A Mouragues
email: arthur.mouragues@u-bordeaux.fr

P Bonneton

D Lannes

B Castelle

V Marieu

Field data-based evaluation of methods for recovering surface wave elevation from pressure measurements

Keywords: wave measurements, pressure sensor, reconstruction methods, Acoustic Surface Tracking, wave-by-wave analysis, nonlinear waves, extreme

We compare different methods to reconstruct the surface elevation of irregular waves propagating outside the surf zone from pressure measurements at the bottom. The traditional transfer function method (TFM), based on the linear wave theory, predicts reasonably well the significant wave height but cannot describe the highest frequencies of the wave spectrum. This is why the TFM cannot reproduce the skewed shape of nonlinear waves and strongly underestimates their crest elevation. The surface elevation reconstructed from the TFM is very sensitive to the value of the cutoff frequency. At the individual wave scale, high-frequency tail correction strategies associated with this method do not significantly improve the prediction of the highest waves. Unlike the TFM, the recently developed weakly-dispersive nonlinear reconstruction method correctly reproduces the wave energy over a large number of harmonics leading to an accurate estimation of the peaked and skewed shape of the highest waves.

This method is able to recover the most nonlinear waves within wave groups which some can be characterized as extreme waves. It is anticipated that using relevant reconstruction method will improve the description of individual wave transformation close to breaking.

Introduction

Pressure sensors have long been used to measure waves in the coastal zone mainly because of their robustness, low-cost aspect and convenience to deploy. However, they do not provide direct measurement of the wave surface elevation.

The widely-used method to reconstruct the wave surface elevation from pressure measurements at the bottom is the so-called transfer function method (TFM; e.g. [START_REF] Bishop | Measuring waves with pressure transducers[END_REF] [START_REF] Bishop | Measuring waves with pressure transducers[END_REF]), based on the linear wave theory. The TFM allows to recover linear wave fields and gives a reliable estimate of the significant wave height (i.e wave energy). [START_REF] Guza | Local and shoaled comparisons of sea surface elevations, pressures, and velocities[END_REF] [START_REF] Guza | Local and shoaled comparisons of sea surface elevations, pressures, and velocities[END_REF] found that the total harmonic variance could be retrieved with error of 7.6 % near the breakpoint. However, for the highest frequencies, the energy density spectra reconstructed from the TFM blows up. To prevent the latter, a cutoff frequency is commonly used (e.g. Lee and Wang, 1984 [3]; [START_REF] Bishop | Measuring waves with pressure transducers[END_REF] [START_REF] Bishop | Measuring waves with pressure transducers[END_REF]).

Contrary to what is generally accepted in the literature for swell reconstruction, the need for such a cutoff is mainly due to wave nonlinearities rather than to pressure measurement noise (Bonneton and Lannes, 2017 [4]). In this paper, we will show that the cutoff frequency is artificial and that the TFM solution is very sensitive to its value.

In shallow water, nonlinear interactions induce the development of high-frequency harmonics which cannot be correctly reproduced by the TFM. Martins et al., 2017 [START_REF] Martins | The influence of swashbased reflection on surf zone hydrodynamics: a wave-by-wave approach[END_REF] found that the TFM fails to recover the peaked and skewed shape of nonlinear waves with individual wave height error up to 30 %. However, well predicting nonlinear waves, especially in the shoaling zone, is of paramount importance for many coastal applications. Indeed, the most nonlinear waves are very often found to be the largest waves. An accurate prediction of these waves is then essential for applications involving extreme wave events, wave submersion studies, or coastal construction projects that need to cope with the height of the most extreme waves. Moreover, an accurate characterization of wave skewness and asymmetry is essential for studying sediment transport (Dubarbier et al., 2015 [START_REF] Dubarbier | Process-based modeling of cross-shore sandbar behavior[END_REF]). Lastly, the surface wave reconstruction is also crucial for the calibration of phase-averaged wave model parameters (e.g. Booij et al., 1999 [7]) and for the validation of phase-resolving wave models (e.g. Zijlema et al. 2011 [START_REF] Zijlema | Swash: An operational public domain code for simulating wave fields and rapidly varied flows in coastal waters[END_REF], Bonneton et al., 2011 [9]).

In the present paper, we review and apply the main methods designed to reconstruct in situ irregular waves. First, we present the commonly-used linear methods as well as recently developed nonlinear methods. Different high-frequency tail correction procedures associated with the TFM are also reviewed. Then, we apply and compare each method with field data, in terms of spectral and temporal parameters, in near-breaking conditions. In such conditions, wave groups contain highly nonlinear waves for which the use of the TFM, based on the linear wave theory, is questionable. More importantly, these waves need to be properly described as they control the break point position and can be characterized as extreme waves. Finally, we conduct a wave-by-wave analysis of the whole dataset in order to compare each method over a large range of nonlinearities.

Reconstruction methods

In this section, the main methods to reconstruct irregular wave surface elevation from in situ pressure measurements at the bottom are reviewed. We focus on unbroken waves propagating outside the surf zone in intermediate to shallow water depth, for which the flow can be assumed irrotational. In this work, the bottom variation contribution is assumed negligible, which is true for many coastal applications. The background current contribution is also assumed negligible, which is true for most wave-dominated coastal areas far from river mouths or tidal inlets. [START_REF] Bonneton | Recovering water wave elevation from pressure measurements[END_REF] [START_REF] Bonneton | Recovering water wave elevation from pressure measurements[END_REF] derived a reconstruction formula which takes into account a background current. However, their method requires additional velocity measurement, which, in most nearshore field campaigns, is rarely collected at the same location as the pressure sensor, and is therefore out of scope of the present paper.

Three main length scales are critical to the problem addressed here : the wave amplitude a, the characteristic horizontal length scale L (k = 1/L the typical wave number) and the mean water depth h 0 . The wave propagation is then controlled by two dimensionless parameters :

= a h 0 , µ = h 0 L 2 = (kh 0 ) 2 , (1)
where is a nonlinearity parameter and µ is a shallowness (or dispersion) parameter; or alternatively the steepness parameter σ :

σ = a L = √ µ. (2)
From a practical point of view, deep water cases (µ >> 1) are disregarded as pressure measurements are not relevant in such water depths. Reconstruction methods are usually based on an asymptotic expansion of the irrotationnal wave equations in terms of the steepness parameter σ, which is a small parameter for most coastal waves. The small steepness regime encompasses the two following scenarios: large amplitude waves ( ∼ 1) in shallow water (µ << 1) and small amplitude waves ( << 1) in intermediate depth (µ ∼ 1).

For the sake of clarity, a two-dimensional wave field associated with the Cartesian coordinates (x,z) is considered, where x corresponds to the horizontal axis along which waves propagate and z is the positive-upward vertical axis (see Fig. 1). The mean water level and the free surface elevation are defined by z = 0 and z = ζ(x, t), respectively. The pressure sensor is located at a distance δ m from the bottom level z = -h 0 and provides the measured pressure For very long waves (very small µ), the surface elevation can be estimated from the hydrostatic equilibrium. The hydrostatic reconstructed elevation, ζ δm H , is then given by :

P m (t) = P (z = -h 0 + δ m , t).
ζ δm H (t) = P m (t) -P a ρg + δ m -h 0 , (3)
where P a is the constant atmospheric pressure, ρ is the water density and g is the gravity. This hydrostatic reconstruction (Eq. 3) gives good results for tides and tsunamis, but cannot be applied to wind waves which have non-hydrostatic characteristics. Some of the most commonly-used non-hydrostatic linear reconstruction methods are introduced below, and the recently developed nonlinear approaches are further presented.

Linear methods

Linear wave theory

The derivation of the following reconstruction method can be applied to threedimensional wave fields but a two-dimensional wave field is considered here for the sake of derivation simplicity (see Fig. 1). The fluid motion is governed by the free-surface incompressible Euler equations. From the irrotational assumption, the horizontal velocity u and the vertical velocity w are given by the velocity potential φ as : u = ∂ x φ and w = ∂ z φ. Neglecting the O(σ) terms in the Euler 95 equations, the following linearized system is obtained :

∂ xx φ + ∂ zz φ = 0 for z ∈ [-h 0 , 0] (4)
∂ t φ + P (z, t) -P a ρ + gz = 0 for z ∈ [-h 0 , 0], (5)
where Eq. 4 is the mass conservation equation and Eq. 5 is the linearized Bernoulli equation. This set of equations is completed by linearized boundary conditions at the bottom and at the surface :

∂ z φ = 0 at z = -h 0 (6)
∂ z φ = ∂ t ζ at z = 0 (7)
P = P a at z = 0. ( 8
)
Evaluating Eq. 5 at z = -h 0 + δ m and z = 0 results in the following expression 100 of ζ δm H and ζ :

ζ δm H (t) = - 1 g ∂ t φ at z = -h 0 + δ m (9) ζ(t) = - 1 g ∂ t φ at z = 0. ( 10
)
The variables ζ, ζ δm H and φ are then decomposed using the following Fourier transform in space :

F X {f }(k) = R f (x)e -ikx dx, (11)
where F X {f } is the Fourier transform in space of the function f : x → f (x).

Combining Eq. 4 and 6, we get the expression of the Fourier transform in space of φ :

F X {φ}(k, z, t) = B cosh(k(z + h 0 )), ( 12
)
where B is an independent function of z.

Plugging Eq. 12 into Eq. 9 and 10, ζ can be expressed as a function of ζ δm H :

F X {ζ}(k, t) = K P (k)F X {ζ δm H }(k, t) (13)
K P (k) = cosh(kh 0 ) cosh(kδ m ) , (14)
where K P is the non-hydrostatic correction factor.

Eq. 13 and 16 are hereafter referred to as the linear formula in space, and the resulting reconstructed surface elevation is hereafter referred to as ζ L,space (see linear formula in space in Tab. 1). In Fig. 2, ζ L,space is compared to the surface elevation computed from the Full Euler equations (Fenton, 2014 [10]) in case of a periodic weakly nonlinear wave field ( = 0.15 and µ = 0.25). The linear formula in space significantly improves the hydrostatic reconstruction in terms of crest elevation and wave shape, even if :: the : crest elevation is still underestimated compared to the Full Euler solution.

