLOG Inc., Hayward, CA, USA) following [START_REF] Garland | Classification and characterization of heterotrophic microbial communities on the basis of patterns of community-level sole-carbon-source utilization[END_REF]. Briefly, 4 g dry weight equivalent of a subsample was added to 50 ml of sterile 0.1% Na-pyrophosphate solution (pH 7). The mixture was then shaken on an orbital shaker for 20 minutes, and centrifuged at 500 g for 10 minutes at 4 ∘ C to obtain a microbial suspension. Exactly 1 ml of supernatant was diluted into 99 ml of sterile saline solution (0.85% NaCl), hand mixed for 30 s and left to stand for 10 minutes. For each sample, a 125-μl aliquot of the diluted solution was added to each of 32 wells (31 substrates plus a control) in a BIOLOG ® EcoPlate. The plates were incubated at 25 ∘ C for 48 hours and the absorbance was read at 590 nm with a microplate reader (Infinite 200, Tecan, Männedorf, Switzerland). To measure the functional diversity (FD), Shannon's diversity index, H ′ [START_REF] Shannon | The Mathematical Theory of Information[END_REF], was calculated [START_REF] Zak | Functional diversity of microbial communities: a quantitative approach[END_REF] from:

H ′ = - s ∑ i=1 p i ln p i , ( 1 )
where p i is the ratio of colour development of well i, i = 1, 2, … , S, to the sum of colour development of all positive wells and S is the number of wells. The average capacity of the microbial cultivable community in each microplate to use the 31 substrates was assessed by calculating the average well-colour development (AWCD) as follows:

AWCD = N ∑ i=1 OD i 31 , ( 2
)
where OD is the optical density for each well and 31 is the number of substrates.

The basal and substrate-induced respirations were measured as described by [START_REF] Anderson | A physiological method for the quantitative measurement of microbial biomass in soils[END_REF]. Microbial respiration was expressed as μg of C formed as CO 2 g -1 dry soil hour -1 . Substrate-induced respiration rates were converted into microbial biomass with equations given by [START_REF] Beare | A substrate-induced respiration (SIR) method for measurement of fungal and bacterial biomass on plant residues[END_REF]. Enzyme activities of tyrosinase, cellulase, protease, urease and lipase were quantified directly in the soil samples. Tyrosinase activity was based on the oxidation of tyrosine to L-Dopa (𝜀 M = 3400 M -1 cm -1 ) following [START_REF] Saiya-Cork | The effects of long term nitrogen deposition on extracellular enzyme activity in an Acer saccharum forest soil[END_REF]. Cellulase activity was measured with the Somogy-Nelson method modified by Farnet et al. (2010b) with carboxymethyl cellulose as substrate. Urease activity was measured as described by [START_REF] Kandeler | Short-term assay of soil urease activity using colorimetric determination of ammonium[END_REF] based on the determination of NH 4 + after urea hydrolysis. Lipase activity was measured in accordance with Farnet et al. (2010a) with p-nitrophenyl (pNP) laurate as substrate. Protease activity was determined using the method of [START_REF] Ladd | Short-term assays of soil proteolytic enzyme activities using proteins and dipeptide derivatives as substrates[END_REF].

Statistical analyses

Two-way analyses of variance (anovas) were performed taking into account geographical location (coastal and inland), agricultural practices (no-tillage, tillage and Fabaceae co-culture) and their interactions, followed by Fisher's least significant difference (LSD) test. A Student's t-test was also used to determine whether means for the same practice in the two geographical areas were significantly different. The normality and homogeneity of the variances were determined on the residuals from the regression model with the Shapiro-Wilk and Levene tests, respectively. Data were transformed to common logarithms, log 10 , when necessary to meet the requirements of normality and homogeneity of variance for anova. Principal component analysis (PCA) was used to explore the effects of agricultural practice and geographical context on the catabolic structure of microbial communities from Biolog ® . Absorbance values from the 31 substrates obtained for AWCD = 0.5 were centred and the PCA was done on the covariance matrix. The proportion of the variance in the data explained by each principal component was calculated and the strongly correlated substrates (r > 0.5 or < -0.5) for a given axis were used to explain its meaning. The mean scores (on PC1 and PC2) of the four plots and standard errors were calculated to define each projection point (centroid). Statistical analyses were carried out with XLSTAT-Pro 7.5 (Addinsoft, Paris, France) and differences were considered significant at P < 0.05.

Results and discussion

The two-way analyses of variance did not show any significant effect of agricultural practice, when considered alone or in interaction with the area, on total N content, C/N ratio and organic matter quality described by NMR data (Table 1). Surprisingly, the type of agricultural practice implemented did not modify the soil chemical properties significantly in the two areas studied. Tillage has been reported to have a negative effect on soil quality, whereas cover crops have the opposite effect, which is good for soil conservation [START_REF] Hernández | Effects of different management practices on soil conservation and soil water in a rainfed olive orchard[END_REF][START_REF] Ramos | Cover crops under different managements vs. frequent tillage in almond orchards in semiarid conditions: effects on soil quality[END_REF][START_REF] Nieto | Conventional tillage versus cover crops in relation to carbon fixation in Mediterranean olive cultivation[END_REF]. In olive orchards, for example, tillage usually enhances mineralization of labile C, leading to a decrease in soil organic carbon in the top few centimetres (0-5 cm), whereas a cover crop results in C storage in the soil [START_REF] Nieto | Uso sostenible de suelos de olivar. Efecto del manejo del suelo en sus propiedades[END_REF]. Cover crops, however, are usually effective in reducing run-off and the loss of soluble nutrients from the soil after precipitation, which modifies the soil ion (such as K + and P + ) concentrations [START_REF] Gómez | The influence of cover crops and tillage on water and sediment yield, and on nutrient, and organic matter losses in an olive orchard on a sandy loam soil[END_REF].

Our results indicate that geographical location strongly affects the chemical characteristics of the soil in the two areas. Conductivity was larger in the soil from the coastal area (130.5 μS cm -1 ± 6.4 (coast), 69 μS cm -1 ± 4.9 (inland)), which indicates the effect on the soil of exposure to sea-spray and the functional processes in soil that result from osmotic stress. Organic matter content also differed depending on distance from the sea (Tables 1 and2): total nitrogen was significantly less (P < 0.05) in the coastal area, whereas there was no difference in organic carbon content between geographical areas. Consequently, the C/N ratio was significantly 1). Salinity can reduce nitrogen fixation in soil; for example, nitrogen concentrations decrease in soil according to the degree of salinity of the water used for irrigation. An effect of salinity on the mineral nitrogen production by microbiological activities in the soil through organic matter transformation has been shown under these conditions [START_REF] Van Hoorn | Effect of salinity on yield and nitrogen uptake of four grain legumes and on biological nitrogen contribution from the soil[END_REF].

Figure 1 shows the results of the PCA for six combinations of area and management (based on the profiles of microbial substrate use) plotted on the first two principal components. The first two principal components (PCs) account for 46% of the variation in the data; PC1 accounts for 29.2% and PC2 for 16.9% (Table 3). Figure 1(a) shows a plot of the principle component scores in the plane of the first two components. It illustrates the discrimination between inland and coastal soil on PC1, which indicates a change in the structure of catabolic diversity that depends on geographical location. The coastal samples (black circles) are on the right-hand side (rhs) of this axis, whereas the inland sites (white circles) are on the left-hand side (lhs).

Figure 1(b) gives the plot of the correlation coefficients within a circle of unit radius in the plane of the first two principal axes. It shows that microbial community level physiological profiles (CLPPs) for tillage in the coastal area (rhs) are characterized by a large use of glycogen (r > 0.5), a labile polysaccharide, whereas in the inland area d-cellobiose and d-mannitol are strongly and positively correlated (r > 0.5) with no-tillage and Fabaceae co-culture (Figure 1b). These changes in CLPP according to the location are associated with the larger functional diversity in the coastal area assessed by the Shannon-Weaver index, H ′ , calculated from Biolog data.

