# Satellite and In Situ Sampling Mismatches: Consequences for the Estimation of Satellite Sea Surface Salinity Uncertainties

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

- -
- Satellite product: CCI v3.2.

- -
- Instead of using SMOS SSS produced by the Centre Aval de Traitement des Données SMOS (CATDS) operational chain, the SMOS SSS were reprocessed with a modified ESA v662 processor [15].
- -
- Instead of using European Center for Medium Weather Forecast (ECMWF) Integrated Forecast System (IFS) fields as auxiliary parameters, the processing used ECMWF ERA5 fields.
- -
- The SMOS vicarious calibration, the so-called Ocean Target Transformation, is estimated using in situ interpolated SSS fields produced by the In Situ Analysis System (ISAS) [18] instead of using a salinity climatology.
- -
- The dielectric constant model is updated as proposed by [5].
- -
- The SMOS SSS affected by instantaneous rain rate are adjusted for RR up to 10 mm h
^{−1}[19] and are sorted out in case of stronger RR. SMAP SSS retrieved with RR larger than 0.5 mm h^{−1}are filtered out. Therefore, in rainy regions the CCI v3 fields are close to bulk salinities. - -
- In the CCI v3 L4 optimal interpolation, a full least square propagation of the errors is implemented, instead of a simplified propagation.
- -
- Representativity uncertainties between the swath measurements of the SMOS and SMAP level 2 SSS and that of the L4 estimated SSS (weekly or monthly fields), corresponding to the temporal variability of the SSS at the SMOS and SMAP resolution (~50 km), within one week or one month, are taken into account as described in [5] and in its Supplementary Information (S3). The equations are the same for CCI version 2 and version 3, but the numerical implementation differs: a full error propagation scheme has been implemented in version 3.

- -
- In-situ products: Argo floats

- -
- Mercator GLORYS reanalysis

#### 2.2. Methods

- -
- Uncertainties balance

- -
- ${\mathrm{x}}_{\mathrm{sat}}-{\mathrm{x}}_{\mathrm{ref}}$

_{ref}corresponding to the Argo measurement points made available in the Pi-MEP database are used as the basis for calculating the normalized differences described in Equation (1). The terms U

_{sat}and U

_{mis}are thus collocated at these locations as described below.

- -
- ${\mathrm{U}}_{\mathrm{sat}}$

- -
- ${\mathrm{U}}_{\mathrm{ref}}$

- -
- ${\mathrm{U}}_{\mathrm{mis}}$

_{mis_glo}).

_{mis_glo}is limited, however, by the resolution of the GLORYS reanalysis (1/12°). This means that the variability at scales smaller than 1/6° present in the point measurements will not be well represented in GLORYS simulations. We take this limitation into account by estimating the variability missed by the GLORYS reanalysis following a wavenumber spectral analysis [12], as summarized below and described with more details in Appendix A.

^{−3}power law expected for a passive tracer under the influence of advection, with the slightly steeper slope being likely attributable to atmospheric processes [12]. Even though this slope is likely to vary slightly from time to time and place to place, we use it to infer an order of magnitude of the ratio between the variability expected in the case where all wavelengths below 50 km are considered, U

_{mis}, and the variability estimated with GLORYS reanalysis resolution, U

_{mis_glo}, using Equation (2). We find the following relation:

- -
- -
- In order to make local analysis, we also compute STD of normalized differences in 2° boxes. However, due to the reduced number of colocations in 2° boxes (as will be seen in Section 3.2.2), they are noisy and it is not possible to estimate a reliable fit of the normalized differences distribution in each box. Nevertheless, we look at to which extent the statistical distribution of STD estimated in 2° boxes is consistent with Gaussian distributions of the normalized differences in each box. This is performed by considering the histograms of the variances, STD
^{2}, multiplied by the number of measurements. Indeed, given X_{1}, ⋯, X_{n}a random sample from a Gaussian distribution (with a μ average and a σ standard deviation) N(μ,σ^{2}), in any of the 2° boxes, with ${S}^{2}=\frac{1}{n-1}{\displaystyle \sum}_{1}^{n}\left({X}_{i}-\overline{X}\right)$, the random variable Y = (n−1)S^{2}/σ^{2}= n Var(X)/σ^{2}follows a χ^{2}_{n}_{−1}distribution (See p. 211 of [27]). This choice of representation allows us to compare the histograms obtained to a theoretical curve that Y should follow if the normalized differences distributions in each box followed a N(0,1) distribution, as is expected if the uncertainties are correctly estimated. The theoretical curve is deduced by cumulating the distributions of the Y term expected for each of the 2° boxes, considering σ = 1.

