[12] After the down-converted and digitized GPS-L1 raw data has been acquired, it is processed by a closed-loop GPS software receiver in the instrument data management unit (see *Ruffini and Soulat* [2004] for more details on the STARLIGHT tracking software). The software receiver initializes the process (that is, finds satellites in view and selects those within the mask), performs the correlations and tracks the delay and Doppler of the direct GPS-C/A signal, feeding this information to the reflected signal correlation module. This operation is typically carried out with 1–10 minute data segments (but limited to 1 minute in HOPE 2). The output of this process is so-called Level 0 data, consisting of the times series of complex waveforms for the direct and reflected signals. Level 0 data can then be used to produce Level 1 data (such as the ICF discussed below or the altimetric lapse discussed by *Ruffini and Soulat* [2004]) and Level 2 geophysical products (such as sea surface height or sea state related parameters, as discussed next). The software receiver carries out other tasks, such as solving for the time and position of the up-looking antenna.

[13] The analysis for sea state begins with the interferometric complex field (ICF), defined at time *t* by *F*_{I}(*t*) = *F*_{R}(*t*)/*F*_{D}(*t*), where *F*_{D} and *F*_{R} are the complex values at the amplitude peaks of the direct and reflected complex waveforms, respectively. The direct signal is thus used as a reference to remove features unrelated to ocean motion, such as any residual Doppler, the navigation bit phase offset, or direct signal power variability. The ICF contains very valuable information on the sea state. More precisely, it is the dynamics of the ICF which is of interest, as we discuss below.

#### 3.1. ICF Coherence Time

[16] As a first step in the analysis of the ICF dynamics we have focused on the coherence time of the ICF, τ_{F}, defined here as the short time width of the ICF autocorrelation function, Γ(Δ*t*) = 〈*F**_{I}(*t*)*F*_{I}(*t* + Δ*t*)〉_{z}.

[17] After removal of the carrier and code, we can use the Kirchhoff Tangent Plane approximation for the scattered field [see *Beckmann and Spizzichino*, 1963],

where Δω is the residual carrier frequency, *n*π the navigation bit, ℛ is the Fresnel coefficient, *k* = 2π/λ, with λ ≈ 19 cm in L1, *r* (*s*) is the distance between the receiver (transmitter) and each point of the sea-surface, the surface normal and = (_{⊥}, *q*_{z}) is the scattering vector (the vector normal to the plane that would specularly reflect the wave in the receiver direction). This vector is a function of the incoming and outgoing unit vectors _{i} and _{s}, = *k*(_{s} − _{i}).

[18] We assume here that · ≈ *q* (small slope approximation, with scattering and/or support only near the specular).

[19] We note here that the exponent in the integrand can be expanded as a power series in the surface elevation *z*, and that higher order terms are suppressed by the other scales in the problem. As an approximation, we can retain only the first order term (as in the Fraunhofer limit). In order to compute Γ(Δ*t*) we now assume a Gaussian probability distribution for the surface elevation and write [see, e.g., *Beckmann and Spizzichino*, 1963]

where = (*x*, *y*) is the horizontal displacement vector from the specular point, σ_{z} is the standard deviation of the surface elevation, ε the scattering elevation angle and *C*(Δ, Δ*t*) the spatio-temporal autocorrelation function of the surface. Using a parabolic isotropic approximation for *C*(Δ, Δ*t*) (valid for small Δ and Δ*t*) and considering for simplicity that spatial and temporal properties of the surface can be separated, we write *C*(Δ, Δ*t*) ≈ 1 − (Δρ)^{2}/2*l*_{z}^{2} − Δ*t*^{2}/2τ_{z}^{2}, where *l*_{z} and τ_{z} are, respectively, the correlation length and correlation time of the surface. Isotropy is a rather strong assumption, and will lead to a coherence time independent of wave direction (directional analysis will be taken up in a future effort).