The results of the two-way anova support Figure 1(a); they indicate a significant effect of geographical location (Table 1) on the index of functional diversity, H ′ , regardless of agricultural practices. This result reveals that the environmental conditions specific to the Mediterranean coastal area have a stronger effect than agriculture practices on soil functional diversity. This is consistent with previous research that has shown the significant contribution of micro-local environmental conditions, such as exposure to salinity in coastal environments, to strong functional diversity [START_REF] Qasemian | How do microlocal environmental variations affect microbial activities of a Pinus halepensis litter in a Mediterranean coastal area?[END_REF]. Similarly, the selective adaptation of microbial populations to drought and osmotic stresses was responsible for the large functional diversity of microbes in a Greek Mediterranean grassland [START_REF] Papatheodorou | The effects of large-and small-scale differences in soil temperature and moisture on bacterial functional diversity and the community of bacterivorous nematodes[END_REF]. Climatic conditions in the Mediterranean are known to select thermo-tolerant and drought-resistant microbial communities in soil; these particular conditions act as strong selective environmental factors [START_REF] Yuste | Strong functional stability of soil microbial communities under semiarid Mediterranean conditions and subjected to long-term shifts in baseline precipitation[END_REF]. Another possible explanation for the effect of geographical location on functional diversity is that the agricultural practices applied in olive tree orchards are not intensive. The catabolic evenness of microbial communities depends on the intensity of soil management; soil under long-term arable cultivation is less resistant to osmotic stress than that under pasture [START_REF] Degens | Is the microbial community in a soil with reduced catabolic diversity less resistant to stress or disturbance[END_REF]. The way that soil functions under olive tree orchards can be compared with that observed under pasture because soil management practices under olive trees are less intensive than those under long-term cropping [START_REF] Gómez | Evaluating the impact of soil management on soil loss in olive orchards[END_REF][START_REF] Sofo | Net CO 2 storage in mediterranean olive and peach orchards[END_REF]. Figure 1(a) also indicates that PC2 can be explained by the effect of tillage, with both Fabacea co-culture and no-tillage separated from tillage; for the coastal site this separation is along PC1 and for the inland site it is along PC2. [START_REF] Lupwayi | Soil microbial response to nitrogen fertilizer and tillage in barley and corn[END_REF] showed that tillage can induce changes in the catabolic potential of soil microbial communities. Conventional tillage disturbs the soil, which results in desiccation, fragmentation of microaggregates (microhabitats that harbour soil microbial biomass and diversity) and long-term compaction [START_REF] Franzluebbers | Water-stable aggregation and organic matter in four soils under conventional and zero tillage[END_REF]. However, modification of the Shannon-Weaver index of diversity did not depend on agricultural practices: a different catabolic structure is evident in the projection in Figure 1(a) by the discrimination between tillage, and no-tillage and Fabaceae for a similar level of functional diversity (assessed by H ′ ). We expected that the different types of agricultural practice selected here would induce specific effects on microbial communities, mainly through organic residue inputs (carbon and nitrogen from Vicia sativa L. var. sativa or natural grass). Greater metabolic diversity in the soil from Mediterranean olive orchards is usually observed because of the organic matter input from self-seeding weeds such as Gramineae L. and Fabaceae Lindley [START_REF] Sofo | Soil microbial diversity and activity in a Mediterranean olive orchard using sustainable agricultural practices[END_REF]. Figure 1(a) indicates a little difference, however, in catabolic structure between cover crops sown after tillage, Fabaceae co-culture, and no-tillage for the inland area because the scores are close to each other (Figure 1a). This result suggests that Vicia sativa L. var. sativa counterbalanced the modifications induced by tillage inland, which led to similar functioning of the soil as when under natural grass. Fabaceae co-culture, therefore, appears to be a good soil management practice to sustain soil quality [START_REF] Hernández | Effects of different management practices on soil conservation and soil water in a rainfed olive orchard[END_REF][START_REF] Gómez | The influence of cover crops and tillage on water and sediment yield, and on nutrient, and organic matter losses in an olive orchard on a sandy loam soil[END_REF][START_REF] Gómez | The effects of cover crops and conventional tillage on soil and runoff loss in vineyards and olive groves in several Mediterranean countries[END_REF][START_REF] Nieto | Conventional tillage versus cover crops in relation to carbon fixation in Mediterranean olive cultivation[END_REF], but our results suggest that this is less so near the coast where the soil is under greater stress.

For the other microbial markers under study, one should note that although the C/N ratio (P < 0.05) was larger in the coastal area, basal respiration (P > 0.05) and biomass (P > 0.05) were similar in both areas (data not shown). In the coastal area, less depletion of N and greater microbial catabolic diversity might have enhanced the efficiency of both nitrogen and carbon resource use and consequently sustained microbial respiration [START_REF] Guénon | Soil organic matter quality and microbial catabolic functions along a gradient of wildfire history in a Mediterranean ecosystem[END_REF].

Figure 2(a) shows that the effect of agricultural practices relates to geographical area for cellulase activity; it is larger (P < 0.05) in the coastal area, but only after tillage. The effect of agricultural practices on cellulase seemed to be strengthened by the link with geographical location, which shows that mineralization of labile carbon might be enhanced under tillage as described previously by [START_REF] Balesdent | Relationship of soil organic matter dynamics to physical protection and tillage[END_REF]. In the coastal environments, tillage favoured the degradation of the major vegetal polymer (i.e. cellulose) in the soil, which consequently induced a loss of soil carbon. Several previous studies have shown that tillage in olive orchards induced the loss of soil organic carbon because of the large rate of mineralization that results from the break-up of soil aggregates; this is a serious threat under Mediterranean conditions [START_REF] Gómez | The influence of cover crops and tillage on water and sediment yield, and on nutrient, and organic matter losses in an olive orchard on a sandy loam soil[END_REF][START_REF] Vanwalleghem | Quantifying the effect of historical soil management on soil erosion rates in Mediterranean olive orchards[END_REF][START_REF] Marquez-Garcia | Improvement of soil carbon sink by cover crops in olive orchards under semiarid conditions. Influence of the type of soil and weed[END_REF]. Moreover, we found that lipase activity was greater in non-tilled soil than after tillage in the coastal area (Figure 2b). This result indicates that the recalcitrant fraction of organic matter was actively degraded under natural grass cover. On the other hand, neither urease nor tyrosinase activities (Table 1) were affected regardless of the factor considered (i.e. geographical location, agricultural practices or their interaction). In our research, the effects of agricultural practice appeared to depend mainly on distance from the sea, which suggests the importance of the 'coastal position' (characterized mainly by hydric, thermic and osmotic stresses amplified because of wind conditions and exposure to sea spray) as a controlling factor in the functioning of soil.

Conclusions

Defining sustainable agriculture is a great challenge, especially in the Mediterranean coastal zones where demographic pressures and human activities are particularly intense. Our study shows that tillage induces negative changes in soil properties (a shift in microbial catabolic profile) that can be reduced by cover such as Vicia sativa L. var. sativa (catabolic profiles were similar for cover crops sown after tillage and no-tillage). Such cover crops are commonly used under olive trees throughout the circum-Mediterranean region, therefore, they can be considered an appropriate conservation practice to sustain soil functioning in these agrosystems. Furthermore, the effect of tillage depends on the geographical location: coastal-specific constraints seem to increase the effect of soil management on microbial functioning. Thus distance from the sea should be considered when selecting sustainable agricultural practices in Mediterranean countries, and more specifically tillage should not be implemented in olive orchards of the coastal area.

Figure 1

 1 Figure 1 (a) Plot of the principal component analysis (PCA) scores in the plane of PC1 and PC2 obtained from a PCA of microbial catabolic profiles from soil in relation to the agricultural practices selected for both inland (open symbols) and coastal (black symbols) areas and (b) correlation circle of strongly correlated C-substrates (arrows), r > 0.6. Substrates with stronger correlations (r > 0.5 or r < -0.5) with PC1 or PC2 were: d-cellobiose (-0.233; -0.752), d-mannitol (-0.137; -0.603), d-malic acid (0.079; -0.633), d-galacturonic acid (0.208; -0.780), d-xylose (-0.080; 0.538), phenylethylenamine (-0.072; 0.562), 𝛼-d-lactose (0.098; 0.808) and glycogen (0.997; 0.032). Centroids (a), representing the average of four replicates, are reported.

Figure 2

 2 Figure 2 (a) Cellulase and (b) lipase of soil sampled from olive orchards under three different agricultural practices in both inland and coastal areas. Bars represent standard errors (n = 4).

Table 1

 1 Summary of the analysis of variance for certain soil chemical and microbial markers to test the effect of the distance from the sea, the agricultural practices and their interaction

	Variable	Source	d.f.	Sum of squares	Mean square	F ratio	P a
	Total N content	Area	1	0.019	0.019	14.039	0.003
		Sites (blocks)	6	0.045	0.007	5.533	0.006
		Practices	2	0.005	0.002	1.812	0.205
		Area × practices	2	< 0.001	< 0.001	0.353	0.709
		Residues	12	0.016	0.001
		Total	23
	C/N ratio	Area	1	30.710	30.713	6.471	0.026
		Sites (blocks)	6	30.740	5.123	1.079	0.427
		Practices	2	16.490	8.244	1.737	0.218
		Area × practices	2	4.670	2.334	0.492	0.623
		Residues	12	56.960	4.746
		Total	23
	Conductivity	Area	1	< 0.001	< 0.001	76.425	< 0.001
		Sites (blocks)	6	< 0.001	< 0.001	4.114	0.021
		Practices	2	< 0.001	< 0.001	2.516	0.126
		Area × practices	2	< 0.001	< 0.001	2.109	0.168
		Residues	11	< 0.001	< 0.001
		Total	22
	Shannon index H ′	Area	1	0.008	0.008	14.666	0.003
		Sites (blocks)	6	0.009	0.001	2.763	0.069
		Practices	2	< 0.001	< 0.001	0.486	0.628
		Area × practices	2	< 0.001	< 0.001	0.055	0.947
		Residues	11	0.006	< 0.001
		Total	22
	Lipases	Area	1	< 0.001	< 0.001	7.814	0.016
		Sites (blocks)	6	< 0.001	< 0.001	0.273	0.939
		Practices	2	< 0.001	< 0.001	3.807	0.052
		Area × practices	2	< 0.001	< 0.001	0.258	0.777
		Residues	12	0.001	< 0.001
		Total	23
	Cellulases	Area	1	1.205	1.205	28.710	< 0.001
		Sites (blocks)	6	0.490	0.082	1.947	0.154
		Practices	2	0.106	0.053	1.259	0.319
		Area × practices	2	0.435	0.217	5.179	0.024
		Residues	12	0.504	0.042
		Total	23

a Values shown in bold are significant for P < 0.05. larger (P < 0.05) in the coastal area (Table

Table 2

 2 Soil chemical properties for inland and coastal areas for the three agricultural practices (means ± standard error, N = 4) Means in inland and coastal areas (n = 12), indicated in bold within each row, are significantly different at P < 0.05, determined by Fisher's least significant difference (LSD) test.

