Análisis de la Variable independiente: Turismo Receptivo

Optimal interpolation (OI) is a robust method of objective analysis extensively used in oceanography. The technique is based on the Gauss-Markov theorem, and determines the esti-mate of the interpolated physical variable with the least mean-square error, given the information about the variances and correlation functions of both the physical variable and the data (Gandin and Hardin, 1965; Bretherton et al.,1976;McIntosh,1990). In other words, it is based on the minimization of the errors between the analyzed field and a true field. The OI methodology has been successfully applied to ocean remote sensing variables as SST (Reynolds and Smith, 1994;Reynolds et al.,2007;Thi´ebaux et al.,2003), SSS (Melnichenko et al.,2014) and sea level anomaly (LeTraon et al.,1998;Ducet et al.,2000).

The analysis at each point x^a will be influenced by all observations y^o that lie within a predefined region of influence. In contrast with the weighted average, OI uses a previous estimate of the analysis or background field x^b and the weight K in Eq. 3.2 depends on the distance between the observation and the point of analysis. In other words, the estimate or analysis is equal to the first guess plus little increments consisting on an interpolation of the first guess of the observation points and an interpolation of the differences between the observed and first guess values back to the analysis point. The gain matrix K is built from matrices that encode the covariances of the field, from the grid point to the observation point and from any observation point to another observation point.

The covariance between variables x and y describe how close are they related in average:

cov(x, y) = E[(x − E[x])(y − E[y])] (3.5) where E[·] is the expectation value. The covariance is larger in absolute value if variables have a close-to-linear mutual dependence, so that equal increases of the value of one variable lead to almost equal increases or decreases of the other. The sign of the covariance tells us if the slope of variations in value of both values are of the same or opposite signs.

When working with vector variables a covariance matrix C needs to be calculated. The element (i, j) of the covariance matrix associated with the random vector x is the covariance between the elements xi and xj.

C = E^h(x − E[x])(x − E[x])^Ti (3.6) When applying objective analysis we always want to estimate an analysis state x^a, which is as close as possible to the true state vector x^t, given a reference field x^b and the observations y^o. It is assumed that both the reference field and the observation have errors:

x^b= x^t+ η^b (3.7)

y^o= Hx^t+ ε (3.8)

where η^b represents the error in the first guess and ε the error of the observations. We also assume that the first guess and the observations are unbiased, what means that the average of

the error is zero:

E[η^b] = 0 (3.9)

E[ε] = 0 (3.10)

However it is necessary to have an a priori knowledge about the magnitude of the observation and the first guess errors, expressed as the error covariances matrices:

E^hη^b(η^b)^Ti= B (3.11)

E^hεε^Ti= R (3.12)

E^hη^bε^Ti= 0 (3.13)

where B is the covariance matrix of the background error, R the covariance matrix of the observation error, and we also assume that the error of the background is independent of the observations error (Eq. 3.13).

The weights that form the gain matrix K (Eq. 3.2) can be thus derived by imposing that departures between the analyzed and reference field are minimized. The solution of this mini-mization problem is:

K = S^TD⁻¹ (3.14)

where D is the observation covariance matrix, and S is a matrix equal to the covariance between each observation point with the reference field:

S = HB (3.15)

The covariance matrix D is divided in two parts, assuming that observational errors are not correlated with the reference field:

D = HBH^T + R (3.16)

where B describes the covariance between pairs of the reference field values at the observation points, and R describes the observational error covariances. In matrix R, the diagonal terms rep-resent the observational error variances and the non-diagonal terms reprep-resent the observational error covariances.