## 3. Results

#### 3.1. The Different Contributions to the Uncertainty Balance

_{sat}from CCI data (Figure 3b) are similar in most open ocean regions, where the natural variability of salinity is expected to be low (below 0.2). The STD of CCI7-Argo SSS seems therefore bounded by the value of the satellite uncertainty U

_{sat}in regions of low variability. High values of STD (CCI7-Argo) (>0.5, going up to 1) are observed in Figure 3a in regions where natural variability is important, for example, in river plumes (notably the Amazon, Congo, Malvinas, Ganga Brahmaputra and Mississippi) and in areas of important fronts (Gulf Stream region, Agulhas return current and Kurushio). In these regions, U

_{sat}is also higher than in low variability regions due to an increased uncertainty in the CCI weekly SSS field related to the temporal undersampling of SMOS and SMAP, but U

_{sat}remains, in these regions, lower than the STD(CCI7-Argo). The difference between the squared terms (STD

^{2}(CCI7-Argo)—mean (U

_{sat}

^{2}), Figure 3c) and U

_{mis}

^{2}(Figure 3d) agree qualitatively well, even though U

_{mis}appears higher in river plumes between 20°N and 20°S. In a few regions, notably near Japan and in the Labrador Current region, we observe large values of STD(CCI7-Argo) that are stronger than U

_{sat}and U

_{mis}. These differences are likely due to the frequent presence of RFIs in these regions [26].

#### 3.2. Detailed Analysis of Uncertainties Balance

#### 3.2.1. Distribution of Reduced Differences

_{mis_glo}on the comparisons between in-situ Argo values and CCI products, we analyze the STD of the differences normalized by:

- (a)
- The satellite uncertainty U
_{sat}only (orange curves); - (b)
- The quadratic mean of U
_{sat}and U_{mis_glo}(red curves); and - (c)
- The quadratic mean of U
_{sat}and U_{mis}(blue curves).

_{sat}only (orange curve), which is too large, to a value of 1.03 much closer to unity with the quadratic sum of U

_{sat}and U

_{mis_glo}(red curve). Using U

_{mis}(blue curve) results in a STD of 0.99, very close to the value of 1 we are looking for.

_{mis_glo}considerably reduces the distribution tails: the values higher than 3.9 are reduced by about one half, and even more with U

_{mis}.

_{sat}, and that the addition of U

_{mis_glo}has little impact on the distribution of differences. On Figure 5a, the STD of the red and blue curves taking into account U

_{mis_glo}are slightly too low, which means that U

_{mis_glo}or U

_{sat}is slightly overestimated in this region.

_{mis_glo}and U

_{mis}allows one to obtain a STD of 1.05 and 1.02, respectively.

_{mis_glo}or U

_{sat}are too low. Taking into account U

_{mis_glo}or U

_{mis}allows one to get closer to a Gaussian of standard deviation of 1, going from a STD = 1.42 for the orange curve to a STD = 1.20 or STD = 1.13 with U

_{mis_glo}and with U

_{mis}, respectively. This corresponds to an improvement of about 30% but remains imperfect. This may be related to remaining seasonal biases observed in this region in the CCI data whose origin remains under study.

_{mis_glo}and U

_{mis}has a too strong impact. This could be related to uncertainty in GLORYS derived variability, as discussed in next section.

_{sat}, we compare the standard deviation values to the distributions obtained with the previous version of the CCI products (v2.3), for the global ocean, in which U

_{sat}was estimated with a simplified approach.

_{mis_glo}and 0.89 with U

_{mis}). In version 2.3, the errors were therefore overestimated by about 12% and particularly too high in regions of low variability. This has been corrected in version 3.2. Histograms for version 2.3 are available in Supplementary Materials.

_{mis_glo}, 1.003 with U

_{mis}). In the Pacific region, the importance of mismatch uncertainty is less, but taking it into account also allows one to approach the STD = 1 curve. The histograms on the variable regions yield the same conclusions as with the weekly products.

#### 3.2.2. Global Distributions

_{mis_glo}(Figure 6c) reduces the very large STD in highly variable regions such as the Gulf Stream or the Agulhas Current retroflection. This reduction is even stronger with U

_{mis}(Figure 6d). This can be seen in the histograms (Figure 7), where the red and blue histograms, which take into account U

_{mis_glo}and U

_{mis}, respectively, are much closer to the black theoretical curve for high values (above 50) corresponding to high variances than the orange histogram, where the differences are normalized by U

_{sat}only. Quantitatively, the correlation coefficient between the χ

^{2}theoretical distribution and the one obtained with U

_{sat}only is 0.95, and, respectively, 0.97 and 0.98 when taking into account U

_{mis_glo}or U

_{mis}.

