HAL is

INTRODUCTION

Steel structures in sea or estuary area are subjected to corrosion process. The phenomenon is very complex due to the nature of the environment and material, and the type of the structure. This phenomenon needs to be characterized and modeled for structural analysis which accounts for the loss of thickness. Moreover, due to the randomness of the corrosion process, probabilistic models are needed for structural reliability [START_REF] Boéro | Timefunction reliability of harbour infrastructures from stochastic modelling of corrosion[END_REF]. Ultrasonic residual measurements allow determining profiles of loss of thickness, identifying the areas which are the most affected by corrosion and stating the assumptions for modeling the random corrosion process. Several inspection campaigns have been performed during the three last decades in some French harbours. Thus a great number of ultrasonic residual measurements are now available for structures in various environments. The protocol is recommended by the CET-MEF (French Center for Maritime and Fluvial Technical Studies :Engineering centre of the French Ministry of Public Works): at a given level, this protocol suggest to perform three geometrical measurements at a given location to assess the loss of thickness as the average of these three readings. Generally, the amount of data is more than 1000 per structure and allows performing a statistical analysis of the spatial distribution of the loss of thickness according to the different areas (mainly tidal and immersion areas). Moreover environmental parameters (temperature, PH, oxygen, salinity, conductivity, nutrients) have been measured since a decade and are available.

This study began within the MEDACHS framework (Marine Environment Damage to Atlantic Coast Historical and transport works or Structures-Interreg IIIB Project funded by EC 2005EC -2007) ) and is performed now in the GEROM framework (Risk management of French harbor structures: stakes, current practices and needs -Experience feedback of owners). The steel structures are sheet-piles, onpile and on-sheet-pile wharves. This paper focuses on sheet-piles seawalls and coffer-dams. More than 35 000 data are available: they correspond to measurements in 4 harbours and concern more than 20 quays (Boero et al. 2009a(Boero et al. , 2009b)). This paper focuses on the spatial dependence of the corrosion process and we present only results of three of them where the distance between data is small (0.20 m horizontally and 0.10 m vertically).

In the first part, the paper presents the data available (Euromarcorr data base) to study the spatio(temporal) fields of corrosion for steel sheet-piles.

The next part presents the geo-statistical modeling of the spatial variability of the steel sheet piles corrosion; and more particularly the study of the stationary of the process of corrosion according to the length.

We suggest a three scale modeling of spatial variability of corrosion base on a study case.

Finally, from identification of parameters of a stationary stochastic field, we suggest an optimization of the inspection.

Assessment of Spatial Variability of the corrosion of steel infrastructures from Ultrasonic measurements: application to coastal infrastructures ABSTRACT: Corrosion is one of the major causes of degradation for harbor structures. The time variant modeling is still a challenge due to the complexity of phenomena and the number of physical and electrochemical equations that govern the corrosion process and that are competing with each other. That emphasizes the role of condition assessment from on-site measurements as the unique input for maintenance decision. The cost of inspection of marine structures being prohibitive, owners required decision aid-tool for optimizing this step of the maintenance process. After a brief description of the existing data-base in France, this paper is devoted to the optimization of ultra-sonic measurements by considering the spatial variability measured from real data for each zone of corrosion. Four harbours are considered and called BO, HA, PL, SE. The distribution of the structures within the studied harbours and according to the type of water variation (dock at constant seawater level, wet dock and tidal dock) is presented on figure 3. Over a total amount of 23 structures, 11 are located in docks at constant seawater level, 9 in tidal docks and 3 in wet docks. In this last case the water level changes with the same period as the tide but with lower amplitude.

The variation of the seawater level in the various docks is detailed on figure 4.

The tide level corresponds to the range between the HAT (Highest Astronomical Tide) and the (LAT (Lowest Astronomical Tide). The variation of the height of water is important in tidal docks of BO and HA harbours. The variation is of course weaker in harbours located along the Mediterranean coast (i.e. PL and SE harbours).

Most of the measurements are made in the immersed zone, including the zone of low seawater level and the mud zone. Only the HA harbour began some measurements in the tidal/spray and in the atmospheric zones. The amount of data according to the studied harbours and for the various exposure zones is detailed on the figure 5. zones is detailed on the figure 5. The structure of the BO harbour was alone the subject of large campaign of measurements: it represents 99 % of all the available measurements.