	Variable	Inland area	Coastal area
	Organic C content / g C kg -1	20.6 ± 1.8	18.2 ± 1.6
	Total N concentration / g N kg -1	2.2 ± 0.1	1.6 ± 0.2
	C/N ratio	9.3 ± 0.5	11.6 ± 0.8
	Alkyl C	24.6 ± 2.1	19.8 ± 2.2
	O-alkyl C	28.6 ± 3.0	26.4 ± 2.1
	Methoxyl C	11.9 ± 1.6	11.5 ± 2.4
	Carboxyl C	9.9 ± 0.4	8.8 ± 0.9
	Aromatic C	25.4 ± 3.3	33.2 ± 3.6
	Humification ratio	0.9 ± 0.1	0.8 ± 0.1
	Aromaticity ratio	0.4 ± 0.1	0.6 ± 0.1
	pH	7.5 ± 0.1	7.1 ± 0.1
	Conductivity / μS cm -1	69.3 ± 4.9	130.5 ± 6.4

Table 3

 3 Eigenvalues from the principal component analysis obtained from Biolog data on microbial catabolic profiles of soil

		PC1	PC2	PC3	PC4	PC5	PC6
	Eigenvalue	0.247 0.143 0.092 0.070 0.052 0.047
	Percentage variance	29.21 16.90 10.92	8.261 6.157 5.519
	explained
	Cumulative / %	29.21 46.11 57.04 65.30 71.46 76.98

© 2016 British Society of Soil Science, European Journal of Soil Science, 67, 470-477

Acknowledgements

This work was supported financially by a PhD grant from the National Council for Scientific Research in Lebanon (CNRSL, Beirut, Lebanon). We thank the Center for Remote Sensing for providing environmental data and maps of the areas studied. We would like to thank all the people who helped us with field experiments and technical assistance. This manuscript benefited from the helpful comments of the Editor-in-Chief, the Associate Editor and the three anonymous reviewers. We are very grateful to Marjorie Sweetko and Patrick Fournier for their assistance with the English language.


Mathieu Aucejo
email: mathieu.aucejo@lecnam.net

Olivier De Smet

An optimal Bayesian regularization for force reconstruction problems

Keywords: Linear inverse problem, Force reconstruction, Bayesian regularization, Generalized Gaussian priors

come

An optimal Bayesian regularization for force reconstruction problems

Since all the necessary information about the problem is available, statistical measures, such as the mean, the median and the mode of the solution, can be easily estimated. In many practical situations, however, one only wants to determine the most probable parameters given the available data. Consequently, it is not relevant to implement a full Bayesian inference to only extract a point estimate. To overcome this potential issue, this paper introduces an optimal Bayesian regularization aiming at computing the Maximum a Posteriori estimate of the Bayesian formulation previously introduced by the authors. In doing so, the most probable parameters are obtained without heavy computations. The validity of the proposed method is assessed numerically and experimentally. In particular, obtained results highlight the

Introduction

In structural dynamics, force identification is a special class of inverse problem aiming at reconstructing the excitation sources acting on a mechanical structure from the knowledge of the vibration field measured over its surface. Unfortunately, this problem is known to be ill-posed, meaning that the existence of a unique stable solution is not guaranteed. A classical solution to bypass this difficulty is to apply regularization strategies, which allows constraining the space of solutions by using some prior information on the noise corrupting the data and the sources to identify. The most widespread and, certainly, the most popular technique is the Tikhonov regularization (a.k.a. 2 -regularization) [START_REF] Tikhonov | Regularization of incorrectly posed problems[END_REF][START_REF] Jacquelin | Force reconstruction: analysis and regularization of a deconvolution[END_REF][START_REF] Leclere | Indirect measurement of main bearing loads in an operating diesel engine[END_REF][START_REF] Liu | Reducing the impact of measurement errors when reconstructing dynamic forces[END_REF][START_REF] Thite | The quantification of structure-borne transmission paths by inverse methods. Part 2 : Use of regularization techniques[END_REF][START_REF] Mao | Experimental study of hammer impact identification on a steel cantilever beam[END_REF][START_REF] Turco | Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives[END_REF]. Although widely used, it is theoretically mainly applicable to the identification of rather smooth excitation fields or excitation signals [START_REF] Boyd | Convex optimization[END_REF], which is not a desirable effect when a localized source or an impulsive excitation signal have to be identified. To remedy this problem, LASSO regularization (a.k.a. 1 -regularization) has been developed to promote the sparsity of the regularized solutions, while keeping the inverse problem convex. For this particular reason, it has focused many of the research efforts in the recent years [START_REF] Ginsberg | New approach for impact detection by finding sparse solution[END_REF][START_REF] Qiao | Sparse regularization for force identification using dictionaries[END_REF][START_REF] Qiao | Impact-force sparse reconstruction from highly incomplete and inaccurate measurements[END_REF][START_REF] Pan | Moving force identification based on redundant concatenated dictionary and weighted 1norm regularization[END_REF]. To unify all these approaches, Aucejo introduced the q -regularization in the context of force reconstruction, which includes Tikhonov and LASSO regularization as special cases [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF]. However, because the norm parameter q can take any value in R + * , it can lead to more accurate solutions than those obtained from standard regularization strategies [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF]. In the procedures described above, the a priori on the sources to identify is global, meaning that poor reconstructions can be obtained, if, for instance, a structure is excited by several sources having different spatial distributions, since the a priori has to reflect a compromise between contradictory distributions. That is why, Aucejo and De Smet extend the q -regularization to take advantage of prior local information available on the sources to identify [START_REF] Aucejo | Bayesian source identification using local priors[END_REF]. More specifically, the proposed formulation relies on the definition of several identification regions, in which the norm parameters q can be set independently for each region.

Although they have proved their efficiency, all these techniques requires a proper tuning of the parameters of the considered formulation, namely the regularization parameter and, potentially, the norm parameters q. For the regularization parameter, this can be done using external, but expensive, automatic selection procedures [START_REF] Hansen | Discrete Inverse Problems: Insight and Algorithms[END_REF][START_REF] Golub | Generalized cross-validation as a method for choosing a good ridge parameter[END_REF], while the choice of the norm parameters relies more on the user's skills or experience. Fortunately, because all these regularization strategies derive from the Bayesian statistics, a possible solution is to implement a full Bayesian inference to estimate the value of each parameters of the problem as well as the related posterior uncertainty using either MCMC techniques [START_REF] Park | The bayesian lasso[END_REF][START_REF] Faure | Vibration source characterization using force analysis technique and a Bayesian regularization[END_REF][START_REF] Aucejo | Bayesian formulations for force reconstruction problems[END_REF][START_REF] Aucejo | On a full bayesian inference for force reconstruction problems[END_REF][START_REF] Li | A hierarchical bayesian method for vibration-based time domain force reconstruction problems[END_REF] or variational Bayes-like approaches [START_REF] Ormerod | Explaining variational approximations[END_REF][START_REF] Teh | A collapsed variational bayesian inference algorithm for latent dirichlet allocation[END_REF][START_REF] Babacan | Variational bayesian blind deconvolution using a total variation prior[END_REF][START_REF] Ishiguro | Averaged collapsed variational bayes inference[END_REF]. Since all the necessary information about the problem is then available, statistical measures, such as the mean, the median and the mode of the solution, can be easily estimated. In many practical situations, however, one only wants to determine the most probable param-eters given the available data. In this respect, implementing a full Bayesian inference appears to be irrelevant, since it is generally time-consuming and computationally expensive. It results that it seems wiser and more efficient, in this case, to calculate directly the Maximum a Posteriori (MAP) estimate of the considered Bayesian formulation. This idea has given rise to the augmented Tikhonov regularization [START_REF] Jin | Augmented tikhonov regularization[END_REF][START_REF] Feng | Simultaneous identification of bridge structural parameters and vehicle loads[END_REF][START_REF] Sun | Identification of traffic-induced nodal excitations of truss bridges through heterogeneous data fusion[END_REF][START_REF] Yan | Impact load identification for composite structures using bayesian regularization and unscented kalman filter[END_REF], to the sparse Bayesian blind deconvolution [START_REF] Amizic | Sparse bayesian blind image deconvolution with parameter estimation[END_REF] or to the adaptive q -regularization [START_REF] Li | Time domain force identification based on adaptive q regularization[END_REF].

The present paper introduces an optimal Bayesian regularization corresponding to the MAP estimate of the Bayesian formulation proposed by the authors in Ref. [START_REF] Aucejo | On a full bayesian inference for force reconstruction problems[END_REF]. More specifically, the considered regularization strategy allows assessing within a single iterative process the most probable parameters of the problem given the measured vibration field. To avoid any misunderstanding, it is important to note that the proposed Bayesian regularization is optimal in the sense of the underlying Bayesian formulation. In other words, the optimality statement should not be regarded as a statement of superiority over the other existing strategies. To properly highlight the main features of the proposed approach, this article is divided into four parts.