[20] Using this expression, it can readily be shown that the autocorrelation of the field can be approximated by

[see also *Ruffini et al.*, 2004]. This equation, valid for small times, states that the autocorrelation of the field is a Gaussian function of Δ*t* and proportional to a coefficient depending on the sea surface elevation standard deviation σ_{z}, surface autocorrelation length, *l*_{z}, geometry and antenna gain *G*_{r}.

[21] The coherence time of the ICF is now given by the width (second order moment) of this Gaussian function,

According to this model, τ_{F} depends on the electromagnetic wavelength and the ratio between the correlation time of the surface and the significant wave height (an inverse velocity). A fundamental product of the instrument is therefore τ_{z}/SWH = πτ_{F} sin ε/λ.

[22] In order to check this model using buoy data (SWH and MWP), we have derived a relation between MWP (available from the buoy measurements) and the sea-surface correlation time τ_{z} (needed to evaluate the right hand side of equation (4), through Monte-Carlo simulations using a Gaussian sea-surface spectrum [*Elfouhaily et al.*, 1997]. Simulating the surface propagation at a given point *z*(*x*_{o}, *y*_{o}, *t*), the MWP was estimated through the Fourier analysis of the time series of *z*(*x*_{o}, *y*_{o}, *t*), while τ_{z} was determined by the width of the autocorrelation function 〈*z*(*x*_{o}, *y*_{o}, *t*)*z*(*x*_{o}, *y*_{o}, *t* + Δ*t*)〉. We obtained, for a well developed sea-state (with inverse wave age Ω = 1), the relation τ_{z} ≈ *a*_{m} + *b*_{m} * MWP (*a*_{m} = 0.07, *b*_{m} = 0.12, with an error of 0.09 s). Using this expression we can write

[23] Based on this analysis, the Level 0 to Level 2 data processing involves two steps. First, the computation of the autocorrelation function of the complex interferometric field is carried out. Then, a Gaussian is fitted around lag zero to provide the estimate of the coherence time (Level 1).

[24] The comparison of the estimated ICF coherence time with the available ground truth (wind speed, SWH and MWP) is made through equation (5). The results are shown in Figure 3 (left). As observed, the measurements correlate well with theory. Note that there is also good consistency between Take 2 and Take 3 data.

[25] It is worth mentioning that the linear relationship relating τ_{z} to MWP has been obtained under the assumption of a fully developed sea. This assumption will not hold in general in coastal areas for the whole range of sea-state conditions.

#### 3.2. ICF and SWH

[26] Coherence time data can also be translated into Level 2 geophysical products such as SWH using a semi-empirical algorithm, as we now discuss. We assume that the correlation time of the surface is itself a function of the SWH and write an expression for the “effective” coherence time, τ′_{F} ≡ τ_{F} sin ε = *f*(SWH), where in the open ocean *f*(SWH) is in general a known function of SWH but which will also depend on the sea state maturity, fetch, bathymetry, etc. In coastal areas, this function will be harder to estimate from theory and a semi-empirical approach is envisioned.

[27] Based again on the *Elfouhaily et al.* [1997] spectrum we have derived a linear relationship between τ_{z} and the SWH: τ_{z} = *a*_{s} + *b*_{s} * SWH (*a*_{s} = 0.167, *b*_{s} = 0.388, and an error of 0.03 s). This relation turns out to be rather independent of wave age. Using it, we can now rewrite equation (4) as

Since the instrument gathered coastal data (within ∼100 m radius), the comparison with open ocean buoy data is not direct. In order to compare open ocean data to coastal measurements, we include a SWH “shift” parameter, SWH_{0} and a scale parameter γ.

[28] The algorithm for translation of effective ICF coherence time to SWH becomes, finally (Oceanpal SWH Algorithm 1),

valid for SWH > SWH_{0}. We have found that a value of SWH_{0} = 0.21 m and γ = 1.8 gives the best fit to the campaign data. Figure 3 (right) plots SWH buoy data against Oceanpal SWH Algorithm 1. The algorithm standard deviation from the buoy data is 9 cm.