Corrosion measurements

Corrosion of steel structures is classically assessed by ultrasonic non destructive testing. In France, a protocol is given by the CETMEF and classically used by several companies to perform inspections of coastal steel structures. The CETMEF is part of the French ministry of building. It is devoted to the diffusion of knowledge, to provide technical and research studies as well as engineering and expert's reports. This protocol consists in performing residual thickness measurements for several heights of the structure using ultrasonic testing. The corrosion products are removed by grinding. By using this technique and considering the harsh conditions for marine inspections, the error on the measurements at a given point on the structure cannot be neglected. Error comes both from the physical measurement (around 0.1 mm) and from the protocol (grinding, link diver-operator, etc.). Thus three measurements are performed on a given location: the three measurements are distributed on a circle, with diameter of about 5 cm, every 120° as shown on figure 1. The average of these three readings is considered as the "true" loss of thickness. For on-pile wharves, in view to analyze the corrosion around the pile, measurements are performed at several cardinal positions of the pile's section (see figure 1). For on-sheet-pilewall wharves, they are performed along the length of the wharf around every 10 meters, which is classically the mean distance between two piles for onpile wharves. Starting from these residual thickness values, and knowing the initial thickness of piles or sheet-piles, we deduce the thickness loss mainly due to corrosion. This protocol involves recommendations only and in some cases, this protocol is not followed because of the experience of owners or the results of the risk based inspection analysis. That is the case for structures considered in the lat part of the structures where the distance between measurements is closer.

Selected structures for spatial variability

The study of the spatial variability of corrosion is performed from the data of three structures extracted from the database. This selection is based on the fact that the distance between horizontal or vertical measurements are much weaker than everywhere else. Consequently, the quantity of information is consequent to perform a spatial analysis of the corrosion process.

In the three cases, it is important to remind that these specific campaigns were devoted to the discovery of perforations on steel sheet piles. In the case of the structure HA7d, the control was also started after the observation of accidental acids discharges.The main technical characteristics of these structures are detailed in tables 1. Table 2 gives more details concerning the type of sheet piles and the auscultation strategy.

Environment of selected structures

The physico-chemical parameters of seawater near the studied structures are described on table 3. Most of them were measured between 1997 and 2007. The physico-chemical parameters of seawater for the structures HA7d and HA8b are identical because they are located in the same dock.

MODELING SPATIAL VARIABILITY FROM MEASUREMENTS

Data pre-processing

The loss of thickness on each location is computed by averaging the three readings corresponding to three positions occupied by the ultrasonic transducer inside the required location (see figure 1). Sometimes, for technical reasons, a single basic measurement is realized in a location. In that case, the loss of thickness which characterizes this location was removed from the database because the error can be significant.

Moreover, perforations were observed on steel sheet piles. The loss of thickness cannot be measured in that case and the corresponding data were also removed for the statistical analysis.

To analyze the influence of the physico-chemical parameters on the corrosion mechanism in marine environment, the following hypotheses will be verified in the future: ! The temporal series of the physico-chemical parameters of the seawater follow a stationarity process (no trend or jump). The hypothesis of the stationarity or non-stationarity can be verified by an appropriate test; ! The absence or low influence of the depth on the physico-chemical parameters. We don't have any measure of these parameters along the wharf length; ! The homogeneous distribution of the measures of the seawater quality during the various seasons is necessary to make sure that the statistical characteristics of the parameters are not biased. It can easily be verified or corrected.

Analysis of the spatial variability of the corrosion process

The "loss of thickness" due to the corrosion is considered here as a stochastic field indexed by the space at the inspection time. Due to the corrosion process, we consider here that the principal directions are known: x, along the wharf length and z, along the wharf depth.

From the available distances between measurements, the objective is here to analyse if the corrosion is or not a second-order stationarity process in R ! i.e. according to the two coordinates x and z (see figure 6). figure 6). For that purpose, three conditions are to be verified: ! (i) The arithmetic mean of the marginal variables must be constant according to the studied direction; ! (ii) The variance of the marginal variables must be constant according to the studied direction too.

If not, the random process is not stationary. If yes and if the first condition is not verified, the field can be centered and the further analysis is made on the remainder; ! (iii) The covariance depends only on the distance between points according to the studied direction!"