Transfer function method

The linear formula (Eq. 13 and 14) involves a Fourier transform in space, which requires the knowledge of ζ δm H (or equivalently P m ) over the whole horizontal 120 space. However, for most coastal applications, P m is only known at one single measurement point. The common practice with the TFM is to replace the Fourier transform in space by a Fourier transform in time, using the linear dispersion relation to express k as a function of the pulsation ω. The TFM writes:

125 F T {ζ}(x, ω) = K P (ω)F T {ζ δm H }(x, ω) (15)
K P (ω) = cosh(kh 0 ) cosh(kδ m ) (16
)

ω 2 = gk tanh(kh 0 ), ( 17
)
where the Fourier transform in time is defined by :

F T {f }(ω) = R f (t)e -iωt dt. ( 18
)
Bonneton and Lannes (2017) [START_REF] Bonneton | Recovering water wave elevation from pressure measurements[END_REF] showed that this method is mathematically justified for linear wave fields but is questionable when applied to nonlinear waves. This is illustrated with the periodic weakly nonlinear wave field presented in Fig. 2. The surface elevation computed from the TFM is hereafter referred to as ζ L,NC (Eq. 15, 16 and 17; see TFM -no cutoff in Tab. 1). Fig. 3 shows that the energy density of the two first harmonics is well predicted by ζ L,space while the energy density of the following harmonics are underestimated. 1. The cutoff harmonic index is indicated by the vertical black dotted line.

Unlike the linear formula in space, the TFM (see ζ L,NC in Fig. 3) leads to an energy density overestimation, even for weakly nonlinear waves, which increases rapidly with the harmonics and leads to a blow up of the TFM solution (overestimation of the fifth harmonic by two orders of magnitude). As described by [START_REF] Bonneton | Recovering water wave elevation from pressure measurements[END_REF] [START_REF] Bonneton | Recovering water wave elevation from pressure measurements[END_REF], it is due to secondary harmonics which are phase locked, or bound, to the fundamental harmonics and travel at a celerity which is much larger than their intrinsic (linear) phase speed. Thus, the linear dispersive relation (Eq. 17) strongly overestimates the wave number of the harmonics leading to the overestimation of K P (Eq. 16).

To overcome this TFM problem, the commonly-used approach is to introduce a cutoff frequency f c . At f = f c , the TFM spectrum is truncated and replaced by the hydrostatic spectrum for f > f c (equivalent to a low-pass filter). The expression of K P then becomes:

K P (ω) = cosh(kh 0 ) cosh(kδ m ) for ω 2π ≤ f c (19)
K P (ω) = 1 for ω 2π > f c (20)
Eq. 20 is hereafter referred to as the sharp cutoff and the associated surface elevation is referred to as ζ L,Sh (see TFM -sharp cutoff in Tab. 1).

It is worth noting that contrary to what is generally accepted in the literature for swell reconstruction, the need for such a cutoff is mainly due to wave nonlinearities rather than to pressure measurement noise. Therefore, f c can be considered as a nonlinear cutoff frequency. Nonetheless, in most coastal applications, two different empirical approaches are generally used to determine the value of f c (Smith, 2002 [11]). The first one consists in setting f c at the frequency where the pressure signal is one order of magnitude higher than the noise floor, which is questionable. In the second approach, f c is set to the frequency where K P is less than 10 to 1000, which value depends on the pressure sensor resolution (Wolf, 1997 [12]). The reconstructed wave characteristics are very sensitive to the subjective value of f c (see Section 4.1.1; Smith, 2002 [START_REF] Smith | Wave pressure gauge analysis with current[END_REF];

Jones and Monismith, 2007 [START_REF] Jones | Measuring short-period wind waves in a tidally forced environment with a subsurface pressure gauge: Measuring waves with pressure gauges[END_REF]). In Fig. 3, the value of f c is optimized i.e.

we set f c by comparing the wave energy reconstructed by the TFM with no nonlinear cutoff frequency, ζ L,NC (see TFM -no cutoff in Tab. 1), and the true wave energy. f c is taken at the frequency where ζ L,NC starts to overestimate the true wave energy (i.e. where K P is too high), here after the second harmonic (see the vertical dashed line in Fig. 3).

In most studies, the direct measurement of the surface wave elevation is not available but is retrieved from pressure measurements and f c cannot be optimized objectively. Most of the time, it is not clear how the cutoff frequency has been set and, for field applications, its value typically ranges between 0.25 and 0.6 Hz (e.g. Guza and Thornton, 1980 [2]; Ruessink et al, 1998 [START_REF] Ruessink | The systematic contribution of transporting mechanisms to the cross-shore sediment transport in water depths of 3 to 9 m[END_REF]; Smith, 2002 [11]; Sénéchal et al, 2004 [15]). However, the way f c is set is crucial and can strongly affect the wave shape. Indeed, the cutoff induces spectral information loss beyond f c that will generate oscillations within the reconstructed time series (see Section 4.1.1). The frequency of these oscillations being of the same order as f c , those are then strongly dependent on the cutoff and are not physical. This kind of oscillations is hereafter referred to as parasite oscillations.

High-frequency tail empirical correction

As introduced above, the widely-used TFM requires a cutoff frequency. Using the sharp cutoff method (Eq. 20; see TFM -sharp cutoff in Tab. 1) will induce spectral information loss for f > f c (see Fig. 3). To limit this loss of information, several empirical formula were derived to artificially fill the high-frequency tail.

Three empirical methods are presented here.

A first method is to replace the high-frequency tail by a Jonswap diagnostic tail. The energy density spectra E(f ) is expressed as a function of f -n where n represents the tail's slope (Eq. 21; see TFM -Jonswap in Tab. 1) :

E(f > f c ) = E(f c ) f f c -n . ( 21
)
The value of n depends on the water depth. n = 5 is usually set in deep water while n = 4 and n = 3 are set for intermediate water and shallow water, respectively.

Eq. 21 allows to better predict spectral wave parameters (e.g. Wolf, 1997 [12]; Smith, 2002 [11]; Jones and Monismith, 2007 [START_REF] Jones | Measuring short-period wind waves in a tidally forced environment with a subsurface pressure gauge: Measuring waves with pressure gauges[END_REF]). However, this method is not able to recover surface elevation time series as the phase signal is not given by this approach.

The two other methods consist in tuning the value of K P beyond f c . Neumeier [16] uses an empirical correction factor K P,L which linearly decreases over an artificial frequency range (Eq. 22 and 23; see TFM -linear cutoff in Tab. 1).

K P (ω) = K P,L for f c < ω 2π < f lin (22)
K P (ω) = 1 for ω 2π > f lin , (23)
where the expression of K P,L and f lin can be found in Neumeier [16].

A steady correction factor can also be applied (Eq. 24; see TFM -steady cutoff in Tab. 1). The correction factor beyond f c is taken as K P (ω = 2πf c ) and stays the same over the whole high-frequency tail.

K P (ω) = K P (ω = 2πf c ) for ω 2π > f c (24)
In Section 4.1.1, the influence of the above high-frequency tail correction methods on wave reconstruction and parasite oscillations will be addressed. A sensitivity study over the typical in situ f c range will be also conducted.

Semi-empirical transfer function method

To avoid introducing a cutoff frequency, several authors have proposed local methods, as opposed to global (spectral) methods, in order to improve the shape and height of individual waves. Nielsen (1986) [START_REF] Nielsen | Local approximations : a new way of dealing with irregular waves[END_REF] was the first to develop such methods called local sinusoidal approximation (LSA) methods.

A local frequency based on the local curvature is defined as:

ω 2 i = - (∂ tt ζ δm H ) i (ζ δm H ) i = - (ζ δm H ) i+1 -2(ζ δm H ) i + (ζ δm H ) i-1 (ζ δm H ) i ∆t 2 (25)
where (ζ δm H ) i is the i th value of time series ζ δm H (Eq. 3) and ∆t = 1 f a , f a being the sampling rate.

Along with a stretched linear theory, Nielsen (1986) [START_REF] Nielsen | Local approximations : a new way of dealing with irregular waves[END_REF] established the following semi-empirical transfer function method (see semi-empirical TFM in Tab. 1):

ζ N = ζ δm H F ω 2 g (h 0 + ζ δm H -δ m ) ( 26
)
where the transfer function F is fitted as

F (x) = exp A δ m h 0
x and the empirical factor A is given by A δ m h 0 = 0.64 + 0.34δ m h 0 .

Fenton (1987) [START_REF] Fenton | Polynomial approximation and water waves[END_REF] introduced local polynomial approximation (LPA) methods in which the complex velocity potential and the surface elevation are given by polynomials and incorporated into the fully nonlinear equations of motion.

Townsend and Fenton (1997) [START_REF] Townsend | A comparison of analysis methods for wave pressure data[END_REF] compared both LSA and LPA and concluded that LSA (Eq. 26) performs better than LPA especially for low δ m /h 0 ratio.

Moreover, LSA requires less computational effort than LPA (Nielsen, 1989 [20];

Townsend and Fenton, 1997 [START_REF] Townsend | A comparison of analysis methods for wave pressure data[END_REF]). Therefore, only the LSA method from Nielsen (1989) [START_REF] Nielsen | Analysis of natural waves by local approximations[END_REF] is considered in this study. [START_REF] Clamond | Recovery of steady periodic wave profiles from pressure measurements at the bed[END_REF]). Nevertheless, all these methods were derived assuming steady water waves propagating at a constant celerity and are therefore not suitable for real coastal applications.

However, Oliveras et al. (2012) [START_REF] Oliveras | Recovering the water-wave profile from pressure measurements[END_REF] derived a heuristic reconstruction method 230 ζ HE as a function of ζ L (here, ζ L is the surface elevation reconstructed from the linear formula in space or from the TFM) that can be applied for irregular waves travelling at different wave celerities. For δ m = 0 (see Vasan and Oliveras, 2017 [START_REF] Vasan | Water-wave profiles from pressure measurements[END_REF] if δ m > 0), ζ HE is written as follows :

ζ HE = ζ L 1 -F -1 T {k sinh(kh 0 )F T {ζ δm H }} , ( 27
)
where k is computed with the dispersion relation (Eq. 17).