For the sake of completeness, the Bayesian formulation behind the proposed regularization strategy is first recalled in section 2 to make the paper selfcontained. This section allows clarifying the assumptions made to derive the formulation and allows defining the main parameters of the corresponding probability distributions. Then, section 3 details the optimal Bayesian regularization by thoroughly introducing its mathematical formulation as well as the algorithm used to solve the problem. Finally, the ability of the pro-posed approach in optimally identifying all the parameters of the problem is illustrated using synthetic and experimental data in sections 4 and 5, where locally sparse sources are reconstructed in the frequency domain. In particular, the proposed validations reveal that the optimal Bayesian regularization is an efficient tool to obtain consistent reconstructions, while avoiding the user to determine an accurate initial solution. In other words, only a rough knowledge of the excitation field to identify is required.

Bayesian formulation of the reconstruction problem

Let us consider the practical situation where the vibration field X, measured over the surface of a structure, is caused by an unknown excitation field F. If the structure is linear and time invariant, its dynamic behavior is completely determined by the transfer functions matrix H, relating the vibration field X to the excitation field F, so that:

X = HF + N, ( 1
)
where N is the noise vector related to the measurement errors as well as modeling errors (provided the latter are small enough).

Because the force reconstruction problem consists in estimating the unknown excitation field F acting on a structure from the knowledge of the measured vibration field X and the transfer functions matrix H, the noise vector N is another unknown of the reconstruction problem. To solve this problem efficiently, the Bayesian framework is adopted. Formally, the Bayesian paradigm consists in considering all the parameters of the problem as random variables. Consequently, the uncertainty about each parameter is modeled by a probability distribution, describing the state of knowledge or the prior on this parameter. From a mathematical standpoint, the standard Bayesian formulation of the reconstruction problem relies on the Bayes' rule:

p(F|X) ∝ p(X|F) p(F), (2)
where:

• p(F|X) is the posterior probability distribution, representing the probability of observing F given a vibration field X. In other words, it defines what it is known about the excitation field F after making vibration measurements;

• p(X|F) is the likelihood function, representing the probability of measuring X given an excitation field F. It reflects the uncertainty related to the measurement of the vibration field X;

• p(F) is the prior probability distribution, representing our knowledge of the unknown excitation field F before measuring the vibration field X.

Generally, the quality of the force reconstruction strongly depends on the choice of the likelihood function and the prior probability distribution. That is why, the probability distributions have to be carefully chosen.

As mentioned previously, the likelihood function reflects the uncertainty related to the measurement of the vibrations field X. By definition, this uncertainty is mainly related to the noise vector N. If the noise is supposed spatially white and due to multiple independent causes, then the likelihood function can be represented by a complex multivariate normal distribution with zero mean and precision parameter τ n , namely:

p(X|F, τ n ) = τ n π N exp -τ n X -HF 2 2 . ( 3
)
where • 2 is the 2 -norm and N is the number of measurement points.

Regarding now the prior probability distribution, it reflects the uncertainty related to the unknown excitation field F. If one supposes that the structure is excited in R different regions by uncorrelated excitations of various types (localized or distributed), then local excitation fields F r can be considered as independent random vectors. As a result, the prior probability distribution can be written as the product of local prior probability distributions p(F r ), that is:

p(F) = R r=1 p(F r ), (4)
where p(F r ) reflects the prior knowledge of the nature of the sources in the region r.

In the present formulation, it is assumed, for practical reasons, that the local excitation vectors F r are real random vectors, whose components are supposed to be independent and identically distributed random variables following a generalized Gaussian distribution with zero mean [START_REF] Nadarajah | A generalized normal distribution[END_REF]. As a consequence, each local excitation field follows a multivariate generalized Gaussian distribution with zero mean. From the mathematical standpoint, the local prior probability distributions are thus written:

p(F r |τ sr , q r ) = q r 2 Γ(1/q r ) Mr τ Mr qr sr exp -τ sr F r qr qr , (5)
where:

• q r is the shape parameter of the distribution in the region r. Its value is defined in the interval ]0, +∞[; • • qr is the q r -norm or quasi-norm, if q r ≥ 1 and q r < 1 respectively;

• τ sr is the scale parameter of the distribution, which can be viewed as a generalized measure of the precision of the distribution;

• M r is the number of reconstruction points in the region r;

• Γ(x) = +∞ 0 t x-1 e -t dt is the gamma function.

It should be noted that the choice of a multivariate generalized Gaussian distribution offers a high flexibility for describing prior knowledge of the sources to identify, since it is allows defining either a sparse (or localized)

prior for q r ≤ 1 or a smooth (or distributed) prior for q r = 2.

From the explanations given above, the Bayesian formulation of the reconstruction problem finally writes:

p(F|X, τ n , τ sr , q r ) ∝ p(X|F, τ n ) R r=1 p(F r |τ sr , q r ). (6)
At this stage, it is clear that the main shortcoming of the standard Bayesian formulation is the choice of the associated parameters, since the quality of the identification is conditioned to the knowledge of the shape and precision parameters. If their values are poorly chosen, then the resulting reconstruction won't be representative of the actual target excitation field. To alleviate this limitation, a possible alternative is to infer the shape and precision parameters by also considering them as independent random variables.

Such an idea gives rise to the complete Bayesian formulation at the root of the proposed optimal Bayesian regularization [START_REF] Aucejo | On a full bayesian inference for force reconstruction problems[END_REF]. From a mathematical standpoint, this particular Bayesian formulation of the reconstruction problem is given by:

p(F, τ n , τ sr , q r |X) ∝ p(X|F, τ n ) p(τ n ) R r=1
p(F r |τ sr , q r ) p(τ sr ) p(q r ), [START_REF] Turco | Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives[END_REF] where p(q r ) is the prior probability distribution of the shape parameters q r , while p(τ n ) and p(τ sr ) are the prior probability distributions of the precision parameters τ n and τ sr respectively.

To finalize the definition of the complete Bayesian formulation, it remains to specify the prior probability distributions of the shape and precision parameters. The choice of the priori probability distributions p(τ n ) and p(τ sr )

is first limited to distribution having a strictly positive support, because the precision parameters τ n and τ sr are real positive numbers. The common choice, made in the literature to use conjugacy properties, is the Gamma distribution [START_REF] Bishop | Pattern recognition and machine learning[END_REF]. Practically, the Gamma distribution is defined by:

G(x|α, β) = β α Γ(α) x α-1 exp(-β x) with α > 0, β > 0, (8)
where α and β are respectively the scale parameter and the rate parameter of the distribution.

The definition of the prior probability distribution of the shape parameters q r relies on the fact that they are positive and bounded. In absence of more precise knowledge on the parameters, the probability distribution is not only chosen to reflect the available information but also for its mathematical tractability. A probability distribution that meets these requirements is the truncated Gamma distribution defined by:

G T (x|α, β, l b , u b ) = Γ(α) γ(α, βu b ) -γ(α, βl b ) G(x|α, β) I [l b ,u b ] (x), (9)
where:

• G(x|α, β) is the Gamma distribution defined in Eq. ( 8);

• I [l b ,u b ] (x) is the truncation function defined between the lower bound l b and the upper bound u b . More precisely, this function simply writes:

I [l b ,u b ] (x) =      1 if x ∈ [l b , u b ] 0 otherwise ; (10)
• γ(s, x) = x 0 t s-1 exp(-t) dt is the lower incomplete Gamma function.

From the above considerations, it results that the proposed complete Bayesian formulation is finally expressed as follows:

p(F, τ n , τ sr , q r |X) ∝ p(X|F, τ n ) p(τ n |α n , β n ) × R r=1 p(F r |τ sr , q r ) p(τ sr |α sr , β sr ) p(q r |α r , β r , l b , u b ), (11)
where (α r , β r ) are the hyperparameters related to the shape parameters q r , while (α n , β n ) and (α sr , β sr ) are the hyperparameters associated to the precision parameters τ n and τ sr respectively.

As a final remark, it is worth mentioning that the previous Bayesian formulation allows emphasizing the specificity of the proposed optimal Bayesian regularization with respect to those at the roots of the Augmented Tikhonov regularization and the adaptive q -regularization used in the context of force reconstruction. First, the proposed Bayesian regularization generalizes to some extent the Augmented Tihkonov regularization by considering the shape parameters q r as random variables and not as deterministic parameters. Second, it extends the adaptive q -regularization, which is based on the formulation proposed by the authors in Ref. [START_REF] Aucejo | Bayesian formulations for force reconstruction problems[END_REF], by defining several identification regions and imposing, if necessary, a greater sparsity of local excitation fields by authorizing the shape parameters to be less than 1.

Optimal Bayesian regularization

This section aims at introducing thoroughly the proposed optimal Bayesian regularization. To this end, its practical implementation is detailed after having introduced the underlying theoretical formulation. In the present work, the identification regions are supposed to be known a priori. In general, a careful analysis of the mechanical system is sufficient to determine the identification regions if the studied system or the operating conditions are rather simple. In case of more complex systems or operating conditions, the definition of the identification regions could be automatically done using machine learning strategies such as Relevance Vector Machine [START_REF] Tipping | The relevance vector machine[END_REF], but this is outside the scope of this paper.