A proof that K minimizes the differences between reference and analyzed field follows: in OI method we chose that the matrix K is the one that minimizes the variance of the analysis P^a. If the total variance is at its minimum, then a slight increment of the gain δK does not modify the value of the total variance in the first order of δK.

tr(P^a(K + δK)) − tr(P^a(K)) = 0

= 2tr(KHBH^TδK^T) − 2tr(BH^TδK^T) + 2tr(KRδK^T)

= 2tr([K(HBH^T + R) − BH^T]δK^T) (3.17)

Since the perturbation of the gain matrix δK is arbitrary, the expression in the brackets have to be zero. The optimal gain, or the gain matrix K is thus:

K = BH^T(HBH^T + R)⁻¹ (3.18)

(analog to Eq. 3.16). One relevant feature of OI is that it also provides an estimate of the analysis error:

P^a= B − KHB

= B − (HBH^T + R)⁻¹HB (3.19)

Thus, granted the above stated assumptions, Eq. 3.2 yields at every point i an analysis or estimate x^a_i which is optimal in the sense that from all estimates that depend linearly on the data supplied, this one has the least error variance. An important element of the method is that adequate estimates of the statistics of the fields should be known considering that the accuracy of the analysis depends on how accurately these statistics are known, on the density of sampling points, and on the variable to be mapped.

3.3.1 Application to SMOS Sea Surface Salinity

To apply optimal interpolation method to generate SMOS SSS OI products it is necessary to first address the following: i) the definition of a suitable correlation model; ii) the characteri-zation of the observational error statistics; and iii) the prescription of a reference or background field.

The first key point of the algorithm is the definition of the spatio-temporal covariance B. In the case of unbiased errors, the spatial structure of the covariance functions is usually expressed in terms of their normalized expression, i.e. the correlation matrix. The covariance matrix B are usually modeled by using analytical functions of the correlation function with certain basic characteristics: i) the correlation is essentially a function of distance, ii) the correlation is one for zero distance and iii) the correlation decreases as distance increases.

The correlation function is often a Gaussian function or a Gaussian function multiplied by a polynomial. The characteristic length scale of the function determines the typical distance over which a data point can be extrapolated. This scale length is the typical scale of the oceanographic processes acting on the field.

The correlation functions used in SMOS-BEC to generate OI maps were originally computed using the results of a numerical model developed byJorda et al. (2011) as there did not exist,

at that time, any satellite SSS data for computing the spatio-temporal correlations necessary to properly describe the required correlation matrix B. Today, with both SMOS and Aquarius satellites on orbit since 2009 and 2011 respectively, new attempts to estimate realistic correlation scales for satellite SSS are being addressed. Taking advantage of this new knowledge, we use correlation functions as the ones calculated by Melnichenko et al. (2014) to produce Aquarius OI products in the generation of the SMOS OI products.

Melnichenko et al. (2014) estimated the correlation functions of SSS from Aquarius data as follows. The L2 SSS data are low-pass filtered and divided into latitudinal regions. A first guess of the reference field (monthly mean SSS from Argo buoys) is subtracted, and mean autocorrelation functions are estimated for repeated swaths over the North Atlantic at each latitudinal region. Figure3.5a displays the mean autocorrelation of SSS; besides autocorrelation functions for ancillary Aquarius data (coming from a HYCOM model solution) are displayed and shown to be in great correspondence (dashed lines). Fig. 3.5b shows the corresponding wavenumber spectra: in the wavelength corresponding to 60 to 300 km, the spectrum follows a power law of the form of k⁻², where k is the wavenumber, what is consistent with the expected shape for the power spectrum.

Figure 3.5: (a) Autocorrelation functions for SSS from Aquarius (solid lines) and from HYCOM SSS (dashed lines) over the SPURS region. The ensemble mean approximations by a Gaussian function with e-folding of 90 km is shown in heavy green line. (b) Corresponding normalized wavenumber spectra. Extracted from published work ofMelnichenko et al.(2014).