235

As remarked in Bonneton and Lannes (2017) [START_REF] Bonneton | Recovering water wave elevation from pressure measurements[END_REF], at order O(σ), this formula is equivalent to :

ζ HE = ζ L - 1 g ζ L ∂ tt ζ L . (28)
Using data from laboratory experiments, the heuristic method was found to significantly improve the wave crest elevation as well as the wave shape compared to the TFM (Oliveras et al., 2012 [22]).

Recently, Bonneton and Lannes (2017) [START_REF] Bonneton | Recovering water wave elevation from pressure measurements[END_REF] and Bonneton et al. ( 2018) [START_REF] Bonneton | A nonlinear weakly dispersive method for recovering the elevation of irrotational surface waves from pressure measurements[END_REF] have derived nonlinear reconstruction methods also suitable for irregular waves.

Bonneton and Lannes (2017) [START_REF] Bonneton | Recovering water wave elevation from pressure measurements[END_REF] performed an asymptotic expansion of the nonlinear wave equations in terms of the steepness parameter σ. For δ m = 0 (see Bonneton and Lannes, 2017 [START_REF] Bonneton | Recovering water wave elevation from pressure measurements[END_REF] if δ m > 0) and neglecting the O(σ 2 ) terms, they obtained a fully-dispersive nonlinear reconstruction method :

ζ NL = ζ L - 1 g ∂ t (ζ L ∂ t ζ L ). (29)
The nonlinear term on the right-hand side of Eq. 29 can be splitted into two

nonlinear terms: (1) -1 g (ζ L ∂ tt ζ L ) and (2) -1 g (∂ t ζ L ) 2 .
Term (1) improves the wave extrema compared to the linear reconstruction ζ L by increasing the crest elevation and flattening the wave trough. Term (2), which is neglected in the heuristic method (see Eq. 28), amplifies the wave skewness and asymmetry.

Both nonlinear methods described above rely on ζ L . The latter can theoretically be computed using the linear formula in space ζ L,space . Fig. 4 shows the surface elevation reconstructed from the two nonlinear methods using the linear formula in space (ζ HE,space and ζ NL,space , respectively; see Tab. Both nonlinear methods accurately reproduce the crest elevation and the wave shape. Nonetheless, ζ NL,space provides a better description of the peaked wave shape as well as the crest elevation compared to ζ HE,space . As explained in the previous section, the measured pressure P m is often available at one single measurement point which implies to use the classical TFM instead of the linear 260 formula in space. Hence, in practice, a cutoff frequency needs to be introduced for computing ζ HE and ζ NL . Bonneton and Lannes (2017) [START_REF] Bonneton | Recovering water wave elevation from pressure measurements[END_REF] have applied these nonlinear reconstructions in case of fully-dispersive nonlinear bichromatic waves (µ = 0.53). Even though the heuristic method is able to properly reproduce the crest elevation, it still underestimates the skewed shape of the largest waves.

265

The fully-dispersive nonlinear method was found to provide a much better de-scription of the peaked and skewed waves than both the TFM and the heuristic reconstructions.

To overcome the need for a cutoff, in weakly-dispersive regime (µ << 1), Bonneton et al. (2018) [START_REF] Bonneton | A nonlinear weakly dispersive method for recovering the elevation of irrotational surface waves from pressure measurements[END_REF] made a Taylor expansion of the nonlinear wave equations with respect to µ. Neglecting the O(µ 2 ) terms, they obtained the following weakly-dispersive linear and nonlinear reconstruction methods (see Bonneton et al., 2018 [26] if δ m > 0):

ζ SL = ζ δm H - h 0 2g ∂ tt ζ δm H ( 30
)
ζ SNL = ζ SL - 1 g ∂ t (ζ SL ∂ t ζ SL ) . (31)
Along with the semi-empirical TFM (see Tab. 1), Eq. 30 and 31 can be applied to ::::: filter ::::::::::::: measurement ::::: noise, ::: as ::::: with :::: the ::::::: Fourier :::::::::: approach. :::::::::: However, ::::: time :::::::::

derivatives :::::::::::: computation :: is ::::: more :::::::: accurate ::::: using ::::::: Fourier :::::::: analysis. :

For practical applications 3. In situ dataset

Field site

In order to assess and compare the ability of the reconstruction methods to recover irregular wave field from pressure measurements, in situ hydrodynamic data was collected at La Salie beach, SW France (see Fig. 5). La Salie beach 295 is a relatively alongshore-uniform gently-sloping sandy beach associated with a meso-macro semi diurnal tidal regime. The relatively wide intertidal region (∼ 200 m in the cross-shore) allows easy and convenient instrument deployment at low tide.

Field experiment

The field experiment was carried out over two periods on April 13-14 2017 Nonetheless, AST is very sensitive to air bubbles as the acoustic signal can be significantly altered within the water column (see Nortek Manual [START_REF]Nortek manual : Principles of operation[END_REF]). Hence, the AST ability to provide reliable measurements under wave breaking can be questioned (Pedersen et al., 2002 [31]; Birch et al., 2004 [32]), but the present study focuses on waves propagating outside the surf zone (see Fig. 6). A video system was set up on the first day of each deployment period to follow the position of the surf zone during the experiment. The system allowed to identify the time evolution of the outer edge of the surf zone, defined as the location of the onset of breaking of the largest waves (see darker points in Fig. 6, within which none of the waves are breaking). The other outer surf zone limits were set by visually checking the AST signal.

Both pressure and AST measurements were divided into 10-minute time series.

Pressure time series was low-pass filtered (1 Hz) to remove instrumental noise.

AST time series was also low-pass filtered to be consistent with pressure time series. The mean water depth h 0 was computed from pressure measurements both outside and inside the surf zone. Each water depth time series was detrended to remove tidal variation. The water level was slowly fluctuating with the infragravity motion. We then define the free surface elevation (in the short-wave frequency band) as :

ζ(t) = h(t) -h infra (t), ( 32
)
where h is the water depth and h infra is the water depth computed over the infragravity frequency range (0.005 Hz -0.05 Hz). Eq. 32 was applied to both pressure and AST measurements yielding the hydrostatic surface elevation ζ δm H (Eq. 3) and the direct measurement of the surface elevation ζ m , respectively.

Outside the surf zone, h 0 ranged from 2.25 m to 3.72 m (see Fig. 6a). For such water depths, we choose to take n = 4 (as in Jones and Monismith, 2007 [START_REF] Jones | Measuring short-period wind waves in a tidally forced environment with a subsurface pressure gauge: Measuring waves with pressure gauges[END_REF]) for the TFM -Jonswap method (Eq. 21). Except for h 0 , parameters in Fig. 6 were calculated using ζ δm H inside the surf zone and using ζ m outside the surf zone. Both experiments were characterized by long and grouped wave conditions with a sea-swell significant wave height H s,short-wave ranging from 0.54 m to 1.08 m (see Fig. 6b) associated with a peak period T P ranging from 8.6 s to 11.5 s (see Fig. 6c). The maximum observed wave height was 1.54 m for LS1 and 1.95 m for LS2. The wave number k was estimated using the linear dispersion relation (Eq. 17) yielding the shallowness parameter µ = (kh 0 ) 2 . The whole dataset features relatively small µ (µ ≤ 0.2; see Fig. 6d) characterizing a weakly-dispersive wave

Data processing

AST processing

Even outside the surf zone, the AST signal was sometimes altered by reflection within the water column. This was caused by the presence of air bubbles that 360 were generated by wave breaking occurring shoreward but close to the instrument, which were occasionally moved by currents above the instrument. This led to the presence of spikes in the surface wave elevation time series that were removed using a gradient thresholds between two consecutive points.

Wave-by-wave analysis

Zero crossing analysis is the most traditional method to determine individual wave characteristics. It first consists in identifying individual waves between each zero-downcrossing or each zero-upcrossing of surface wave elevation. Wave crests and troughs are then respectively defined as the maximum and minimum of surface elevation between two consecutive crossings. In intermediate depth to shallow water, low-frequency motions can be strong and might potentially lead to crests under the mean sea level and troughs above the mean sea level, making the zero-crossing method irrelevant. In addition, filtering-out low-frequency motions is not a reliable option as it can critically transform the wave extrema and the wave shape (Power et al., 2010 [33]). A different method based on a local maxima analysis was implemented (Power et al. 2010 [START_REF] Power | Nearshore wave height variation in unsaturated surf[END_REF]; Power et al.

2015 [START_REF] Power | A novel method for tracking individual waves in the surf zone[END_REF]; Martins et al., 2017 [START_REF] Martins | The influence of swashbased reflection on surf zone hydrodynamics: a wave-by-wave approach[END_REF]). A wave is identified between two consecutive crests. The wave trough is taken as the minimum of the surface elevation between the two consecutive crests. Wave height and period criteria are set to avoid detecting small oscillations (with amplitude < 0.1 m and period < T P /4).

Results and discussion

In this section, the different reconstruction methods presented above are applied to LS1 and LS2 dataset and further compared (see Tab. 1 and 2). We first assess each method in near-breaking conditions. These conditions are characterized by the presence of nonlinear waves for which the validity of linear reconstruction methods to recover individual wave characteristics is questionable. Then, the ability of each method to recover waves within wave groups which contain highly nonlinear and extreme waves is addressed. Finally, we present a wave-by-wave analysis over the whole dataset.

Near-breaking conditions

In this subsection, we focus on a 10-minute time series from LS1 characterized by highly nonlinear waves, i.e. waves just before the onset of breaking (see the blue point at the first outer surf zone limit of LS1 in Fig. 6; h 0 = 2.25 m; µ = 0.075). This time series is characterized by a peak wave period of 11.5 s, a significant wave height of 0.71 m and a maximum individual wave height of 1.4 m. The latter yields a maximum nonlinearity parameter of 0.31 corresponding to strong in-situ nonlinearities.