Theoretical formulation

The basic idea of the optimal Bayesian regularization is to determine the MAP estimate of the complete Bayesian formulation given by Eq. ( 11), without performing a full Bayesian inference. Consequently, the solution of the source identification problem is sought such that:

F, τ n , τ sr , q r = argmax (F,τn,τsr,qr) p(F, τ n , τ sr , q r |X). ( 12
)
The solution of the previous optimization problem can be classically solved by applying the first-order optimality condition to the dual minimization problem. However, to render the presentation of the optimal Bayesian regularization more didactic, one can notice that an alternative way of solving this optimization problem consists in maximizing the full conditional probability distributions associated to each parameters. Consequently, the MAP estimate of the complete Bayesian formulation can be found by solving the following set of optimization problems:

q r = argmax qr p(q r |X, F, τ n , τ sr ), (13a)
τ sr = argmax τsr p(τ sr |X, F, τ n , q r ), (13b)
τ n = argmax τn p(τ n |X, F, τ sr , q r ), (13c)
F = argmax F p(F|X, τ n , τ sr , q r ). (13d)
Practically, it is generally easier to solve the dual minimization problem, which consists in finding the value of the parameter minimizing the opposite of the logarithm of the corresponding full conditional probability distribution. For the complete Bayesian formulation, the full conditional probability distributions are for:

• the shape parameters q r :

p(q r |X, F, τ n , τ sr ) ∝ τ Mr qr sr Γ(1/q r ) Mr q αr+Mr-1 r exp -β r q r -τ sr F r qr qr I [l b ,u b ] (q); (14)
• the precision parameters τ sr :

p(τ sr |X, F, τ n , q r ) ∝ G τ sr α sr + M r q r , β sr + F r qr qr ; (15)
• the precision parameters τ n :

p(τ n |X, F, τ sr , q r ) ∝ G τ n α n + N, β n + X -HF 2 2 ; (16)
• the force vector F:

p(F|X, τ n , τ sr , q r ) ∝ exp -τ n X -HF 2 2 - R r=1 τ sr F r qr qr . (17)
Consequently, finding the MAP estimate is equivalent to solve the following set of minimization problems, defining the optimal Bayesian regularization1 :

q r = argmin qr f (q r |τ sr , F r ) for q r ∈ [l b , u b ], (18a)
τ sr = argmin τsr τ sr β sr + F r qr qr -α sr + M r q r -1 log τ sr , (18b)
τ n = argmin τn τ n β n + X -HF 2 2 -(α n + N -1) log τ n , (18c)
F = argmin F X -HF 2 2 + R r=1 λ r F r qr qr . ( 18d
)
where f (q r |τ sr , F r ) = M r log Γ(1/q r )-Mr qr log τ sr -[α r + M r -1] log q r +β r q r + τ sr F r qr qr and λ r = τ sr /τ n .

Resolution algorithm

The analysis of the set of equations [START_REF] Faure | Vibration source characterization using force analysis technique and a Bayesian regularization[END_REF] suggests that the optimal Bayesian regularization can only be solved in an iterative manner, since the optimal value of a given parameter explicitly depends on the value of the others. It results that the resolution process is divided into three main steps:

1. Set k = 0 and initialize q (0) r , τ f. monitor the convergence end while 3. return F, τ n , τ sr and q r . The practical implementation of the previous resolution procedure requires a particular attention, since each step has to be carefully and properly defined to converge to the optimal solution. For this particular reason, implementation issues are detailed in the next sections.

Initialization of the resolution algorithm

The initialization step can be one of the keys of the convergence of the iterative procedure when the functional to minimize exhibits local minima.

To properly address this issue, it has been decided to initialize the procedure from a starting point having a reasonably high probability. In the present case, the initial force vector

F (0)
is obtained from the MAP estimate of the standard Bayesian formulation, which is defined by [see Ref. [START_REF] Aucejo | Bayesian source identification using local priors[END_REF] for more details]:

F (0) = argmax F p F|X, τ (0) n , τ (0) sr , q (0) r = argmin F X -HF 2 2 + R r=1 λ (0) r F r q (0) r q (0) r , (19)
where λ

(0) r = τ (0) sr / τ (0)
n is the regularization parameter in the zone r.

To simplify somewhat the calculation, it is supposed that the precision parameters τ (0)

sr are equal to a unique constant value τ (0)

s . Accordingly, Eq. ( 19) becomes:

F (0) = argmin F X -HF 2 2 + λ (0) R r=1 F r q (0) r q (0) r , (20)
where

λ (0) = τ (0) s / τ (0) n .
To obtain a relevant initial force vector, it is necessary to determine reasonable values of the initial shape parameters q n . Practically, the values of the shape parameters q

(0)
r lie in the interval ]0, 2] for force reconstruction problems. They can be chosen from subjective considerations by recalling that smoothed solutions are promoted for q (0) r = 2, while sparse solutions are promoted for q (0) r ≤ 1 [START_REF] Boyd | Convex optimization[END_REF][START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF]. On the contrary, near-optimal values of the precision parameters are difficult to assess a priori, i.e. without any calculation, because they are strongly related to the optimization problem given in Eq. ( 20) through the regularization parameter λ (0) , whose optimal value is partly conditioned to q (0) r . In addition, since the shape parameter q (0) r can take any value in the range ]0,2], the solution of the optimization problem has generally no closed-form expression.

Incidentally, the optimal value of the regularization parameter λ (0) can not be directly estimated from the marginalized MAP [START_REF] Pereira | Empirical bayesian regularization of the inverse acoustic problem[END_REF], the L-curve principle [START_REF] Hansen | Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion[END_REF] or the Generalized Cross Validation [START_REF] Golub | Generalized cross-validation as a method for choosing a good ridge parameter[END_REF] before solving the minimization problem [see Ref. [START_REF] Aucejo | A multiplicative regularization for force reconstruction[END_REF] for details]. To this end, the optimization problem is solved iteratively using an adapted Iteratively Reweighted (IR) algorithm, which allows determining in an iterative manner the regularized force vector as well as the optimal regularization parameter associated to the minimization problem.

The proposed IR algorithm is derived by applying the first-order optimality condition to Eq. [START_REF] Aucejo | On a full bayesian inference for force reconstruction problems[END_REF]. It results from what precedes that the solution at iteration j of the iterative process is given by:

F (0,j) = H H H + λ (0,j) W (j) -1 H H X, (21)
where the regularization parameter λ (0,j) is updated from an automatic selection procedure such as the L-curve principle [START_REF] Hansen | Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion[END_REF], while W (j) is a diagonal global weighting matrix depending explicitly on F (0,j-1) r and q (0) r and defined from local weighting matrices W r such that:

W (j) = diag[W (j) 1 , . . . , W (j) r , . . . , W (j) R ]. (22)
In the previous equation, each local weighting matrix W (j) r is a diagonal matrix, expressed as:

W (j) r = diag w (j) r,1 , . . . , w (j) r,i , . . . , w (j)
r,Mr ,

(j) r,i = q (0) r 2 max r , f (j-1) ri q (0) r -2 , ( (23) with: w
) 24
where f

(j-1) ri
is the i th component of the vector

F (j-1) r
and r is a small real positive number acting as a damping parameter. It allows avoiding infinite weights when f (j-1) ri → 0 and q (0) r < 2. Practically, the damping parameter is automatically selected from the cumulative histogram of F

r . More precisely, its value is chosen so that 5% of the values of F (0) r are less than or equal to r [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF][START_REF] Rodriguez | Efficient minimization method for a generalized total variation functional[END_REF].

After convergence of the iterative process, one obtains the optimal force vector F (0)

, the global weighting matrix W, as well as the optimal value of the regularization parameter λ (0) . At this stage of the initialization procedure, it remains to determine the values of τ (0) n and τ (0) s . For this purpose, we follow the approach proposed by Pereira et al. [START_REF] Pereira | Empirical bayesian regularization of the inverse acoustic problem[END_REF] consisting in finding the the most probable values of τ n [see Ref. [START_REF] Aucejo | On a full bayesian inference for force reconstruction problems[END_REF] for details]:

τ (0) s = N X H λ (0) I + HW -1 H H -1 X and τ (0) n = τ (0) s λ (0) . ( 25
)
As a final remark, it is worth noting that the proposed initialization procedure implies that only the initial shape parameters q

(0)
r must be defined by the user, since the other parameters are computer from Eqs. ( 20) and (25).

Computation of the optimal shape parameters q (k) r

The computation of q (k) r requires a special attention, since the function to minimize has no analytical solution. However, because the function to minimize, f (q r |τ sr , F r ), is known analytically [see Eq. ( 18)], it is possible to calculate its gradient with respect to q r . In the present case, the gradient, g(q r |τ sr , F r ), is expressed as:

g(q r |τ sr , F r ) = β r + τ sr Mr i=1 (|F ri | qr log |F ri |) - α r + M r -1 q r + M r q 2 r [log τ sr -ψ(1/q r )] , (26)
where F ri is the i th component of the force vector F r and ψ(x) is the digamma function.