_{sat}only, the STD obtained is clearly greater than 1 and often even greater than 2 in a large majority of the 2° boxes. Taking into account U

_{mis_glo}or U

_{mis}gives quite similar maps (Figure 6c,d), which again shows the importance of mismatch uncertainties in the validations, especially for monthly products. The histograms obtained for the monthly products also show this clear improvement, which is strengthened with U

_{mis}. The correlation coefficient between the χ

^{2}theoretical distribution and the one obtained with U

_{sat}only is 0.81, and, respectively, 0.97 and 0.98 when taking into account U

_{mis_glo}or U

_{mis}.

^{2}curve, with a very large number of points at low values of n Var(X) indicating that the U

_{sat}of version 2 was overestimated, mostly because of the simplified propagation of errors in version 2 instead of the full least square propagation of the errors implemented in version 3. The correlation coefficient between the χ

^{2}theoretical distribution and the one obtained with U

_{sat}only is 0.71 and, respectively, 0.75 and 0.76 when taking into account U

_{mis_glo}or U

_{mis}for CCI v2.

## 4. Discussion

#### 4.1. Estimation of the Sampling Mismatch Uncertainty

^{−3.3}constant in time and space, but this slope could vary slightly according to regions and time. We note, however, a weak variation of the coefficient according to the slope of the spectrum: +(−)2% for a spectrum of slope of k

^{−3.2}(k

^{−3.4}), compared to the used spectrum of slope k

^{−3.3}.

_{mis_glo}is close to the one used by [13] to estimate the Representation Error (RE), except that [13] uses higher resolution (1/48°) simulations. We have compared both estimates, made within an area of 50 km radius for U

_{mis_glo}and within 40 km for RE. Maps of the median of U

_{mis_glo}over the 3 years of our study (available in Supplementary Materials) are qualitatively in agreement with the 40 km RE (Table S5 of [13]).

#### 4.2. Sampling Mismatch Uncertainty and Representativity Uncertainty Included in U_{sat}

_{sat}quantifies the uncertainty on the weekly-50 km smoothed SSS related to the temporal undersampling of SMOS and SMAP. This temporal variability, taken into account in the CCI temporal optimal interpolation, is deduced from the GLORYS SSS fields smoothed to 50 km. Therefore, U

_{sat}does not include information on the spatial variability at scales smaller than 50 km nor on the expected variability between a point in time measurement and a weekly field, contrary to U

_{mis}. Nevertheless, a flaw in GLORYS SSS fields that would affect SSS fields at scales both larger and smaller than 50 km would lead to correlated errors in U

_{sat}and U

_{mis}. We cannot rule out this kind of error near river plumes between 20°N and 20°S.

#### 4.3. Vertical Near-Surface Variability

## 5. Conclusions

_{mis_glo}or U

_{mis}in the validation allows the STD of the normalized differences to get very close to a χ

^{2}theoretical distribution (Figure 7; correlation coefficient of 0.95 with U

_{sat}only and of 0.98 when taking into account sampling mismatches).

_{sat}and U

_{mis}has a standard deviation very close to 1 over the global ocean, slightly too high (1.05) in the variable regions and a little too low (0.98) in the region of low variability of the Pacific. These results are significantly better, up to 30% better in the Gulf Stream, than when the sampling mismatch uncertainty is not taken into account.

_{sat}, as was commented on in the discussion, the calculation of U

_{mis_glo}in the river plume regions may be too high. Future studies should address this issue by comparing the variabilities obtained from GLORYS to the ones estimated from other models or with other in situ datasets.

_{sat}is too low there. This could be related to remaining seasonal variation of biases at high northern latitudes possibly linked to ice contamination, and to calibration flaws related to sun or antenna temperature or RFIs impacts, which are not well represented in the U

_{sat}values.

_{sat}, the satellite measurements uncertainties and the representativity uncertainties is specific to each instrument and strongly differs for the Aquarius measurements. Aquarius SSS’ uncertainties are lower than the uncertainties of SMOS and SMAP level 2 SSS, while its representativity uncertainties are larger due to the spatial integration over 100–150 km relative to 50 km and to a narrower swath leading to an increased undersampling. The work on level 4 CCI+SSS data presented here could also be applied to other types of data, by adapting the spatio-temporal scales on which the uncertainties due to sampling mismatches are calculated.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{mis}, based on a spectral analysis of surface salinity. This method has been developed in [12], Appendix A.

_{n}is the Nyquist wavenumber.

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**Figure 1.**Examples of SSS measured in three regions, in February 2016.

**Top**: ARGO floats,

**Middle**: CCI,

**Bottom**: GLORYS Reanalysis; Column 1: West African coast; Column 2: Gulf Stream; and Column 3: Amazon plume.

**Figure 2.**Standard deviation of GLORYS data within one week and 50 × 50 km

^{2}surface (U

_{mis_glo}) at 5 m depth for the 15 February 2017. Orange (purple) boxes correspond to the low (high) variability regions that are selected for our tests (see regions definition in Table 1).

**Figure 3.**Terms of Equation (1) integrated for 2016–2018: (

**a**) observed STD of the differences between weekly CCI and Argo salinity; (

**b**) quadratic mean of U

_{sat}; (

**c**) the difference of squared terms shown in (

**a**,

**b**), STD

^{2}(CCI7-Argo)—mean (U

_{sat}

^{2}); and (

**d**) quadratic mean of U

_{mis}. The statistics are reported in 1° boxes containing at least 3 Argo measurements (boxes in Figure 3a,b with less than 3 measurements are in white).

**Figure 4.**Distribution of differences between Argo floats and CCI data normalized with U

_{sat}(orange), with U

_{sat}and U

_{mis_glo}(red) and with U

_{sat}and U

_{mis}(blue), and corresponding Gaussian fits, over the global ocean. The black line corresponds to a theoretical Gaussian distribution with a standard deviation of 1. Corresponding STD are reported in Table 2.

**Figure 6.**Standard deviations of CCI-Argo differences normalized by the various sources of uncertainties, in 2° boxes. Only boxes with more than three Argo measurements are considered. (

**a**,

**c**,

**d**) Maps of the STD of the differences normalized Equation (1) by (

**a**) U

_{sat}; (

**c**) U

_{sat}and U

_{mis_glo}; and (

**d**) U

_{sat}and U

_{mis}. (

**b**) Number of collocated points in each 2° box.

**Figure 7.**Histograms (bar graph) of number of points multiplied by variance in the pixels shown in Figure 6: (a) (orange), (c) (red) and (d) (blue). Points above 300 are grouped in one single class. This histogram is expected to follow a Chi2 distribution (dashed line) if Equation (1) holds for each pixel of the maps.

**Figure 8.**Comparison between GLORYS data standard deviation and in-situ variability. (

**a**) Merchant ships averaged over 1° boxes; (

**b**,

**c**) histograms of Merchant ships STD (blue) and GLORYS STD (red) (

**b**) globally, and (

**c**) in very variable areas defined Table 1.

**Figure 9.**Difference between STD of GLORYS at surface minus STD of GLORYS at 5 m, computed over 2016–2018 period.

**Table 1.**Definition of regions shown on Figure 2.

Region | Latitude (°) | Longitude (°) | Very Variable Area | ||
---|---|---|---|---|---|

Gulf Stream | 30.52 | 57.64 | −74.83 | −29.70 | X |

Amazon plume | −3.43 | 14.37 | −59.27 | −34.89 | X |

Agulhas return current | −54.84 | −31.43 | 5.84 | 91.43 | X |

South Pacific Ocean | −60.27 | −40.10 | −172.35 | −81.57 |

**Table 2.**STD of the distributions of the normalized differences between Argo and CCI v3.2 weekly products, in the regions defined in Table 1.

Region | U_{sat} | U_{sat} + U_{mis_glo} | U_{sat} + U_{mis} |
---|---|---|---|

Global | 1.158 | 1.029 | 0.988 |

Gulf Stream | 1.424 | 0.904 | 1.138 |

Amazon plume | 1.050 | 1.202 | 0.859 |

Agulhas return current | 1.163 | 1.059 | 1.022 |

South Pacific Ocean | 1.017 | 0.978 | 0.964 |

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## Share and Cite

**MDPI and ACS Style**

Thouvenin-Masson, C.; Boutin, J.; Vergely, J.-L.; Reverdin, G.; Martin, A.C.H.; Guimbard, S.; Reul, N.; Sabia, R.; Catany, R.; Hembise Fanton-d’Andon, O.
Satellite and In Situ Sampling Mismatches: Consequences for the Estimation of Satellite Sea Surface Salinity Uncertainties. *Remote Sens.* **2022**, *14*, 1878.
https://doi.org/10.3390/rs14081878

**AMA Style**

Thouvenin-Masson C, Boutin J, Vergely J-L, Reverdin G, Martin ACH, Guimbard S, Reul N, Sabia R, Catany R, Hembise Fanton-d’Andon O.
Satellite and In Situ Sampling Mismatches: Consequences for the Estimation of Satellite Sea Surface Salinity Uncertainties. *Remote Sensing*. 2022; 14(8):1878.
https://doi.org/10.3390/rs14081878

**Chicago/Turabian Style**

Thouvenin-Masson, Clovis, Jacqueline Boutin, Jean-Luc Vergely, Gilles Reverdin, Adrien C. H. Martin, Sébastien Guimbard, Nicolas Reul, Roberto Sabia, Rafael Catany, and Odile Hembise Fanton-d’Andon.
2022. "Satellite and In Situ Sampling Mismatches: Consequences for the Estimation of Satellite Sea Surface Salinity Uncertainties" *Remote Sensing* 14, no. 8: 1878.
https://doi.org/10.3390/rs14081878