The spatial meridional scales of mesoscale SSS variability (defined in their study as the scale associated with the first zero crossing of the corresponding correlation function) are displayed in Figure 3.6. They found the structure of the correlation functions to be similar to a high degree in all latitude bands (see also Figure 3.6) and approximate the observed correlation functions using a simple Gaussian curve, given by:

b(r) = e^−r2/R2 (3.20)

where r is the spatial lag and R = 90km is the decay scale. The Gaussian function with such decay scale (green curve in Figure 3.5) represents well the observed results. The Gaussian model assumes that the correlations are isotropic, which it is non true for tropical latitudes where salinity structures are strongly anisotropic, and zonal scales are larger than meridional

scales. Then, the Gaussian model is modified to account for two scales of decorrelation:

b(rx, ry) = e^{−r2x/R2x−r2y/R2y} (3.21) where rx and ry are the spatial lags in the zonal and meridional directions and Rx and Ry are the associated zonal and meridional decorrelation scales. The meridional scale is set to Ry = 90 km while the zonal scales varies from 180 km in the equator and 90 km as follows:

R_x(y) = 180e^−y2/324.6,0^o≤ y ≤15^oN (3.22)

Figure 3.6: The variance and correlation length scales (the lag of the first zero crossing of the spatial correlation function) of mesoscale SSS variability as seen by Aquarius in the North Atlantic. Extracted from published work ofMelnichenko et al.(2014).

The influence radius determining the number of observations that will influence the anal-ysis at each grid point is set to 192 km to allow sufficient observations to be weighted using correlation matrix B as defined above. The error of the observations are here assumed to be un-correlated to each other, so the observation error covariance R becomes a diagonal matrix. This hypothesis is not guaranteed for satellite SSS data, since the errors in satellite retrievals are of different types and are spatially correlated (Lagerloef and Coauthors,2013;Melnichenko et al., 2014). A procedure to reduce the noise and the noise correlation is spatio-temporal averaging of observations instead of working with all observations.

With this in mind, we chose to apply the optimal interpolation method to the L3 SSS fields produced as explained in section3.2. The downside of this procedure is that the spatial resolution of the data is also reduced and that small-scale structures visible in the original data are not present in the binned observations (Barth et al., 2008). The dispersion in terms of standard deviation of the L2 measures computed at each bin (Figure3.4) is used as an approximation of the observational error R.

Finally, we use a climatological field as a background field, obtained from the time average of historical data. Climatological fields constitute proper first guesses from the statistical point of view (the residuals have a zero statistical mean by definition) and they ensure that there will be no discontinuities in the background field used for the analysis of the neighboring points.

The first background or reference field used to apply the OI methodology is the WOA09 climatology (Antonov et al., 2010). This climatology is linearly interpolated into 0.25 spatial grid and daily time resolution. The error variance of this first guess is supposed to be 0.09 (ε in Eq. 3.8). An example of this climatological SSS field for the month of June is shown in Figure 3.23. It must be noted, however, that using WOA09 as a background field is not theoretically

correct, as this climatology is being used as ancillary data to retrieve SMOS SSS, and in OI approach it is a requirement that the reference field is an independent source of information.

The impact on using different background fields is assessed in section 3.5.

Figure 3.7: Background surface salinity World Ocean Atlas for month of June.

Figure 3.8: Number of L3 measures of 0.25^◦×0.25^◦used in each analysis point for the first 9 days of June 2012.

The map with the number of L3 measurements used to create the optimal analysis at each grid point for an example of 9-day map for June 2012 is shown in Figure 3.8. On average, 50 observations are used to interpolate the field at each analysis point for a 9-day map.

The application of the OI scheme to interpolate the weighted averaged SMOS SSS data for 1st-9th of June 2012 and for the entire month of June 2012 are shown in Figure3.9. These maps

have better spatial coherence compared to the used observations (Figures 3.2). Moreover, the resulting field contains signal coming from observations that was not in the background field (Figure 3.23). Comparing the results with the monthly climatology, the SSS is fresher in the equatorial zone, and saltier in the subtropical gyres.

Figure 3.9: Map of surface salinity applying optimal interpolation on SMOS data for first nine days (top) and the entire month of June 2012 (bottom).