The knowledge of the function and its gradient allows considering several possibilities for finding the optimal values of the shape parameters q r at iteration k within the interval [l b , u b ]. Indeed, the minimization problem defined by Eq. (18a) can be solved using either a bound-constrained minimization method [START_REF] Byrd | A limited memory algorithm for bound constrained optimization[END_REF][START_REF] Nash | Newton-type minimization via the lanczos method[END_REF] or a bounded root-finding algorithm applied to the gradient [START_REF] Brent | Algorithms for Minimization Without Derivatives[END_REF] or even a brute-force approach.

It should be noted here that, even if the lower and upper bounds l b and u b can theoretically take any positive value, the values of the shape parameters q r generally lies in the interval ]0, 2] [see section 3. Finally, to complete this section, a comment must be added on the choice of the hyperparameters α r and β r . As evoked previously, the choice of the truncated Gamma distribution has been made for mathematical convenience, because other continuous truncated distribution could have theoretically been used. To avoid biasing the optimization process, the shape of the prior distribution needs to be weakly informative [START_REF] Hoff | A First Course in Bayesian Statistical Methods[END_REF]. According to Gelman, a prior distribution is said weakly informative if it is proper but is set up so that the information it does provide is intentionally weaker than whatever actual prior knowledge is available [START_REF] Gelman | Prior distributions for variance parameters in hierarchical models[END_REF]. Here, this leads us to set the hyperparameters such that α r = 1 and β r → 0.

Computation of the optimal precision parameters τ

(k) n and τ

(k) sr

The computation of the precision parameters τ

(k) n and τ (k)
sr is straightforward, insofar as the solutions of minimization problems given by Eqs. (18b) and (18c) can be analytically calculated by applying the first-order optimality condition. In doing so, it readily comes that:

τ (k) sr = α sr + M r q r (k) - 1
β sr + F (k-1) r q (k) r q (k) r and τ (k) n = α n + N -1 β n + X -H F (k-1) 2 2 . ( 27
)
As for the computation of the shape parameters, the choice of the hyperparameters (α sr , β sr ) and (α n , β n ) is questionable, since the Gamma distribution has been chosen for mathematical convenience and does not reflect any real prior information on the precision parameters, except their positiveness.

That is why, the prior distributions on τ n and τ sr should be as minimally informative as possible. To this end, one sets, as previously, α n = α sr = 1

and β n = β sr → 0.

Computation of the optimal force vector

F (k)
By definition, the force vector at iteration k is solution of the following minimization problem:

F (k) = argmin F X -HF 2 2 + R r=1 λ (k) r F r q (k) r q (k) r , (28)
where λ

(k) r = τ (k) sr / τ (k)
n is the regularization parameter at iteration k.

Because this minimization problem has the same form as the one used to compute the initial force vector F

, the IR algorithm, briefly described in section 3.2.1, is also implemented to compute the optimal force vector F

(k) at iteration k. In this case, however, all the regularization parameters λ (k) r are considered during the resolution process [START_REF] Aucejo | On a full bayesian inference for force reconstruction problems[END_REF].

Convergence monitoring

As any iterative procedure, the proposed algorithm must be stopped after either a certain stopping criterion is less than some tolerance or a certain number of iterations fixed by the user is reached. In the present paper, the stopping criterion is related to the relative error of the force vector between two successive iterations. Mathematically, the relative error δ is defined such that:

δ F (k-1)
, F

=

F (k) -F (k-1) 1 F (k-1) 1 . (29)
The choice of the tolerance has a strong influence on the overall performance of the optimal Bayesian regularization, since a too small tolerance can lead to unnecessary calculations, while a too large tolerance can lead to inaccurate solutions. This particular point will be thoroughly studied in the next section.

Numerical validation

The present numerical study intends to demonstrate how the proposed optimal Bayesian regularization (OBR) operates with respect to the choice of the initial solution, the tolerance of the iterative process and the measurement noise level. For the sake of completeness, the proposed optimal Bayesian formulation is also applied for different prior structures and comparisons with results obtained from the corresponding full Bayesian inference are proposed.

Finally, the possibility of a parameters reduction is studied. To properly simulate experimental measurements, the exact vibration displacement field X exact is first computed from a FE mesh of the plate made up with 187 shell elements, assuming that only bending motions are measurable. Then, in absence of contradictory information, the exact displacement field is supposed to be corrupted by an additive Gaussian white noise with a signal-to-noise ratio (SNR) equal to 35 dB.

Finally, the transfer functions matrix H is computed from a FE model of the plate with free boundary conditions, assuming that only bending motions are measured. In other words, the computed transfer functions matrix H is dynamically condensed over the measurable degrees of freedom only [START_REF] Aucejo | Structural source identification using a generalized Tikhonov regularization[END_REF][START_REF] Ibrahim | Structural force identification at unmeasured locations[END_REF].

The main interest in using free boundary conditions to model the dynamic behavior of the plate is to allow the identification of external excitations acting on the structure as well as reaction forces at boundaries [START_REF] Renzi | Vibratory source identification by using the finite element model of a subdomain of a flexural beam[END_REF].

Application

To numerically validate any force reconstruction strategy, it is first necessary to define the reference force vector F ref that could serve as a proper benchmark. This reference force vector is computed from the transfer functions matrix H and the exact displacement field X exact thanks to the following relation:

F ref = H -1 X exact . (30)
This numerical study is focused on the identification of the excitation field at 350 Hz, i.e. outside the resonance frequencies of the plate. As shown in Fig. 1, the reference force vector corresponds to the description of the test case given in the previous section, since it exhibits smooth reaction forces at boundaries of the plate as well as a unit point force F 0 at (x 0 , y 0 ) = (0.42 m, 0.25 m).

Consequently, the analysis of the reference force vector suggests that two identification regions can be defined to apply the OBR: (i) a central region associated to the shape parameter q 1 and containing the point force only and

(ii) a region associated to the shape parameter q 2 and corresponding to the boundaries of the plate [see Fig. 2].

Regarding the application of the OBR to the previous synthesized data, it should be noted that the tolerance for stopping the iterative is set to 10 if no contradictory information is given. Finally, to quantify the accuracy of identified solutions with respect to the considered operating conditions, the relative error (RE) and the peak error (PE) are evaluated. Formally, the relative error δ is defined in Eq. ( 29) and corresponds, in the present situation, to the relative error between the reference excitation field F ref and the excitation field F reconstructed from the OBR. As a result, the relative error must be written δ(F ref , F). Practically, the relative error is a global indicator of the reconstruction quality. The peak error is a local indicator that allows assessing the reconstruction quality of the point force amplitude.

Mathematically, it is defined such that:

δ p = F p -F ref p F ref p , (31)
where F ref p is the point force amplitude associated to the reference force vector F ref , while F p is the point force amplitude associated to the identified solution F.

Influence of the choice of the initial parameters

In section 3.2.1, it has been noted that only the definition of the shape parameters q (0) r is required to initialize the OBR, since all the other parameters are computed accordingly. In this context, it is interesting to assess the convergence of the OBR with respect to the initial choice of the shape parameters. To this end, it is supposed in the following that q (0) 1 = q (0) 2 = q 0 with q 0 = {0.5, 1, 2}. Fig. 3 presents the excitation fields reconstructed from the OBR for the three q (0) previously defined. At first sight, the identified excitation fields are in a relatively good agreement with the reference one. This qualitative observation is confirmed by the results listed in Table 1. In particular, it should be noted that the values of the identified parameters are very close to each other. It is also worth noting that the point force is properly recovered in all cases. It should however be noted that the overall reconstruction error is about 41%, indicating that the reconstruction of the reaction forces is not as good as expected. This confirms the visual inspection of Fig. 3. Finally, the only noticeable difference appears in the number of iterations performed by the algorithm to reach the convergence.

The main conclusion of this convergence study is that the OBR is almost insensitive to the choice of the initial shape parameter q 0 . OBR for q 0 = 0.5, (-• -) OBR for q 0 = 1 and (• • • ) OBR for q 0 = 2

Influence of the choice of the convergence tolerance

As any iterative procedure, the identification accuracy of the OBR is partly related to the value of tolerance used to stop the iterative process.

For this reason, it is important to study the convergence of the OBR with respect to the value of tolerance defined by the user. Table 2 gathers the results obtained with the OBR for tolerance values ranging from 10 -3 to 10 -8 and q 0 = 2. The analysis of the results show that the OBR converges quickly to the optimal solutions. Consequently, one can choose the value of the tolerance so as to define a compromise between the solution accuracy and the computational efficiency.

Influence of the measurement noise level

In the previous sections, it has been supposed that the vibration field is corrupted by an additive Gaussian white noise having an SNR equal to 34 dB. However, it is legitimate to wonder whether the quality of the identified solutions remains the same when the SNR gets lower. To answer this question, the OBR is applied to vibration data having SNR ranging from 35 dB to 10 dB. For this numerical experiment, the initial shape parameter q 0 is set to 2.

The results listed in Table 3 indicates that the OBR performs well whatever the SNR values considered, which is an indicator of the robustness of the proposed regularization strategy.

Comparison with Bayesian inference results

At this stage of the paper, it is interesting to compare the results obtained in the previous section with those obtained from the full Bayesian inference as described in Ref. [START_REF] Aucejo | On a full bayesian inference for force reconstruction problems[END_REF].