Spatial variation according to z

This question is under investigation since a decade now in France first by [START_REF] Memet | La corrosion marine des structures métalliques portuaires : étude des mécanismes d'amorçage et de croissance des produits de corrosion[END_REF] and more recently by [START_REF] Boéro | Port infrastructure reliability: innovative approach of analysis and probabilistic modeling of inspection data -Application to the corrosion of metallic structures[END_REF]. It was identified already in the 50's by Larrabee and LaQue in USA [START_REF] Laque | Behaviour of metals and alloys in sea water[END_REF][START_REF] Larrabee | Corrosion-Resistant Experimental Steel for Marine Applications[END_REF]. This is described more in detail in [START_REF] Boéro | Statistical analysis of corrosion process along French coast[END_REF]. For the moment, the complexity of phenomena involved has been identifies in each of the 7 zones i.e. from top to bottom: Aerial, High level of tide, Tide, Low level of tide, Immersion, Mud, and Soil. Due to the number of random influencing factors varying with time no complete modeling is actually available: for instance, in the area "low level of tide" the abrasion can be due to the sand mining during a specific length of time. Phenomenological models suggested by [START_REF] Melchers | Corrosion uncertainty modelling for steel structures[END_REF] and [START_REF] Melchers | The critical involvement of anaerobic bacterial activity in modelling the corrosion behaviour of mild steel in marine environments[END_REF] or statistical models suggested by [START_REF] Boéro | Timefunction reliability of harbour infrastructures from stochastic modelling of corrosion[END_REF] are probably the best ones for long term modeling of the corrosion of harbor structures. Figure 7 provides the usual form of the corrosion [START_REF] Boéro | Timefunction reliability of harbour infrastructures from stochastic modelling of corrosion[END_REF]) with 6 area from the 7 presented upon: only "High level of tide" is absent (around depth -2m). Note that due to the lack of knowledge of the modeling in each area a piecewise stationary process can be suggested, except in the immersion area (from depth -6 to -11.5m) for which we now that the corrosion is decreasing with depth due to the diminished dissolved di-oxygen availability. As the corrosion peak locations are known and the depth limited to several meters, the inspection with depth is not so challenging than the inspection along the length for structures of several hundred meters length. That is why we focus on this aspect mainly in the following.

The 3 scales spatial variation according to x

Three scales of spatial variation have been identified [START_REF] Boéro | Statistical analysis of corrosion process along French coast[END_REF]): the scale of the shape of the sheet pile (U type), the regular part of the wall and special areas as the joints area between cofferdams or ends of the wharf. We focus in this paper on the regular part.

Assessment of stochastic field properties from measurements

Properties (i) and (ii) have already been proved in [START_REF] Boéro | Statistical analysis of corrosion process along French coast[END_REF] and we focus in this paper on the assessment of the auto-correlation ρ TT . In view to optimize the inspection we need to model the field.

If the stationarity is confirmed (first studies by [START_REF] Boéro | Statistical analysis of corrosion process along French coast[END_REF]) several methods are available to identify the parameters of a correlation function. Two major procedures have been reported in the literature for the estimation of δ for a spatially variable property from a digitized record of data. In the first procedure, reported by [START_REF] Li | Effect of spatial variability on maintenance and repair decisions for concrete structures[END_REF], the Maximum Likelihood Estimate method (MLE) is used in which different values for the model parameter of the proposed ACF model is assumed and the value that maximizes the corresponding MLE is taken as the model parameter. In the second procedure, proposed by [START_REF] Vanmarcke | Random fields: analysis and synthesis[END_REF] and applied in Schoefs et al. (2011b), a proposed ACF model can be adjusted to provide the best fit to the actual sample correlation coefficients ρ(Δx) thereby providing estimates of the corresponding model parameter (i.e. b in ( 1)). This last one can lead to a biased estimation when data are lacking. In this paper, we select an exponential ACF (Eq. ( 1)) and we use the MLE for the estimation of b (2).