Linear methods and semi-empirical transfer function

Surface elevation energy density spectra computed from the TFM with different high-frequency tail corrections and the semi-empirical TFM (see Tab. 1) are presented in Fig. 7. Here, the nonlinear cutoff frequency f c is set at 0.32

Hz which corresponds to frequency up to the third harmonic (around 0.28 Hz).

As explained in Section 2, the cutoff frequency associated with spectral reconstruction methods is optimized. Indeed, it is set by comparing the wave energy reconstructed from the TFM -no cutoff (ζ L,NC ) and the true wave energy (ζ m ).

f c is taken at the frequency for which the TFM correction starts exceeding the measured wave energy. Again, results are very sensitive to f c . Depending on its value, the computed wave surface elevation can be significantly altered, which will be addressed at the end of this section. The spectra have been averaged over 1/66 Hz.

The relative error of the spectral significant wave height H m0 = 4

√ m 0 (where m 0 is the zero-th spectral moment calculated between 0 and 1 Hz), the maximal crest elevation (ζ c ) max and the skewness parameter For the sake of clarity, we only display in Fig. 8 the water depth time series reconstructed from the TFM -sharp cutoff and the TFM -steady cutoff (the latter giving the best results among all linear reconstructions). In line with the errors shown in Tab. In most studies involving surface elevation recovery from pressure sensors, the rationale for choosing a particular f c value is often unclear. To assess the cutoff frequency sensitivity, we set f c at 0.6 Hz, which corresponds to the highest value found in the literature. Fig. 9 shows the effect on the wave energy spectrum.

S k =< ζ 3 > /(< ζ 2 >)
With such cutoff frequency, the section of the wave energy between 0.32 Hz and 0.6 Hz computed from ζ L,SH is overestimated. As the wave energy of :::: This These :::::::::: oscillations :::::::: severely ::::::::: transform : the shape of the surface wave elevation, particularly within the back face and the trough of the highest waves (see for instance at t = 236 and 252 s in Fig. 10). 0 0.2 0.4 0.6 0.8 These results show that the classical TFM predicts the significant wave height with reasonable accuracy even when waves are nonlinear (see H m0 in Tab. 3).

470

The different high-frequency tail correction methods associated with the TFM (see Tab. 1) lead to lower H m0 error by artificially amplifying the high-frequency wave spectrum. However, in terms of individual wave characteristics, all linear reconstruction methods reviewed here show similar skill. They significantly underestimate the crest elevation of the highest waves as well as its skewed shape.

In the following, only the TFM with a sharp high-frequency tail correction (see TFM -sharp cutoff in Tab. 1), hereafter referred to as ζ L , is used for systematic comparison with nonlinear methods.

Nonlinear methods

Reconstructed surface elevation energy and time series from each nonlinear method (except for ζ NL for the sake of clarity) are presented in Fig. 11, 12 and 13. As explained in Section 3.2, pressure time series (equivalent to ζ δm H Eq. 3) were low-pass filtered to remove instrumental noise. The cutoff frequency associated with this filter is the cutoff frequency f c,noise set to 1 Hz here. This cutoff frequency is applied to compute the weakly-dispersive methods ζ SL and ζ SNL . f c,noise is much higher than the nonlinear cutoff frequency, used for the fully-dispersive methods (f c = 0.32 Hz). This makes the fully-dispersive methods much more restrictive than the weakly-dispersive methods and the highfrequency tail is better predicted by ζ SL and ζ SNL than by ζ L and ζ HE (see Fig. 11). As both the heuristic method and the fully-dispersive nonlinear method rely on the TFM which requires a nonlinear cutoff frequency, ζ HE and ζ NL do not improve enough ζ L and lead to larger errors compared to ζ SNL (see Tab. 4).

Even if ζ SL and ζ SNL slightly underestimate the second and third harmonics, the energy distribution in the highest frequencies is well evaluated leading to an accurate calculation of H m0 for both methods (error of 6.3 % and 4.2 %, respectively; see Tab. 4). Taking nonlinear effects into account, the SNL method accurately reproduces the energy over a large number of harmonics compared to the SL method. Beyond 0.6 Hz, ζ SNL is considerably better than the classical TFM ζ L (equivalent to ζ δm H in this frequency range) by two orders of magnitude. The third harmonic computed from the heuristic method is slightly overestimated leading to a smaller H m0 error than the TFM (see Tab. 4) but the energy distribution beyond f c is poorly computed in the highest frequencies as it relies on the TFM. In terms of H m0 , ζ SNL has the lowest error of all reconstruction methods reviewed in this article (see Tab. 4). 0 0.2 0.4 [START_REF] Symonds | Two-dimensional surf beat: Long wave generation by a time-varying breakpoint[END_REF]). It is also well known that infragravity waves can result in coastal erosion and inondation events during extreme wave conditions (Roelvink et al., 2009 [37]; Baumann et al., 2017 [38]; Bertin et al., 2018 [39]).

Well predicting wave groupiness is then of paramount importance for coastal applications. Along with the measured infragravity waves, Fig. 14a shows the measured wave envelope computed as the low-pass-filtered Hilbert transform of the short-wave signal (Battjes et al., 2009 [40]), with the corresponding time series of reconstructed dimensionless crest and trough elevation shown in Fig. 14b. The dimensionless crest elevation,

i = ζ c h 0 (33)
can be considered as a local nonlinearity parameter (where ζ c is the crest elevation of each individual wave). Fig. 14b shows the time series of both measured and reconstructed i (( i ) m , ( i ) L and ( i ) SNL , respectively). ( i ) m has an average value of 0.16 but peaks at much higher values (between 0.35 and 0.51) within the three wave groups (see at t = 250 s, 400 s and 760 s in Fig. 14b). Waves within these groups are highly nonlinear and also meet the following criteria:

ζ c H s > 1.25 (34)
where H S is the significant wave height, here defined as four times the standard deviation of the surface elevation. Criteria 34 is commonly used for identifying extreme waves (Dysthe et al., 2008 [41]). These highly nonlinear extreme waves also correspond to waves where the SNL correction is the most skillful compared to the linear (TFM) correction in terms of dimensionless crest elevations. The average of the highest one-tenth dimensionless crest elevations ( i ) 1/10 is underestimated by 30.0 % and by 2.5 % for the linear and SNL method, respectively.

By visually checking images recorded from the video system, the most nonlinear wave over this 10-min time series ((( i ) m ) max = 0.51) is just before the onset of breaking. Over the whole dataset, the highest value of ( i ) m is 0.53 outside of the surf zone, corresponding to a wave that is even closer to breaking. Hence, well predicting these waves is crucial for estimating the break point position which is a key parameter to many coastal applications and wave propagation models. At the individual wave scale, its prediction can significantly differ using 550 the linear or the SNL reconstruction method (see wave groups in Fig. 14b). As pointed out above, the linear and weakly-dispersive nonlinear methods show similar skill to retrieve the significant wave height (see H m0 error in Tab. 3 and 4). However, at the scale of individual waves, the two methods show strongly different reconstruction, especially for highly nonlinear waves in the groups, within which some waves can be characterized as extreme (see criteria 34). Accordingly, in the following section, a wave-by-wave analysis of the whole dataset is conducted in order to identify these nonlinear extreme waves and to further conclude on the overall ability of each method.

A wave-by-wave analysis of the whole dataset

In this section, a wave-by-wave analysis of the entire dataset is performed.

where (H m ) RMS is the measured root-mean-square wave height. AST measurements allow to detect the wave crest of all individual waves of the entire dataset. Of note, in a limited number of 10-minute time series some surface elevation was missing locally in the front faces. For these time series, S k could not be calculated properly.

For the linear reconstruction, the relative error (ζ c ) RMS increases with increasing m . For low m (< 0.10), the RMS crest elevation error (see Fig. 15a) is less than 10 %. The results of SNL method are roughly equivalent to those of the linear reconstruction, even though the SNL method gives slightly smaller errors for low m . As nonlinearities increase, the difference between both methods becomes stronger with SNL error varying around 5 to 10 % while TFM error hovers around 15 to 25 % for the highest m (> 0.14).

This pattern is strengthened for (ζ c ) 1/10 (see Fig. 15b). For low m , the SNL method is better than the linear method by 3 to 5 %. Both methods quickly deviate for moderate to strong nonlinearities. For the highest m , the SNL method is significantly better than the linear method by 13 to 28 %.

In terms of S k (see Fig. 15c), the linear reconstruction fails to correctly describe the skewed wave shape with a scattered S k error between 16.6 % and 51.7 % and an average error of 29.4 %. Indeed, parasite oscillations induced by the 585 cutoff can strongly modify the shape of the most nonlinear waves which worsen S k prediction. Unlike the linear method, the SNL method skillfully recovers the wave shape with a S k error systematically lower than 20 % with an average of 8.5 %. values are relatively low (between 0.05 and 0.2). These linear waves correspond to 78 % of the whole dataset and are well predicted by both reconstruction methods (see Fig. 16a and Fig. 16b).

However, for higher values of ( i ) m (> 0.25), the two methods show different results. The linear method considerably underestimates the highest one tenth dimensionless crest elevation (average error of 20.3 %; see Fig. 16c) while the weakly-dispersive nonlinear method is able to recover the crests of the most nonlinear waves (average error of 6.9 %; see Fig. 16d). Blue crosses represent waves that meet the extreme wave criteria (Eq. 34). These extreme waves correspond to 0.7 % of all detected waves and correspond to some of the most nonlinear waves of our dataset (0.30 < ( i ) m < 0.53), which are correctly recovered by

ζ SNL , only. The linear reconstruction underestimates the dimensionless crest elevation of the detected extreme waves with an average error and a maximum error of 27.9 % and 36.6 %, respectively, against 5.4 % and 16.7 % for the SNL method.