After initially drawing 7000 samples, the values of the parameters corresponding to the mode of posterior distributions of each parameter of the formulation are given in Table 4 for SNR values ranging from 10 dB to 35 dB. Overall, the most probable values of the parameters inferred from the proposed MCMC procedure are in line with the OBR results. However, the analysis of the PE and RE indicators shows that, while the estimation of the point force amplitude is quite satisfying, the reconstruction of the reaction forces differs significantly when the SNR decreases. A possible explanation of these contrasted results is that a normal approximation has been used to draw samples from the full conditional probability distribution p(F|X, τ n , τ sr , q r )2

[see Ref. [START_REF] Aucejo | On a full bayesian inference for force reconstruction problems[END_REF] for details]. Finally, it worth mentioning that the OBR is, as expected, computationally more efficient than the full Bayesian inference, since the solution is obtained in 1 s on average for the OBR, while about 6 min are required on average for the full Bayesian inference.

Influence of the choice of the prior distributions

The proposed OBR being based on a particular prior structure, it is interesting to study the influence of the choice of the prior distributions on the

Prior distributions of the shape and precision parameters

As a first modification, let us assume that the shape and precision parameters are uniformly distributed. Formally, this imposes that:

p(τ n ) ∝ I R + , p(τ sr ) ∝ I R + and p(q r ) ∝ I [l b ,u b ] . (32)
From a practical standpoint, the corresponding OBR is obtained by setting α n = α sr = α r = 1 and β n = β sr = β r = 0 in Eqs. (18a) and [START_REF] Feng | Simultaneous identification of bridge structural parameters and vehicle loads[END_REF].

Given this particular formulation, obtained estimated parameters are listed in Table 5 for SNR values ranging from 10 dB to 35 dB. As expected, the results are identical to those presented in Table 3. Another possibility is to define a Jeffrey's prior for the precision parameters τ n and τ sr , while keeping the prior distribution of the shape parameters defined as a truncated Gamma distribution. It results that the prior distributions of the precision parameters are defined such that:

p(τ n ) ∝ 1 τ n and p(τ sr ) ∝ 1 τ sr . ( 33
)
From a practical standpoint, the corresponding OBR is obtained by setting α n = α sr = 0 and β n = β sr = 0 in Eq. ( 27). In doing so, one obtains the results presented in Table 6. Contrary to the previous situation, the results only slightly differ from those listed in Table 3. Consequently, it can be concluded that regulation results are quite insensitive to the choice of the prior distributions of the shape and precision parameters provided that the latter are weakly informative or non-informative.

Prior distribution of the force vector

As for the shape and precision parameters, the choice of the prior distribution of the force vector to identify is not unique. The main advantage of the proposed prior distribution is to be flexible enough to deal with various distributions such as Gaussian and Laplace distributions by properly setting the value of the shape parameters. When the shape parameters are supposed to be deterministic, the corresponding Bayesian formulation is given by:

p(F, τ n , τ sr |X, q r ) ∝ p(X|F, τ n ) p(τ n |α n , β n ) R r=1
p(F r |τ sr , q r ) p(τ sr |α sr , β sr ).

A classical choice made in the literature is to suppose that the components of the force vector are normally distributed. In the present case, this implies that q r = 2 for all the considered regions. This formulation gives rise to an Augmented group Tikhonov regularization. As shown in Fig. 4 and Table 7, this particular formulation is unfortunately unable to properly identify the point force excitation, while the reconstruction of the reaction forces is quite satisfying. When the components of the force vector are supposed to follow a Laplace distribution (i.e. q r = 1 for each region), the results are not much better as indicated by the analysis of Fig. 5 and Table 8.

These disappointing results can be explained by the fact that the prior on the force vector is not well adapted to the reconstruction of both localized and distributed sources at the same time.

Parameters reduction

When dealing with parameter optimization, it could be interesting to reduce the number of parameters to constrain the space of admissible solutions by eliminating some degrees of freedom. In the present work, two possibilities are studied. The first one consists in defining a unique precision parameter τ s for all the identification region, which leads to define a single regularization parameter for computing the optimal force vector. From the theoretical standpoint, the corresponding Bayesian formulation is written:

p(F, τ n , τ s , q r |X) ∝ p(X|F, τ n ) p(τ n |α n , β n ) p(τ s |α s , β s ) × R r=1 p(F r |τ s , q r ) p(q r |α r , β r , l b , u b ). (35)
Regarding the resolution algorithm, this assumption leads to change the calculation of the precision parameter τ s at step 2.c of the pseudo-code described in section 3.2, whose closed-form expression is given by Eq. [START_REF] Feng | Simultaneous identification of bridge structural parameters and vehicle loads[END_REF]. Under this assumption, the precision parameter τ (k) s at iteration k is expressed as:

τ (k) s = α s + R r=1 M r q (k) r R r=1 F (k-1) r q (k) q (k) . ( 36
)
As shown in Table 9, the results obtained for a unique precision parameter τ s are in a very good agreement with those listed in Table 3. Consequently, it is possible to employ only one precision parameter τ s for all the considered regions. In the light of the previous observations, it is legitimate to wonder whether it is possible to limit the number of the shape parameters. To explore this 1. The OBR being almost insensitive to the choice of the initial shape parameters from which all the other initial parameters are derived, it is recommended to set q (0) 1 = q (0) 2 = 2. This allows computing the initial parameters in an efficient manner;

2. The convergence study, presented in section 4.2.2, has demonstrated that the OBR converges quickly to the optimal solution. Consequently, it is recommended to set the tolerance of the iterative process to 10 -3 , which leads to a reasonable compromise between the solution accuracy and the computational efficiency; 3. As shown in section 4.2.6, it is possible to limit the number of parameters of the optimization problem. Indeed, a parameter reduction allows constraining more the space of admissible solutions, which can help for finding an optimal solution. However, our results suggests to reduce the number of the precision parameters τ sr instead of the shape parameters q r .

Experimental validation

This section aims at confirming the main conclusions drawn in the previous section by extending the analysis to a real-world application. Since the proposed method is a natural extension of the work presented in Refs. [START_REF] Aucejo | Bayesian source identification using local priors[END_REF][START_REF] Aucejo | On a full bayesian inference for force reconstruction problems[END_REF],

we have decided to perform this validation using the same experimental setup.

Description of the experimental set-up

The structure under test is a steel parallelepiped box, excited on one of its faces by a shaker fed by a white noise signal and equipped with a force sensor [see Fig. 6]. The parameters of this experimental validation are given in Table 11.

Measurements of the vibration field were carried out with a scanning laser vibrometer on a grid of 19×22 points along y and z directions respectively using the force signal as phase reference. In all the subsequent identifications, the measured vibration velocity field is normalized to the force signal delivered by the force sensor. In doing so, the identified point force F 0 should be equal to 1.

Regarding the FE mesh used to model the dynamic behavior of the plate, it has been designed to perfectly match the measurement mesh. Hence, it consists of 378 shell elements, making the model theoretically valid up to Location of the force (y 0 , z 0 ) = (0.10 m, 0.09 m) 5000 Hz. Then, the corresponding FE model with free boundary conditions has been used to compute the transfer functions matrix H, considering the bending motions as the only available data.

Application

A careful analysis of the experimental set-up suggests the definition of two identification regions. The first region associated to the shape parameter q 1 contains the point force only, while the second region, associated to the shape parameter q 2 , corresponds to the boundaries of the plate [see Fig. 7]. In the experimental application, the OBR is applied at 525 Hz, which is a frequency lying outside the resonance frequency of the plate for a conver-gence tolerance equal to 10 -3 and q (0) = 2. The aim of this experimental validation is to compare the OBR implemented from the full Bayesian formulations given by Eq. ( 11) and ( 35) respectively. In other words, we are going to compare the OBR considering either two precision parameters τ sr (case 1) or a single precision parameter τ s (case 2). For the sake of completeness, these results are also compared with the MAP solution estimated after initially drawing 7000 samples from the full Bayesian inference introduced in Ref. [START_REF] Aucejo | On a full bayesian inference for force reconstruction problems[END_REF]. Fig. 8 presents the excitation fields reconstructed from the OBR for the two considered parametrizations. This figure shows that the reconstructed excitation fields are qualitatively similar. More precisely, the location and the amplitude of the point force amplitude are recovered. This observation is confirmed by the analysis of the results listed in Table 12. Indeed, the estimates of the shape parameters are consistent with our expectations and the peak error indicates that the point force amplitude is properly estimated. Furthermore, it is interesting to note that, in the present experimental validation, the proposed OBR results are fully consistent with those estimated from the full Bayesian inference.

Conclusion

In the present paper, an optimal Bayesian regularization has been introduced for reconstructing mechanical sources acting on a structure. The purpose of the proposed approach is to determine the most probable excitation field as well as the most probable parameters associated to the full Bayesian formulation previously published by the authors and recalled in this contribution. One of the most interesting features of the proposed strategy lies in its ability in computing optimal solutions from a minimal amount of prior information on the sources to identify. A set of recommendations has been proposed from the numerical and experimental validations we have conducted. It has also been shown that the proposed approach is robust against the initialization, the tolerance set to stop the iterative process and the measurement noise level. Finally, it must be said that the optimal Bayesian regularization is quite general, since it can be directly used to tackle identification problems in frequency or time domains, provided that the reconstruction model is established accordingly.

; 2 .