ρ(Δx)=exp( - Δx b ); 0 < b (1) L = 1 2π exp - ν i 2 2 " # $ % & ' " # $ % & ' i=1 k ∏ = 1 2π " # $ % & ' k exp - ν i 2 i=1 k ∑ 2 " # $ $ $ $ % & ' ' ' ' (2)
where ν i is the i th component of the vector of independent standard values obtained from equation:

ν = C -1 T -µ T σ T " # $ $ % & ' ' ( 3
)
where T is the vector of realizations after inspection of the random variable Z and C a lower triangular matrix such that CC T = ρ and ρ the autocorrelation matrix. Beside, maximize L is equivalent to minimize L 1 :

∑ = = k i i L 1 2 1 ν (4) 4 PRACTICAL STUDY CASE FOR STOCHASTIC FIELD MODELLING

Identification of a regular part

We select here the cofferdam BO1 structure due to the low spacing between measurements (15 to 20 cm), 25 years old for which long-term corrosion mechanisms is installed, the its length of 1000 meters and the flat shape of the sheet-piles: that avoid the local effects due to U-shape sheet piles. A sketch map is presented on Figure 9 where we highlight in the red box an end-effect. Figure 10 pots the corrosion measures along the whole wharf in the Tidal zone where we observe this end-effect in the red box. Let us focus on the rest of the structure. It is 22 meters high and is built with 42 gabions of steel sheet-piles and 42 joints between them, called festoons (Fig. 10). When analyzing the corrosion in the tidal and immersion zones on a gabion, we observe end-effects too. Figure 11 gathers the average corrosion trajectories T(x, 25, $ i ) at 7 positions i in depth for a gabion. The position of the Low seawater zone (ZL) is +1,50 m and the Immersion zone (ZI) is below +1 m. It appears (in green and red circles of Figure 10 and 11) that ends-effects due to joint and presence of festoons are particularly pronounced in ZL, which is the zone of higher level of corrosion. Due to the complexity of corrosion phenomena in this area (current, intensive abrasion,…) we dismiss its analysis in this paper and we focus on what we can now name regular part presented on Figure 10. can now name regular part presented on Figure 10. Note that the mean and the standard deviation can be assumed as constant from the number of measurement 10 to 45: that represents 2/3 of the wharf and a length of about 7 meters for the regular part.

Identification of the stochastic field T

We focus on the auto-correlation in the following. It is plotted on Figure 12 for each of the 7 trajectories presented upon. The shape presents a decreasing with x and can be fitted by a function of x leading to the conclusion that the third property (iii) in section 3.2 is verified. Table 1. Gives the values of b obtained from the fitting of the auto-correlation plots according to section 3.5. It shows values of similar order of magnitude from 0.3 to 0.6 with an average value of 0.4. After the previous identification part, the objective of this part is to reach the best inspection strategy in the regular part for a given goal. The stochastic field being second order stationary we consider a Karhunen-Loève decomposition to generate trajectories and simulate inspections on these trajectories. We select in this paper a confidence interval of both the mean µ and the standard deviation % expressed as a percentage & of µ and % respectively. We carry out Monte-Carlo simulations to estimate the bounds of the confidence interval for a target probability for the inspection respectively P ti,µ and P ti,% . In a reliability study, P ti will be provided by the requirements on the accuracy of the probability of failure assessment. Thus we aim to find the minimum number of inspection along a trajectory and the number of trajectories that allow us to reach the objectives:

P T µ = P(! T ! (1"" µ )! T ;(1+" µ )! T # $ % & (4)
P T ! = P(! T ! (1"! " )" T ;(1+! " )" T [ ] (5)
In the application, target probabilities P ti,µ =P ti,% =0.85, &µ=0.1 and &%=0.2. These objectives can be reached only if enough measurements are performed (statistical limitation) and if independant measurements are obtains (correlation limitation). The first one is well known and we focus here on the second one. By knowing the correlation function fitted on experimental data in the previous section, the objective will be to get fairly correlated measurements in view to obtain a sample: that allows to compute (4) and ( 5). In view to sample independent events for T, we need to inspect the trajectories with a "sufficiently high distance L c " to get fairly correlated events (Schoefs et The condition L> L b is essential to characterize the spatial variability. According to the length of the regular part and results on Table 1, that is the case for the present application. The IDT is defined by assuming that after a given distance, the events measured from an inspection can be assumed as sta-tistically independent. A Spatial Correlation Threshold SCT of the spatial correlation gives this weak correlation. We consider in the following SCT=0.3 that is shown to be optimal by Schoefs et al. (Unpubl.) and seems to be consistent with Figure 12. According to (1), we have the relationship (6) and we obtain IDT=1.2 m and we take L c =IDT in the following.

IDT = b. ln(SCT ) (6)

Results and analysis

The objective is now to get the optimal quantity of inpections that allows us to reach the objectives of 5.1. We will have to optimize the total number of discrete inspections N=Np*Ns*Nt where Np is the number of repetitive tests for reducing the error of inspection, Ns is the number of inspected sections and Nt the number of trajectories. Here the error of inspection is neglected and Np is not consider. In view to satisfy (4) and ( 5), the optimal number N opt is solution of equation [START_REF] References Boéro | a. Technical management of French harbor structures. Part 1: Description of built assets[END_REF].

N opt = argmin N max(N ! µ ; N ! " ( ) (7)
Figure 13 gives the couple of minimum required values for Ns and Nt for the mean value (5) on two positions and the second in the immersion area (-0.5 m) and the second in the tide zone (+2.5 m). Correlations function being similar (Table 1), the curves are very close. 4), Fig. 14) and % (Eq. ( 5), Fig. 15). For z=-0.5, we obtain a total number of 48 with 6 trajectories imposed by the condition on µ and verified by the condition on %. For z=+2.5, we obtain a total number of 44 with 4 trajectories imposed by the condition on µ and verified by the condition on %. For this structure, the effect of the corrosion mechanism along z is of minor influence (10 %). Future developments should integrate specific requirements for P ti governed by a reliability analysis that inte- grates the influence of the position of corrosion (see [START_REF] Boéro | Timefunction reliability of harbour infrastructures from stochastic modelling of corrosion[END_REF]).

CONCLUSION

This paper investigates the scale of corrosion for sheet piles on harbour structures and the requirement for an optimal inspection strategy that accounts for spatial variability. Two directions of spatial variability are highlighted: the position in depth with several zones from sol to spray and the position along the length of the wharf. We focus in this paper on this last one were 3 scales of variability are identified: the scale of the sheet-pile (except for flat-shape), scale of the structure (identification of a regular part) and scale of the wharf with end-effects. By focusing on the regular part, it is shown that the stochastic field is second order. This property allows us to generate easily trajectories of this field and simulate inspections. A probabilistic-oriented criteria of the quality of inspection is suggested on the form of a confidence interval for both the mean and the standard deviation. Virtual inspection are then carried out and their minimum number required to reach this goal is suggested.

  Medium and long-term orrosion data -baseCorrosion measurements are collected on two types of marine structures: on-pile wharves and on-sheet pile wharves. Structures are located on the French coasts, i.e. Atlantic, Channel and Mediterranean coasts. In this paper, only corrosion measurements associated to steel sheet piles are presented.Almost all the studied structures were built between 1955 and 1985. Only three structures are older than 1955 and none is more recent than 1985. The owners began NDT measurements of residual thicknesses from 1990. Consequently most of the structures are 10 to 40 years old at the inspection time: medium an long-term corrosion can be analysed. Figure2presents the distribution of the age of the structures at the first, second, third and more inspection times.

Figure 2 .

 2 Figure 2. Age distribution of the studied structures at inspection time.

Figure 3 .

 3 Figure 3. Harbours and docks distribution.

Figure 4 .

 4 Figure 4. Harbours and docks distribution.

Figure 5 .

 5 Figure 5. Distribution of data (loss of thickness) in the various exposure zones.

Figure 1 .

 1 Figure 1. CETMEF protocol: cardinal points for pile (left) and location for sheet pile (right).

Figure 6 .

 6 Figure 6. Study of the stationarity of the corrosion process according to the direction x (a) and the direction z (b).

Figure 7 .

 7 Figure 7. Mean value and 95% quantiles of steel thickness loss in each zone at time 10, 25 and 50 years.

Figure 8 .Figure 9 .

 89 Figure 8. Sketch map of cofferdam BO1.

Figure 11 .

 11 Figure 10. Current part of the cofferdam with gabions, festoons and joints.

Figure 12 .

 12 Figure 12. Experimental auto-correlation at 7 positions along zaxis.

  al. Unpubl.): it is called IDT for Inspection Distance Threshold. Thus L c should satisfy: L c # ]IDT, L[ where L is the potential length of the trajectory, i.e. the maximum inspectable length .

Figure 13 .

 13 Figure 13. Number of required Ns and Nt to ensure Pti,µ = Pti," = 0.95 (L c =1.2 m).. Figures 14 and 15 present the number of total inspections required as a function of available trajectories (several position along z-axis with a distance of $z if available: $z >L c ) respectively for the condition on µ (Eq. (4), Fig.14) and % (Eq. (5), Fig.15). For z=-0.5, we obtain a total number of 48 with 6 trajectories imposed by the condition on µ and verified by the condition on %. For z=+2.5, we obtain a total number of 44 with 4 trajectories imposed by the condition on µ and verified by the condition on %. For this structure, the effect of the corrosion mechanism along z is of minor influence (10 %). Future developments should integrate specific requirements for P ti governed by a reliability analysis that inte-

Figure 14 .

 14 Figure 14. Required total number of inspection as function of Nt to ensure Pti,µ = 0.95 (L c =1.2 m)..

Figure 15 .

 15 Figure 15. Required total number of inspection as function of Nt to ensure Pti," = 0.95 (L c =1.2 m)..

  . Schoefs & T.V. Tran LUNAM Université, Université de Nantes, Institute for Research in Civil and Mechanical Engineering, CNRS UMR 6183, Nantes, France

F

Table 1 .

 1 Characteristics of the selected structures.

	______________________________________________
	Structure ______________________________________________ BO1 HA7d HA8b
	Function	General	General		General
		cargo	cargo		cargo
	Type	Cofferdam Seawall		Seawall
	Sheet piles	Flat	U-sections U-sections
	Width (m)	0.5	0.5		0.5
	Length (m)	758 (1230*) 345		187
	Age (Years)	25	28		26
	Sea	Channel	Channel		Channel
	Dock type	Tidal	Constant	Constant
			seawater	seawater
			level		level
	Seawater
	variation level 9.1 _____________________________________________ 0.6 0.6
	*wrap around distance for Cofferdam
	Table 2. Corrosion measurements strategies of the selected
	structures. ______________________________________________
	Structure ______________________________________________ BO1 HA7d* HA8b
	Location of measurements
	on steel sheet piles	-	In & out Out-pans
				pans
	Length of the measured
	part of the structure (m)	991.3 35.0	123.0
	Height of the measured
	part of the structure (m)	3.0	9.6	6.1
	Number of measurements
	following length		5988 71	36
	Mean horizontal spacing
	between measurements (m)	0.15	0.5	3.5
			to 0.2
	Number of measurements
	following depth		7	13	48
	Mean vertical spacing
	between measurements (m)	0.5	0.8	0.1
	Number of loss thickness
	measurements		32238 685	504
	Measurements location in
	several exposure zones** ______________________________________________ T, L, I L, I L, I
	Tidal zone ZT ______________________________________________
	Number of measurements
	following depth		2	-	-
	Number of loss thickness
	Measurements		11073 --
	Percentage of
	the total amount of
	measurements ______________________________________________ 34 --
	Low seawater zone ZL ______________________________________________
	Number of measurements
	following depth		1	4	2
	Number of loss thickness
	Measurements		5350 220	24
	Percentage of

Table 3 .

 3 Mean values and ranges of physico-chemical parameters of seawater near the studied structures.

	______________________________________________
	Structure	BO1 _______________	HA7d & HA8b ______________
	Parameter ______________________________________________ Mean Min Max Mean Min Max
	Temperature
	(°C)	13.3	7.2 19.5	13.7	8.1 20.7
	pH	8.0	7.8 8.1	8.1	7.7 8.5
	Conductivity
	(mS/cm)	49.0	46.8 50.5	37.2	33.7 41.4
	Salinity (g/l)	32.9	31.5 33.7	25.4	23.9 27.7
	O2 (mg/l)	8.7	6.4 11.2	8.7	6.9 11.0
	SM* (mg/l)	9.9	4.7 17.7	8.3	3.3 15.0
	NH4 (mg/l)	0.5	0.2 0.9	0.3	0.1 0.6
	NO2 (mg/l)			0.2	0.1 0.3
	NO3 (mg/l)			6.4	2.6 8.8
	PO4 (mg/l) _____________________________________________ 0.4 0.1 0.6
	*SM = Suspended materials.