Discussion

In the above work, the weakly-dispersive nonlinear reconstruction (ζ SNL Eq. 31) was found to be essential to accurately recover the highest waves, especially in the shoaling zone. However, we have focused on weakly-dispersive waves outside the surf zone (µ ≤ 0.2; see Fig. 6). In this way, further investigations need to be carried out to identify a threshold above which the fully-dispersive reconstruction (ζ NL Eq. 29) must be used instead of the weakly-dispersive one. Although AST did not enable an accurate measurement of broken waves in our experiment, it has still provided an approximate sight of the shape and crest elevation of waves inside the outer surf zone. The weakly-dispersive nonlinear reconstruction was found to correctly recover such waves. Inside the inner surf zone, the sawtooth wave shape comes from the balance between the nonlinear distortion of the wave field and the turbulent dissipation within the wave front.

These rotational processes cannot be described with irrotational approaches.

Furthermore, the pressure distribution under such waves is mainly hydrostatic (Lin and Liu, 1998 [42]). Nonlinear shallow-water equations are able to predict waves inside the inner surf zone and the swash zone as they accurately reproduce the distortion of nonlinear waves (Bonneton, 2007 [43]). Hence, the hydrostatic reconstruction (ζ δm H Eq. 3) would tend to be the most suitable method to recover the surface elevation of broken waves.

Concluding this section, Fig. 18 shows the range of validity of fully-dispersive, weakly-dispersive and hydrostatic reconstruction methods. An accurate direct measurement of the surface elevation of fully-dispersive waves outside the surf zone and broken waves inside the surf zone is still required for identifying two thresholds: one for using ζ NL or ζ SNL and one for using ζ SNL or ζ δm H (see Fig. 18).

Conclusion

We have applied and compared different methods to reconstruct the surface elevation from pressure measurements in case of irregular weakly-dispersive waves (µ < 0.2) propagating outside the surf zone (h 0 < 4 m). The commonly-used transfer function method (TFM) was found to give a reliable estimate of the significant wave height (H m0 ) with error not exceeding 7 % in near-breaking conditions which feature highly nonlinear waves. However, this method requires the use of a cutoff frequency which restricts the reconstruction of the most nonlinear waves. The TFM solution is very sensitive to the value of this cutoff frequency, especially the reconstructed surface wave elevation. The latter can be affected by the presence of parasite oscillations that strongly alter the shape of the highest waves. Associated with the TFM, several high-frequency tail correction procedures were tested and found to slightly improve H m0 prediction. Nonetheless, these procedures still fail to describe the energy distribution in the highest frequencies leading to an underestimation of the crest elevation of the highest wave and the skewness parameter. On the contrary, the recently developed weakly-dispersive nonlinear reconstruction method (SNL) was found to correctly reproduce the wave spectrum over a large number of harmonics which allows an accurate estimation of the peaked and skewed shape of the highest waves. More importantly, unlike the TFM, this method is able to recover the most nonlinear waves within wave groups. Some of these waves can be characterized as extreme waves and are still accurately predicted by the SNL method (average relative error of 5.4 %) compared the TFM (average relative error of 27.9 %). Well predicting these waves is essential for many coastal applications, in particular those that require a correct estimation of the highest waves such as studies on wave submersion, but also for predicting the break point position which is crucial for the calibration and the validation of wave propagation models.

Acknowledgements

The Ph.D. thesis work of A. Mouragues is cofunded by a DGA-Région Nouvelle Aquitaine scholarship. This study has received financial support from both Région Nouvelle Aquitaine (contract n • 2017-1R20107) and SHOM ::: was ::::::: carried ::: out ::: as :::: part ::: of ::: the ::::::: project ::::::::: MEPELS : (contract n • 18CP05), :::::::::: performed :::::: under ::: the :::::::: auspices :: of :::: the :::::: DGA, :::: and ::: led ::: by ::::::: SHOM. The authors wish to thank all those who contributed to the field experiment at La Salie Beach: Detandt G., Bonneton N., Dubarbier B. and Poncet P.A..

Figure 1 :

 1 Figure 1: Definition sketch of the physical variables. x and z are the horizontal (wave propagation axis) and vertical axis, respectively. z = 0 is the mean water level and -h 0 is the constant bottom elevation. ζ(x, t) is the surface wave elevation, a is the characteristic wave amplitude and δm represents the distance from the bottom where the pressure sensor is located.

Figure 2 :

 2 Figure 2: Surface elevation of a periodic weakly nonlinear wave, = 0.15, µ = 0.25, δm = 0. Full Euler solution (black line); hydrostatic reconstruction: ζ δm H (blue line); linear formula in space: ζ L,space (magenta dashed line); see the associated equations in Tab. 1.

Figure 3 :

 3 Figure 3: Surface elevation energy density spectra E(n) of a periodic weakly nonlinear wave, = 0.15, µ = 0.25, δm = 0. Full Euler solution (black circle); hydrostatic reconstruction: ζ δm H (blue crosses); linear formula in space: ζ L,space (magenta crosses); TFM -no cutoff: ζ L,NC (green circles); TFM -sharp cutoff: ζ L,Sh (green crosses); see the associated equations in Tab.

Figure 4 :

 4 Figure 4: Surface elevation of a periodic weakly nonlinear wave, = 0.15, µ = 0.25, δm = 0. Full Euler solution (black line); hydrostatic reconstruction: ζ δm H (blue line); linear formula in space: ζ L,space (magenta dashed line); fully-dispersive nonlinear reconstruction in space: ζ NL,space (red line); heuristic reconstruction in space: ζ HE,space (yellow line); see the associated equations in Tab. 1 and 2.

  locally in time and a cutoff frequency is not necessarily needed, unlike for the computation of fully-dispersive methods (TFM, ζ HE and ζ NL ). ::::::::: Bonneton :: et ::: al.::::::(2018) ::::::::::[START_REF] Bonneton | A nonlinear weakly dispersive method for recovering the elevation of irrotational surface waves from pressure measurements[END_REF] applied :::: Eq. ::: 31 :::::: locally :: in ::::: time :: by ::::::::::: discretizing ::::: first-::: and :::::::::::: second-order :::: time :::::::::: derivatives :::::::: involved :: in :::: Eq. ::: 30 :::: and :::: 31. ::::: Such ::::: time :::::::::::: discretization :::::::: requires ::

Figure 5 :

 5 Figure 5: (a) Location map with the field site of La Salie indicated by the black circle. (b) Unmanned aerial vehicle photo of the field site at mid-tide during the experiment. A video system was installed on the pier shown in the left-hand side of the image. The yellow star and the red star show the location of the video system and the instrument, respectively, during the experiment.

(

  LS1) and May 17-18 2018 (LS2) and aimed at characterizing nonlinear waves in intermediate and shallow depth. A Nortek Signature 1000 kHz current profiler was deployed at low tide. The Signature 1000 kHz vertical beam allows a high-frequency direct measurement of the surface wave elevation using Acoustic Surface Tracking (AST). Besides AST, it also provides pressure measurements. Signature 1000 manufacturer (Nortek) reports pressure-derived elevation and AST measurements with accuracy of ± 1 mm and ± 2 cm, respectively. The instrument recorded at 8 Hz sampling rate and pressure was measured at 0.7 m above the bottom (δ m = 0.7 m). :::: The :::::::::::: characteristic ::::::: bottom ::::: slope ::: was :::::::::: σ b = 0.015 :: at ::: the :::::: sensor :::::::: location. :AST is a relatively new ADCP feature that has mainly been validated for waves propagating in deep water (> 20 m) and for sampling rate not exceeding 4 Hz(Pedersen and Nylund, 2004 [27];Pedersen and Lohrmann, 2004 [28]). Martins et al. (2017)[START_REF] Martins | High-resolution monitoring of wave transformation in the surf zone using a lidar scanner array[END_REF] collected high-frequency surface elevation measurements and found a very good agreement between the surface elevation measured from the Signature 1000 kHz and from a LIDAR scanner (root mean square error RMSE of 0.05 m) for an undular tidal bore propagating in the Garonne river ( = 0.08; µ = 5.85; the mean water depth at low tide was 2.8 m).

  regime. : It :: is :::::: worth :::::: noting ::::: that, :::: the ::::::::: parameter :::::: √ µσ b ::::: being ::::: very ::::: small ::: for :::: the ::::: whole ::::::::::: experiment, ::: the :::::::: bottom ::::::::::: contribution :::: can ::: be :::::::: neglected :::::::::: (Bonneton ::: et ::: al., ::::2018 :::::[START_REF] Bonneton | A nonlinear weakly dispersive method for recovering the elevation of irrotational surface waves from pressure measurements[END_REF]). :

Figure 6 :

 6 Figure 6: Wave and tide conditions for both experiments (LS1 in blue and LS2 in red). (a) Mean water depth h 0 ; (b) Short-wave significant wave height H s,short-wave (circles) and infragravity significant wave height H s,infra computed over 20 min (diamonds); (c) Spectral peak period T P ; (d) Shallowness parameter µ. Dark-colored points and light-colored points show data outside and inside the surf zone, respectively. Transitions between both areas represent the outer surf zone limits. In the present study, we only focus on the dark-colored points.

  As expected, ζ δm H correctly reproduces the low-frequency spectrum as well as the first harmonic (around 0.09 Hz) but strongly underestimates the energy of all subsequent harmonics. The semi-empirical TFM, ζ N , provides a good estimate of wave energy up to the second harmonic (around 0.18 Hz) but then slowly starts underpredicting all the subsequent harmonics as well. For f < f c , the TFM -sharp cutoff, ζ L,SH , (equivalent to ζ L,L , ζ L,ST and ζ L,J ) properly reproduces the energy spectrum compared to ζ δm H . For f > f c , ζ L,SH (equivalent to ζ δm H ) strongly underestimates the energy by two to three orders of magnitude at the highest frequencies. The TFM -Jonswap, ζ L,J , is able to reproduce the high-frequency tail's slope but does not reconstruct any harmonics. In the other hand, both the TFM -linear cutoff and the TFMsteady cutoff, ζ L,L and ζ L,ST respectively, improve ζ L,SH by correctly recovering one extra harmonic (fourth harmonic around 0.36 Hz), even though the energy is slightly underestimated. Unlike ζ L,ST that keeps correcting ζ L,SH over all frequencies, ζ L,L fades into ζ L,SH around 0.7 Hz.

Figure 7 :

 7 Figure 7: Surface elevation energy density spectra E(f). AST measurements: ζm (black line); hydrostatic reconstruction: ζ δm H (blue line); TFM -sharp cutoff: ζ L,SH (green line); TFM -linear cutoff: ζ L,L (magenta line); TFM -steady cutoff: ζ L,ST (red line); TFM -Jonswap: ζ L,J (red dashed line); TFM -no cutoff ζ L,NC (black dashed line); semi-empirical TFM: ζ N (yellow line); see the associated equations in Tab. 1 fc = 0.32 Hz (vertical black dotted line).

  ζ L,L ; TFM -steady cutoff: ζ L,ST ; TFM -Jonswap: ζ L,J ; semi-empirical TFM: ζ N ; see the associated equations in Tab. 1. fc = 0.32 Hz. The same trend is observed for temporal parameters (ζ c ) max and S k . In terms of these parameters, ζ N performs roughly the same as ζ L,SH . The TFM is slightly improved using the linear cutoff and the steady cutoff approaches (ζ L,L and ζ L,ST , respectively). Nonetheless, ζ L,ST , which gives the best agreement with ζ m , still considerably underestimates both (ζ c ) max and S k by 26.5 % and 37.2 %, respectively (see Tab. 3).

Figure 8 :

 8 Figure 8: Water depth time series of a wave group. AST measurements: h 0 + ζm (black points); hydrostatic reconstruction: h 0 + ζ δm H (blue line); TFM -sharp cutoff: h 0 + ζ L,SH (green line); TFM -steady cutoff: h 0 + ζ L,ST (red line); see the associated equations in Tab. 1. fc = 0.32 Hz and h 0 = 2.25 m.

:Fig. 9 ,

 9 Fig. 9, this is particularly noticeable for ζ L,ST which overestimates the harmonic around 0.61 Hz. As the reconstructed energy is higher, the H m0 error is much lower (1.4 % for ζ L,SH and 0.4 % for ζ L,ST ). Similarly, as shown ::As :::::: shown ::: in : Fig.10, :::due :: to :::: the ::::::: energy :::::::::::::: overestimation, the crest elevation of

Figure 9 :

 9 Figure 9: Surface elevation energy density spectra E(f). AST measurements: ζm (black line); hydrostatic reconstruction: ζ δm H (blue line); TFM -sharp cutoff: ζ L,SH (green line); TFMlinear cutoff: ζ L,L (magenta line); TFM -steady cutoff: ζ L,ST (red line); TFM -Jonswap: ζ L,J (red dashed line); TFM -no cutoff ζ L,NC (black dashed line); semi-empirical TFM: ζ N (yellow line); see the associated equations in Tab. 1 fc = 0.6 Hz (vertical black dotted line).The spectra have been averaged over 1/66 Hz.

Figure 10 :

 10 Figure 10: Water depth time series of a wave group. AST measurements: h 0 + ζm (black points); hydrostatic reconstruction: h 0 + ζ δm H (blue line); TFM -sharp cutoff: h 0 + ζ L,SH (green line); TFM -steady cutoff: h 0 + ζ L,ST (red line); see the associated equations in Tab. 1. fc = 0.6 Hz and h 0 = 2.25 m.

Figure 11 :

 11 Figure 11: Surface elevation energy density spectra E(f). AST measurements: ζm (black line); hydrostatic reconstruction: ζ δm H (blue line); weakly-dispersive linear reconstruction: ζ SL (magenta line); weakly-dispersive nonlinear reconstruction: ζ SNL (red line); heuristic reconstruction: ζ HE (yellow line); see the associated equations in Tab. 2. fc = 0.32 Hz (vertical black dotted line) and f c,noise = 1 Hz. The spectra have been averaged over 1/66 Hz.

Figure 12 :

 12 Figure 12: Water depth time series of a group of waves. AST measurements: ζm (black points); hydrostatic reconstruction: ζ δm H (blue line); weakly-dispersive linear reconstruction: ζ SL (magenta line); weakly-dispersive nonlinear reconstruction: ζ SNL (red line); heuristic reconstruction: ζ HE (yellow line); see the associated equations in Tab. 2. fc = 0.32 Hz, f c,noise = 1 Hz and h 0 = 2.25 m.

Figure 13 :

 13 Figure 13: Water depth time series of the highest wave. AST measurements: ζm (black points); hydrostatic reconstruction: ζ δm H (blue line); weakly-dispersive linear reconstruction: ζ SL (magenta line); weakly-dispersive nonlinear reconstruction: ζ SNL (red line); heuristic reconstruction: ζ HE (yellow line); see the associated equations in Tab. 2. fc = 0.32 Hz, f c,noise = 1 Hz and h 0 = 2.25 m.

Figure 14 :

 14 Figure 14: (a) Measured wave envelope (black line) and measured infragravity surface elevation (black dashed line). (b) Dimensionless wave crests (filled circles) and troughs (empty circles) elevation. AST measurements: ζm (black circles); TFM -sharp cutoff ζ L (blue circles); weakly-dispersive nonlinear reconstruction ζ SNL (red circles); see the associated equations in Tab. 1 and 2. fc = 0.32 Hz, f c,noise = 1 Hz and h 0 = 2.30 m.

  For each time series, relative errors for both linear and SNL reconstruction methods are computed in terms of three parameters: the root-mean-square crest elevation (ζ c ) RMS , the average of the highest one-tenth crest elevation (ζ c ) 1/10 and the skewness parameter S k . Those are represented in Fig.15 as a function of the average nonlinear parameter m computed as : m = (H m ) RMS /2 h 0

Figure 15 :Fig. 16

 1516 Figure 15: Relative error (%) of (a) the root-mean-square crest elevation (ζc) RMS , (b) the average of the highest one-tenth crest elevation (ζc) 1/10 and (c) the skewness parameter S k as a function of the average nonlinear parameter m. TFM -sharp cutoff: ζ L (blue circles); weakly-dispersive nonlinear reconstruction: ζ SNL (red circles). fc = 0.32 Hz and f c,noise = 1 Hz.

Figure 16 :

 16 Figure 16: Reconstructed dimensionless crest elevations versus measured crest elevations for all detected waves ((a) and (b)) and for the highest one-tenth dimensionless crest elevations ((c) and (d)). The color of each point represents its density (computed as the number of neighbooring points within a 0.015 m radius). Blue crosses show the detected extreme waves (Eq. 34). (a) and (c): TFM -sharp cutoff ζ L . (b) and (d): weakly-dispersive nonlinear reconstruction ζ SNL ; see the associated equations in Tab. 1 and 2. fc = 0.32 Hz and f c,noise = 1 Hz.

First

  Fig. 17).

17

 17 Fig. 17).

  Fig. 17).

Figure 17 :

 17 Figure 17: Shallowness parameter µ as a function of peak period T P and mean water depth h 0 . µ < 0.3 corresponds to weakly-dispersive regime (horizontal dotted line) and µ > 0.3 corresponds to fully-dispersive regime (vertical dotted line).

Figure 18 :

 18 Figure 18: Range of validity of reconstruction methods. x and z are the cross-shore axis and the vertical axis, respectively. µ = (kh 0 ) 2 is a dispersion parameter (Eq. 1).

Table 1 :

 1

	Surface elevation	Reconstruction method	Associated	Nonlinear cutoff	High-frequency
	name		equations	frequency f c	tail correction
	ζ δm H	Hydrostatic (equivalent to Eq. 3	No	-
		pressure measurements)
	ζ L,space	Linear formula in space	Eq. 13, 14	No	-
	ζ L,NC	TFM	Eq. 15, 16 and 17 No	-
	ζ L,Sh	TFM -sharp cutoff	Eq. 15, 17	Yes	Eq. 19, 20
	ζ L,J	TFM -Jonswap	Eq. 15, 17	Yes	Eq. 19, 21
	ζ L,L	TFM -linear cutoff	Eq. 15, 17	Yes	Eq. 19, 22, 23
	ζ L,St	TFM -steady cutoff	Eq. 15, 17	Yes	Eq. 19, 24
	ζ N	semi-empirical TFM	Eq. 26	No	-

Overview of the hydrostatic, TFM and semi-empirical TFM reconstruction methods studied in this article. -means that no high-frequency tail correction is applied.

2.3. Nonlinear methods

Over the past few years, several authors have gone to great lengths studying nonlinear surface wave reconstruction from pressure measurements (e.g. Decon-225 ick et al., 2012 [21]; Oliveras et al., 2012 [22]; Constantin, 2012 [23]; Clamond and Constantin, 2013

Table 2 :

 2 Overview of the nonlinear reconstruction methods studied in this article.

	monochromatic waves ( = 0.65), bichromatic waves ( = 0.37) and in situ
	waves ( max = 0.31 where max is the nonlinear parameter of the highest wave).

:: In :::: this :::: way, recovering ζ SL and ζ SNL (and ζ N ) can still require ::: still :::::::: requires : a cutoff frequency f c,noise in order to remove pressure measurement noise. However, this cutoff frequency is much higher than the nonlinear cutoff frequency f c introduced earlier. Accordingly, ζ SL and ζ SNL (and ζ N ) are computed accounting for much higher frequency spectral information which is crucial to correctly reconstruct the surface elevation of nonlinear waves. Bonneton et al. (2018) [26] found a good agreement for weakly-dispersive nonlinear waves (µ < 0.3) between ζ SNL and direct ζ measurements in case of

Table 3 :

 3 [START_REF] Bishop | Measuring waves with pressure transducers[END_REF] [START_REF] Bishop | Measuring waves with pressure transducers[END_REF]). ζ N provides the same H m0 as ζ L,SH . Both ζ L,L and ζ L,ST are better than ζ L,SH by roughly 2 %. As ζ L,J strongly underestimates the fourth and fifth harmonics (around 0.36 and 0.45 Hz in Fig.7), the computed H m0 error is slightly higher than all other linear reconstructions. Among all reconstructions, ζ L,ST has the lowest H m0 error. Spectral significant wave height H

	3/2

m0 , highest crest elevation (ζc)max and sea surface skewness S k relative error (%). hydrostatic reconstruction: ζ δm H ; TFM -sharp cutoff: ζ L,SH ; TFM -linear cutoff:

  3, both reconstructions are able to recover the smallest waves but they strongly underestimate the peaked and skewed shape of the highest waves within the group, even though ζ L,ST slightly improves the reconstructed crest elevation compared to ζ L,SH .

		3.2
		2.9
	h (m)	2.6
		2.3
		2
		210 1.7	220	230	240	250	260	270	280
					t (s)

Table 4 :

 4 Spectral significant wave height H m0 , highest crest elevation (ζc)max and sea surface skewness S k relative error (%). TFM -sharp cutoff ζ L ; heuristic reconstruction ζ HE ; fullydispersive nonlinear reconstruction ζ NL ; weakly-dispersive linear reconstruction ζ SL ; weaklydispersive nonlinear reconstruction ζ SNL ; see the associated equations in Tab. 1 and 2. Among all reconstruction methods presented in this work, ζ SNL is found to provide the best agreement with the measured surface elevation ζ m regarding spectral wave parameters (H m0 ) and more importantly regarding individual wave characteristics ((ζ c ) max and S k ). In the wave groupiness section below, only the commonly-used transfer function method ζ L (TFM -sharp cutoff) and the weakly-dispersive nonlinear method ζ SNL are used.

		ζ L	ζ HE	ζ NL	ζ SL	ζ SNL
	H m0	7.1	5.7	5.9	6.3	4.2
	(ζ c ) max	33.7	28.6	26.5	25.5	7.1
	S k	49.5	34.8	29.9	37.8	7.5


Sander Van Der Hoog

Christophe Deissenberg
email: christophe.deissenberg@univmed.fr

Herbert Dawid
email: hdawid@wiwi.uni-bielefeld.de

Production and Finance in EURACE

EURACE is a major FP6 STREP project aiming at constructing an exhaustive agent-based model of the European economy, populated by a very large number of sophisticated, autonomous agents. The EURACE model, which has an explicit spatial structure, includes all the major markets considered in quantitative macroeconomic modelling (consumer goods, investment goods, labour, credit and finance). It offers a unique opportunity for studying, from a new perspective, the empirically observed but theoretically poorly understood link between the real and the financial sphere of a modern economy. After summarily presenting the main features of EURACE, this paper describes in more detail the newly developed financial management module that intermediates between the real and the financial spheres in EURACE. In a nutshell, this module defines the link between the hiring and investment behavior of the firms as a function of the revenues they obtain by selling their products, of the money they can raise on the credit and financial markets, of their dividend policy, and other major aspects of financial decision-making.

Introduction

An important question in economics is to explain the link between the real and financial sphere of the economy. Empirically speaking, the macroeconomic and financial variables are correlated along the business cycle. The existing theoretical models, however, have difficulty explaining the observed correlations.

This paper documents recent work carried out to address this problem from an agent-based perspective. This work is part of a major FP6 STREP project, EU-RACE, that aims at developing a comprehensive agent-based model of the European economy. In this paper, we sketch the broad structure of the EURACE model and present in more detail the model of financial policy decisions by the firm that links the real and the financial spheres within the model.

This paper is organized as follows. In the second section, we succinctly present the EURACE project. The third section presents the financial management module. A final fourth section concludes.

The EURACE project

Most of the existing agent-based models cover only a single industry, one restricted geographical area, or a unique market, and involve relatively small populations of agents. In contrast, EURACE aims at creating an integrated agent-based model of the European Union, linking all the various markets one typically encounters e.g. in a large dynamic CGE model of a multi-national economy. The thus defined artificial economy is to be populated with a very large number (possibly tens of millions) of fairly sophisticated agents.

The model has an explicit spatial structure mimicking the regional statistical units used by Eurostat. It includes various (typically, regional) artificial markets for real commodities (that is, consumption goods, investment goods and labour), and markets for financial assets (such as loans, bonds and stocks). For a general overview of the EURACE model, see Deissenberg et al. (forthcoming).

The three-years EURACE Project started in September 2006. It includes economists and computer scientists from eight research centres in Italy, France, Germany, the UK, and Turkey, as well as the 2001 Nobel laureate in economics, Joseph Stiglitz. More institutional and scientific details can be found on the project's web page: www.eurace.org.

By its scope and complexity, the effort is unsurpassed and needs to cover much terra incognita, among others concerning the conceptual and computational architecture of the model, its numerical implementation, its validation, and the exploitation of the simulation results. In particular, running such a large model will necessitate using massively parallel computing clusters, using pioneering software such as the Flexible Large-scale Agent Modelling Environment (FLAME).

FLAME

Developed by Simon Coakley, Mike Holcombe, and others at the University of Sheffield (see www.flame.ac.uk for a more complete presentation and references), FLAME provides a computational framework allowing modellers to easily create, exchange, include and couple models written in a high-level modelling language. Other important aspects include the development of parallelisation techniques, the distribution of agents over many processors, and the inclusion of testing methods to verify developed models. All these elements are vital to agent-based models in general and to EURACE in particular. The framework has been adapted to enable it to run on parallel computing platforms. It has been previously used to study the behavior of a number of biological systems -at the molecular, cellular, tissue and social levels -and has been instrumental in uncovering a number of new biological properties that have been confirmed experimentally by [START_REF] Coakley | From molecules to insect communities -how formal agent-based computational modelling is uncovering new biological facts[END_REF].

The approach followed in FLAME is to define each agent as a so-called Stream X-Machine [START_REF] Laycock | The theory and practice of specification based software testing[END_REF]. That is in a nutshell: as an automaton described by a finite number of states, transitions between those states, and actions, and endowed with both an internal memory and the ability to interact by sending and receiving messages.

In the FLAME framework, agents act within contexts, that is preeminently, within markets. The agents can have different roles in different contexts, e.g., an agent can be a buyer on one market, a seller on another. Table 1 lists the main classes of agents in EURACE, the contexts in which they operate and the main messages they exchange.

The real sector

Somewhat simplifying, the real sector is composed of an investment goods sector, a consumption goods sector, and a labour market. The investment goods sector, however, is not agent-based, but modelled as a passive entity whose behavior is determined by simple rules. In a nutshell, it provides an infinite supply of investment goods at exogenously given prices. The productivity of the investment goods increases over time according to a stochastic process. The amounts paid for the investment goods are channeled back into the economy. Together with labour, the investment goods are used in the consumption goods sector to produce consumption goods. These goods are sold to the households. The consumption goods firms (the firms thereafter) follow plausible, to the largest possible extend empirically grounded rules for investment, production, stocking, pricing, hiring and firing, dividend payment and/or debt making. Likewise, the households follow plausible rules for saving, consuming different types of products, looking for another job while being employed or trying to find a job while unemployed.

The workers are characterized by (i) a general skill level; and (ii) specific skills. The specific skills are acquired on the job to fully exploit the technological potential of the physical capital being used in the production process. The general skill level is obtained through schooling. The higher the worker's general skill level, the faster it acquires the specific skills associated with a given job.

To capture spatial effects, the economy is divided into regions. Consumption occurs locally within each of these regions. The firms send the goods they produce to diverse local outlet malls. The local prices may differ from one outlet mall to the next, i.e. from region to region. Households and firms are distributed across the regions and are allowed to migrate. Workers can apply for jobs in any region, but working outside of their own region of residence is associated with commuting costs that have to be subtracted from the wage.

The real-financial interaction

Figure 1 sketches the place of the financial management in the interaction between the real and financial spheres.

The financial management module serves to reconcile the production and investment goals of the firm with its payout policy and other financial commitments. The firms receive revenues from the sales on the consumption goods market. This income is used to remunerate the workers they hire on the labour market and to buy investment goods. It is also used to pay the firm's prior financial commitments, i.e. to service the debt (interest and debt installment payments) and to pay taxes. Part of it may also be used to pay out dividends or to repurchase shares on the financial market. Alternatively, the firm can borrow on the credit market and/or issue stocks on the financial market to raise financial capital.

The financial management module

All firms in EURACE are corporations. Therefore we use insights from corporate finance and capital management theory to model their financial decision-making. The behavioral rules are kept as close as possible to those used in reality. We define them based, among others, on the seminal contribution by [START_REF] Cyert | A Behavioral Theory of the Firm[END_REF]) on a Behavioral Theory of the Firm and the work of [START_REF] Myers | The capital structure puzzle[END_REF] on Pecking Order Theory, a behavioral theory of corporate finance. We also draw information from Tirole (2006, Ch.2), [START_REF] Allen | Corporate finance: Handbook of the Economics of Finance[END_REF][START_REF] Allen | Corporate finance: Handbook of the Economics of Finance[END_REF][START_REF] Brav | Payout policy in the 21 st century[END_REF], who report empirical evidence on payout policies.

General assumptions

The following general assumptions are captured by the financial management module.

• The firm's production plan is chosen before the financial policy is decided on.

Thus, in the case of newly issued equity, the new shareholders do not influence the production decision for the current period. Hence, preferred stocks and com-mon stocks are indistinguishable and the price of equity is the same on the primary and secondary equity markets. • All dividend payments are made in cash, not in stocks. Thus, there is no dilution of the stock value. • Firms can repurchase their own stocks, but cannot buy stocks of other firms.

• Clearing of the equity-and bond markets occurs through a clearinghouse or a limit-order book mechanism. • The credit and capital markets are imperfect and subject to transaction costs.

Thus, a non-trivial trade-off exists between different financing instruments. • Firms can be credit rationed on the credit market and consequently may need to resort to corporate bonds or equity for financing. • In accordance to the Pecking Order Theory, the firm prefers internal financing since it is the least risky. This is followed by debt financing, corporate bonds and, finally, equity financing. • The wages and investment costs are paid before production takes place, see [START_REF] Dawid | Agent-based Models of Skill Dynamics and Innovation[END_REF]; [START_REF] Gallegati | Agent-Based Models of Goods, Labour and Credit Markets[END_REF]. • Firms pay dividends at the beginning of each operating cycle, based on the results of the previous operating cycle. The total dividend payout may vary, but there is an effort to maintain a constant dividend to earnings ratio (see, e.g. the survey by [START_REF] Brav | Payout policy in the 21 st century[END_REF].

Over the course of the business day (one iteration) the markets open and close in the following sequence:

• Credit market • Financial market • Labour market • Investment goods market • Consumption goods market
Each firm is active on the first four markets during one day every month. This activation day varies from firm to firm and is chosen randomly at the start of the simulation. During their activation day, the active firms complete a so-called operating cycle, as described below. The selling of output however occurs over the course of the entire month that follows the activation day.

The operating cycle

During its operating cycle, the firm executes four main routines:

1. Production planning routine 2. Financial planning routine 3. Production routine 4. Accounting routine It also checks at different points whether or not it is bankrupt, as explained in more detail later on.

Production planning routine

The operating cycle starts with production planning: the firm calculates the production level it would like to realize, the workforce it would like to hire, and the investments in new physical capital it would like to purchase in the absence of financial constraints. It computes the corresponding estimated costs. It also decides upon the dividends to be paid. However, it does not contractually commit to hire respectively to buy or to pay dividends since at this point of its decision-making process it is not sure that the required financing will be available.

Since, as previously assumed, the wages and investment costs are paid before the production takes place, the firm may need to obtain a bank loan or other external capital before being able to start producing.

Financial planning routine

After completing the production planning routine, the firm computes all financial commitments from the previous period: interest payments, debt installment payments. These, together with the production and investment costs and the dividend payout previously calculated, determinate the total liquidity needs for the current operating cycle. These needs are to be financed by internal and external means before production starts.

As previously stated, the firm's first attempt is to use internal financing since this is the least risky alternative. If the current internal financial resources are insufficient they try to complete them on the credit market by applying for bank loans with individual banks. If successful, the firm implements its current production plan. If it fails, it tries to obtain the remaining external financing it needs from the financial market. This starts the firm's financial market routine.

Financial market routine

Here, the firm tries to raise financial capital through the emission of new equity (stocks, bonds) on the primary equity market. If successful, all external financing needs are satisfied and the firm can continue with the actual production routine. Otherwise, the firm needs to revise its production and investment plans. It re-runs the production planning routine subject to the total financial resources available. The new production plan automatically satisfies the budget constraint of the firm, which therefore does not have to run again the financial planning routine. See figure 2 for a decision flowchart.

Production routine

After the production planning routine and the financial planning routine have been completed, the firm starts with the actual production cycle. It first visits the labour market, then the investment goods market, in order to obtain the required factor inputs. At this stage the firm solidifies its production plans, i.e. labour and investments are contracted.

If there is rationing on the labour or investment goods market, the firm spends less than planned and thus has unneeded cash after fulfilling all its contractual obligations. These involuntary savings are deposited in an account at a given bank (different firms can have accounts at different banks). It is deposited there at the end of the firm's operating cycle, i.e., at the end of the day when the firm is active, and will be available for the next production cycle. In that way, all money remains in the system and is available to the banks for making loans to other firms. This prevents inefficiencies due to improductive cash holdings. Incidentally, a similar mechanism is used for modelling the households' left-over consumption budget: all unused cash holdings flow back to the banks immediately in the form of deposits.

The same holds when prices turn out lower than expected: the savings are deposited. If prices turn out higher than expected the firm is rationed in its labour and/or investment demand as well, but there are no savings. The quantity of factor inputs is lower than planned, so also the actual production output will be lower than planned.

After going to the labour market and the investment goods market, the firm starts the actual production process. The output is assumed to be available instantaneously once the financing and factor inputs have been contracted.

The firm distributes the output from a central inventory to the local outlet malls where it tries to sell its merchandise. As previously stated, the selling of output occurs over the course of an entire month, starting from the activation day of the firm. The local outlet malls record and transmit the regional sales revenues. This signals the end of the production cycle, and starts the accounting routine.

Accounting routine

The firm compiles an income statement and a balance sheet after each production cycle.

The income statement includes the monthly cumulative revenues from the sales of goods and services and the total costs: costs of sales, operating expenses, taxes, service of the debt, dividends, etc. (see Table 2).

The balance sheet lists the total assets and total liabilities (see Table 3). The new balance sheet is computed by updating the current cash holdings with the incoming and outgoing cash flows as listed in Table 4.

After the balance sheet has been updated, but before executing any payments, the firm first checks if it is financially solvable by checking a bankruptcy condition. If financially solvable, the firm executes its payments and this signals the end of the operating cycle. Otherwise bankruptcy is declared. The precise modelling of events when a bankruptcy occurs is discussed below.

Bankruptcy conditions

In order to closely mimick managerial practice, the firm checks for bankruptcy at several places during the operating cycle. The first check occurs directly after the balance sheet computation, in order to verify whether the firm can keep its financial commitments from the previous period and pay for the production costs of the upcoming period.

The liquidity needs of the firm are defined as follows:

Financial liquidity needs: FLN = interests + installments + taxes, Production liquidity needs: PLN = labour costs + investment costs, Total liquidity needs: TLN = FLN + PLN + dividends.

Based on these definitions, we define a bankruptcy state, a financial crisis state, and a normal state of affairs: Cash < FLN: bankruptcy state FLN ≤ Cash < TLN: financial crisis state Cash ≥ TLN: normal state of affairs

In the bankruptcy state, the firm cannot keep its prior financial commitments. Its equity is negative and it has not been able to raise sufficient amounts of financial capital. It sends a bankruptcy message to all the banks with which it has outstanding loans, with content: the credit that is refunded, the bad debt that will not be refunded, and the residual value-at-risk. This information allows the bank to update its balance sheet, which will affect its future lending activities.

In the financial crisis state, the firm did not raise sufficient funds to pay for its total liquidity needs, but can possibly salvage the situation by down-scaling the dividends (in a worst-case scenario, to zero). The down-scaled dividend is given by: Div = max{0, Cash -FLN -PLN}.

(1)

This leads to two possible two sub-states:

1. Cash ≥ FLN + PLN and Div ≥ 0. The payment account is sufficient to pay for the financial commitments and production costs. In this case the equity of the firm is positive and the financial crisis is resolved. The firm respects its financial commitments and starts the production cycle. 2. Cash < FLN + PLN and Div = 0.

The payment account is sufficient to pay for the financial commitments, but not for the production costs. The equity of the firm remains negative, even after decreasing the dividends to zero. The firm has not yet resolved the financial crisis.

It needs to down-scale its production before proceeding further. We assume that the firm pays the financial payments before recalculating production costs. Thus, the firm first pays its financial commitments from its current liquidities. It then recalculates the production quantity such that the corresponding production costs are equal to the remaining liquidities:

Set prod. costs s.t. PLN ≤ Cash -FLN.

(2)

If the firm succeeds in lowering production costs sufficiently, it has resolved the financial crisis and produces the decreased quantity. In a worst-case scenario, production costs have to be lowered to zero. The firm does not produce anything in the current period.

In the normal state of affairs the equity of the firm is positive. The firm can respect its financial commitments and pay for the production costs as planned. There is no need for a down-scaling of dividends or production. Business as usual continues with the firm entering the labour market and the investment goods market and producing as planned.

Conclusion

Together with its financial management module, EURACE offers a powerful tool to investigate the yet little understood interaction between real and financial phenomena, including timely questions such as the real impact of credit shortening in a heterogenous, spatially differentiated world. The ability it provides to track the micro-macro interaction and the dynamics into the finest detail may lead to the discovery of transmission mechanisms and phenomena that are obscured by the more traditional representative agent approach. The first simulations we conducted using the financial module indicate that, indeed, the harvest is likely to be rich. 2 Interface diagram between the firm's financial management role and its credit market and financial market roles. This flowchart shows the conditional functions that to run to execute the firm's external financing decisions. It first runs the credit market routine and then the financial market routine. If the firm cannot satisfy all its external financing needs, it decreases its dividends (possibly to zero). If it then still cannot satisfy all its liquidity needs, it should re-run the production planning routine to re-optimize the planned production quantity such that the total production costs (i.e., the costs for labour and capital) can be financed.

Fig. 1

 1 Fig.1Financial management as intermediary between the real and financial spheres.

Firm

  Fig.2Interface diagram between the firm's financial management role and its credit market and financial market roles. This flowchart shows the conditional functions that to run to execute the firm's external financing decisions. It first runs the credit market routine and then the financial market routine. If the firm cannot satisfy all its external financing needs, it decreases its dividends (possibly to zero). If it then still cannot satisfy all its liquidity needs, it should re-run the production planning routine to re-optimize the planned production quantity such that the total production costs (i.e., the costs for labour and capital) can be financed.

Table 1

 1 Main agents, contexts, roles, and messages in the model.

	Agent	Context	Role	Messages
	Household	Consumption goods market Buyer	units demanded
		Labour market	Worker application, accept/reject job
		Credit market	Depositor cash holdings
		Financial market	Investor stock/bond orders
	Firm	Investment goods market Buyer	units demanded
		Consumption goods market Seller	price, quality
		Labour market	Employer vacancy, job offer
		Credit market	Borrower loan request
		Financial market	Issuer	stock/bond orders
	Investment Goods Firm Investment goods market Seller	price, productivity
		Labour market	Employer vacancy, job offer
	Bank	Credit market	Lender credit conditions
	Government	Financial transactions		tax payments
	Central Bank	Credit market	Regulator

Table 2

 2 Firm income statement.

	revenues from sales of goods and services
	-cost of sales
	-operating expenses
	earnings before interest and taxes
	-interest payments
	earnings before taxes
	-tax payments
	net earnings
	-debt installment payments
	-total dividend payment
	-share repurchases
	retained earnings
	Table 3 Firm balance sheet.
	Assets		Liabilities
	cash holdings		total debt
	total value physical capital stock	shareholders' equity
	total value local inventory stocks
	Table 4 Firm cash flow.
	Positive cash flows	Negative cash flows
	revenues from sales cost of sales
		interests
		tax payments
		debt installments
		total dividends
		share repurchases
	total income	total costs and expenses

Research funding by the European Commission as part of the FP6-STREP project EURACE (under contract no. 035086) is gratefully acknowledged. 1