 2 while convergence is not reached a. k = k + 1; b. for each region r, compute q (k) r from Eq. (18a) given τ (k-1) sr and F (k-1) r ; c. for each region r, compute τ (k) sr from Eq. (18b) given F

  vibration field X. The application of this idea to the proposed Bayesian formulation leads to the following values for τ

  2.1]. For this particular reason, we set l b = 0.05 and u b = 2 for all the validations presented in the rest of the paper.

4. 1 .

 1 Description of the test case In the present numerical validation, one seeks to identify a point force of unit amplitude (i.e. F 0 = 1 N) acting on a thin simply supported steel plate with dimensions 0.6 m × 0.4 m × 0.005 m. The coordinates of the point force, measured from the lower left corner of the plate, are (x 0 , y 0 ) = (0.42 m, 0.25 m). Practically, this configuration allows studying the influence of the definition of local regularization terms, since the present excitation field exhibits two types of spatial distribution over the structure, namely a smooth distribution of the reaction forces at boundaries and a singular distribution around the location of the point force.

Figure 1 :

 1 Figure 1: Reference force vector F ref at 350 Hz

Figure 2 : 2 (

 22 Figure 2: Definition of the identification regions -(•) region 1 (Point force), (×) region 2 (Reaction forces) and (♦) location of the point force

Figure 3 :

 3 Figure3: Reconstructed force vector F at 350 Hz from the OBR with respect to the initial of the shape parameter q 0 . (a) Surface plotq 0 = 0.5, (b) Surface plotq 0 = 1, (c) Surface plotq 0 = 2 and (d) Section view at y 0 = 0.25 m -(-) Reference, (--)

Figure 4 :

 4 Figure 4: Reconstructed force vector F at 350 Hz when the components of the force vector are normally distributed -(a) SNR = 10 dB and (b) SNR = 35 dB

Figure 5 :

 5 Figure 5: Reconstructed force vector F at 350 Hz when the components of the force vector follow a Laplace distribution -(a) SNR = 10 dB and (b) SNR = 35 dB

Figure 6 :

 6 Figure 6: Experimental set-up

Figure 7 : 2 (

 72 Figure 7: Definition of the identification regions -(•) region 1 (Point force), (×) region 2 (Reaction forces) and (♦) location of the point force

Figure 8 :

 8 Figure 8: Reconstructed force vector F at 525 Hz (a) OBR for two precision parameters τ sr (case 1), (b) OBR for a single precision parameter τ s (case 2) and (c) MAP estimated from MCMC samples

Table 1 :

 1 Convergence study of the OBR with respect to the choice of the initial shape parameters -N it : Number of iterations of the algorithm

		Initial shape parameter q 0
		0.5	1	2
	q 1	0.51	0.51	0.51
	q 2	2	2	2
	τ s1	32.02	31.07	31.54
	τ s2	19.74	19.71	19.72
	τ n	2.78×10 16 2.80×10 16 2.79×10 16
	PE (%)	0.31	0.24	0.27
	RE (%)	41.43	41.88	41.66

Table 2 :

 2 Convergence study of the OBR with respect to the tolerance fixed to stop the iterative process -N it : Number of iterations of the algorithm

	Tolerance

Table 3 :

 3 Convergence study of the OBR with respect to the measurement noise level corrupting the data -N it : Number of iterations of the algorithm

	SNR

Table 4 :

 4 Most probable parameters inferred from MCMC with respect to the measurement 81×1013 1.88×10 14 5.89×10 14 1.89×10 15 5.26×1015 1.70×1016

	noise level corrupting the data
				SNR
		10	15	20	25	30	35
	q 1	0.62	0.71	0.59	0.60	0.48	0.47
	q 2	1.99	1.99	2	2	1.98	1.99
	τ s1	29.54	27.10	30.92	30.68	34.27	34.18
	τ s2	0.45	1.06	2.07	3.37	5.09	6.24
	τ n 5.PE (%)	-4.92	-1.57	-4.40	-2.62	-2.94	-2.20
	RE (%)	99.90	95.35	71.85	66.89	47.79	41.63
	estimation of the optimal solution.

Table 5 :

 5 Convergence study of the OBR with respect to the measurement noise level corrupting the data when the shape and precision parameters are uniformly distributed -N it : Number of iterations of the algorithm

	SNR

Table 6 :

 6 Convergence study of the OBR with respect to the measurement noise level corrupting the data when the precision parameters follow a Jeffrey's prior -N it : Number

	of iterations of the algorithm
				SNR
		10	15	20	25	30	35
	q 1	0.36	0.37	0.39	0.39	0.44	0.51
	q 2	2	2	2	2	2	2
	τ s1	40.44	39.34	37.15	36.16	32.99	31.46
	τ s2	24.24	22.11	20.75	19.90	19.43	19.35
	τ n	6.13×10 13 2.02×10 14 6.75×10 14 2.26×10 15 7.82×10 15 2.77×10 16
	PE (%)	-0.62	2.80	2.25	1.50	0.67	0.34
	RE (%)	28.80	30.39	33.10	36.10	40.40	41.69
	N it	15	15	18	12	14	23

Table 7 :

 7 Convergence study of the OBR with respect to the measurement noise level corrupting the data when components of the force vector is normally distributed -N it :

	Number of iterations of the algorithm
				SNR
		10	15	20	25	30	35
	τ s1	6.35×10 58 5.45×10 36 4.62×10 35 3.20×10 3 2.32×10 3 1.43×10 3
	τ s2	25.84	22.11	23.48	21.39	20.90	20.04
	τ n	4.32×10 13 9.05×10 13 1.39×10 14 1.74×10 15 5.74×10 15 2.09×10 16
	PE (%)	-100	-100	-100	-93.76	-91.17	-85.97
	RE (%)	36.54	34.11	31.12	53.24	54.94	57.71
	N it	9	10	15	13	13	17

Table 8 :

 8 Convergence study of the OBR with respect to the measurement noise level corrupting the data when the components of the force vector follow a Laplace distribution -N it : Number of iterations of the algorithm

	SNR

Table 9 :

 9 Convergence study of the OBR with respect to the measurement noise level corrupting the data for a unique precision parameter τ s -N it : Number of iterations of the

	algorithm
				SNR
		10	15	20	25	30	35
	q 1	0.34	0.35	0.36	0.34	0.41	0.51
	q 2	2	2	2	2	2	2
	τ s	32.62	30.63	28.98	28.52	26.06	24.55
	τ n	6.10×10 13 2.02×10 14 6.73×10 14 2.25×10 15 7.88×10 15 2.89×10 16
	PE (%)	1.39	3.39	2.43	1.72	0.94	-0.37
	RE (%)	26.73	27.25	29.13	30.53	36.73	42.11
	N it	14	14	16	15	12	15

Table 10 :

 10 Convergence study of the OBR with respect to the measurement noise level corrupting the data for a unique shape parameter q -N it : Number of iterations of the

	algorithm
				SNR
		10	15	20	25	30	35
	q	0.28	0.28	0.32	0.35	0.44	0.52
	τ s1	28.17	30.82	26.90	28.95	30.47	33.84
	τ s2	10.97	9.55	8.61	7.47	6.01	5.30
	τ n	5.96×10 13 1.97×10 14 6.67×10 14 2.25×10 15 7.87×10 15 2.72×10 16
	PE (%)	4.42	6.45	3.74	2.01	0.83	0.27
	RE (%)	122.40	103.05	104.60	86.48	90.32	81.71
	N it	55	31	100	31	26	30

Table 12 :

 12 Comparison of the OBR when considering two precision parameters τ sr (case 1) or a single precision parameter τ s (case 2) with the MAP estimated from MCMC samples -N it : Number of iterations of the algorithm

		case 1	case 2	MCMC
	q 1	0.26	0.26	0.40
	q 2	2	1.40	1.99
	τ s1	64.50	62.79	55.78
	τ s2	458.77	-	6.11
	τ n	7.99×10 8 8.25×10 8 9.01×10 8
	PE (%)	-0.6	0.8	-0.1
	N it	30	70	-

By setting α r = α sr = α n = 1 and β r = β sr = β n = 0, the resulting optimal Bayesian regularization is equivalent to that obtained assuming that the prior probability distributions on the shape and precision parameters are uniform.

It should be mentioned that Metropolis-Hastings and HMC updates have also been implemented without success due to convergence issues (low acceptance rate).

possibility, let us defining a unique shape parameter q for all the identification region. From the theoretical standpoint, the corresponding Bayesian formulation is written:

p(F r |τ s , q) p(τ sr |α sr , β sr ). [START_REF] Hansen | Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion[END_REF] Regarding the resolution algorithm, this assumption leads to change the calculation of the shape parameter q at step 2.b of the pseudo-code described in section 3.2. Under this assumption, the shape parameter q (k) at iteration k is solution of the following minimization problem:

where f (q|τ sr , F r ) = βq -α -1 + R r=1 M r log q + log Γ(1/q)( R r=1 M r ) -

As highlighted in Table 10, the reduction of the number of the shape parameters leads to inconsistent solutions, since the solver converges to the sparsest solution, which is not a desirable effect. However, these results are consistent with the results presented in Tables 7 and8 indicating that a reduction of the number of shape parameters is detrimental to the solution accuracy.

Recommendations

In the light of the previous results, we propose the following recommendations to properly apply the OBR: