It is challenging to derive the quantitative relationship between the SSS and the SST because of the complexity of the governing equations. The multi-fractal theory provides a new approach for quantifying the relationship between the SSS, the SST and other ocean scalars from a geometric perspective (Turiel et al., 2005; Isern-Fontanet et al., 2007; Nieves et al., 2007; Olmedo et al., 2016; Turiel et al., 2008; Umbert et al., 2014). Justification of this theory has been provided by Umbert et al. (2014) and Olmedo et al. (2016). A concise overview of the theoretical basis of multi-fractal fusion now follows.

The association between the geometric structure of the SSS and the SST at any point $ \vec x $ can be expressed as [see Olmedo et al., (2016) for more details]

where $ {T_\psi }\left| {\nabla S} \right|(\vec x,r) $($ {T_\psi }\left| {\nabla \theta } \right|(\vec x,r) $) represents the modulus of the gradient of the SSS (SST) projected on a given wavelet function $ \psi $; $ {\alpha _S}(\vec x) $($ {\alpha _\theta }(\vec x) $) denotes the amplitude factor of the SSS (SST); r is the non-dimensional scale parameter. The singularity exponent h is a dimensionless quantity depicting the fractal structure. Note that h is the same for the SSS and the SST and represents the correspondence of their fractal structures.

According to Umbert et al. (2014), a proportional relationship can be derived based on Eq. (1) via a smooth, slow-varying function $ {\text{Φ}} (\vec x) $.

where $ {\text{Φ}} (\vec x) $ is considered to be a matrix, however the integration of the gradient of the SSS would introduce large systematic errors (Olmedo et al., 2016). As a result, the $ {\text{Φ}} (\vec x) $ has to be simplified as a scalar $ a(\vec x) $ quantity and Eq. (2) can thus be integrated directly giving Eq. (3).

where the intercept b is calculated by $ b(\vec x) = \left\langle {S(\vec x)} \right\rangle - a(\vec x)\left\langle {\theta (\vec x)} \right\rangle $, where $\langle \cdot \rangle $ is the local average operator, namely, a local 2D filter. Then Eq. (3) can be transformed into Eq. (4).

The right part of Eq. (4) consists of two terms. The first term is characterized by the difference between the SST map and the filtered SST map. It is this term that transmits the frontal or singular structures of the SST map to the SSS map. Determined by the feature resolution of the template, this singular term is the guarantee that MFF will improve the resolution of the SSS map. The second term is the background SSS, where both the small-scale signals and the noises are removed.

Although multiple SSS products are available, only the SMOS-based products can provide the long record of data from 2010 to the present. Two SMOS-based SSS products are thus considered in this paper.

The SMOS BEC L3 binned SSS product is retrieved from the non-Bayesian algorithm (Olmedo et al., 2017). This product blends the SSS data measured by SMOS and binned onto a 0.25° × 0.25° grid. The temporal resolution of 9-d running. A series of reprocessing has been employed, including a geophysically-consistent filtering criterion, an empirical correction to reduce the biases, an interpolation scheme to improve the near-coast values and an estimate of the uncertainty (Olmedo et al., 2021). Consequently, the systematic errors have been effectively removed and the effect of noises has been mitigated. There are two versions of BEC L3 SSS products, namely the “high-resolution version” (hr) and the “low-resolution” (lr) version. The hr version retains the potential mesoscale features from SMOS retrievals, while the lr version has removed part of mesoscale features through the 50 km Gaussian filter. The hr version is selected as the source data considering the fusion of a truly high-resolution SSS product.

The climate change initiative (CCI) SSS dataset is produced by the European Space Agency (ESA) CCI project. This product blends the data from three existing SSS missions including SMOS, Aquarius and SMAP. It has been sampled spatially on a 25 km equal area scalable earth (EASE) grid with a sampling time of 1 d. The blending is based on the optimal interpolation and a series corrections and calibrations, e.g., the correction of latitudinal biases and the correction of individual SSS. The comparisons with in situ data indicate much better performances than the ones obtained with a single satellite data product (Boutin et al., 2020).

Two types of sensors can measure the SST from space, i.e., the microwave sensor and the infrared sensor. The microwave sensor is unaffected by cloud cover but the resolution is lower (about 25 km), while the infrared sensor can measure the near-coastal values and the resolution is higher (about 9 km or 1 km). Although multiple SST products blend the data from multi-sensors, the differences in the specific data used and the fusion scheme can result in a divergence in data quality. Four SST products are considered in this paper and are now introduced.

The optimum interpolation SST (OISST) is a (1/4)° daily product that can provide a long-term record of the SST. This dataset incorporates multi-source observations including those via satellites, ships, buoys, and Argo floats. The advanced very high resolution radiometer (AVHRR)-only product is employed here.

The OSTIA SST is a daily product and has a nominal spatial resolution of 0.05° × 0.05° (Donlon et al., 2012). The OSTIA program is run by the Met Office and the dataset is provided by the Copernicus Marine Environment Monitoring Service (CMEMS). This product fuses in situ and satellite data from both infrared and microwave radiometers by means of objective analysis.

The office of satellite and product operations (OSPO) SST analysis is a daily product on a global 0.054° grid. This analysis blends the data from AVHRR, the visible infrared imager radiometer suite (VIIRS), the geostationary operational environmental satellite (GOES) imager, the Japanese advanced meteorological imager (JAMI) and in situ data.

The remote sensing systems (REMSS) MW_IR OI SST product is also employed. The product blends microwave data and infrared data (MW_IR) at 9 km resolution. The spatial resolution is 0.1° and the temporal resolution is daily.

The near surface salinity for the Argo floats of the global argo observational data set from the China Argo Real-time Data Center (Li et al., 2019) was used for validation. This data set is the quality re-controlled product of all of the profiles in the Global Data Assembly Center (GDAC). Only the data with quality control (QC) flag = 1 (“the highest quality”) are considered. To match the daily SSS map, the shallowest Argo salinity above 10 m in a 9-d temporal window was selected. The SSS field was then interpolated onto the positions of these Argo floats by means of the bilinear interpolation. In this way, a pair of SSS measurements together with their co-located Argo near surface salinity (NSS) observations were obtained for comparison purposes.

The tropical atmosphere ocean (TAO) mooring array can provide time series for salinity. This data set undergoes a series of QC measures and only those data whose QC flag = 1 (“good”) are used. The daily observations of 1-m salinity are employed as the NSS.

The Argo profiles of global argo data set (V3.0) can be available by the official website: http://www.argo.org.cn/index.php?m=content&c=index&a=lists&catid=26. The buoys data of TAO are available at http://www.pmel.noaa.gov/gtmba/. The BEC SSS is available at http://bec.icm.csic.es/ocean-global-sss/. The CCI SSS can be obtained from https://catalogue.ceda.ac.uk/uuid/aea368301bff448bbbea7e90e66819e5. The OISST is available at https://psl.noaa.gov/data/gridded/data.noaa.oisst.v2.highres.html. The OSTIA data set (SST_GLO_SST_L4_NRT_OBSERVATIONS_010_001) is disseminated by Copernicus Marine Service: https://resources.marine.copernicus.eu/. The OSPO data set can be retrieved from https://www.ncei.noaa.gov/access/metadata/landing-page/bin/iso?id=gov.noaa.nodc:GHRSST-Geo_Polar_Blended-OSPO-L4-GLOB. The REMSS data set is available at https://www.remss.com/measurements/sea-surface-temperature/oisst-description/.

As shown in Eq. (4), the feature resolution of the fused SSS product relies mainly on the template of the SST. Consequently, the selection of the SST template is an essential step in the fusion of the high-resolution SSS product. The wavenumber spectra for the four candidate SST templates were computed to decide the best template. The selection criterion is the best agreement between the feature resolution and the nominal resolution. Four regions were considered, i.e., the Southern Indian Ocean (SIO; 10°−50°S, 70°−100°E ), the Gulf Stream (GS; 30°−50°N, 30°−80°W), the Kuroshio Current (KC; 25°−50°N, 140°−180°E) and the Tropical Pacific (TP; 30°S−30°N, 120°−180°W). Among these four regions, the GS and the KC are two of the most energetic regions that are characterized by strong currents and abundant eddies, while the SIO and TP are characterized by salinity fronts. As shown in Fig. 1, the spectrum for each product cuts off at the Nyquist frequency according to the grid resolution. It can also be seen that in the four selected regions, the PSD of the OISST is almost the highest of the four products at large scale, but it decays rapidly at 100−150 km and the slope drops to −5. This demonstrates that the feature resolution of the OISST is far coarser than the nominal resolution of about 50 km. Only those motions whose scales are larger than 150 km can be effectively reflected. The spectrum of the OSTIA at 50−150 km is very close to that of the OISST and is also characterized by the sharp drop of the PSD. It turns out that the OSTIA is such an over smoothed product that its feature resolution is at the same level as that of the OISST, despite the far higher nominal resolution (0.05°) of the former. The spectrum of the OSTIA flattens out at about 75 km in the TP (Fig. 1d) and at about 35 km in the SIO, GS, and KC (Figs 1a-c). In this case, the scale that is finer than 35 km is totally dominated by white noises. The PSD of the OSPO is clearly higher than that of the OSTIA at the mesoscale band. The slopes of the OSPO in the SIO (Fig. 1a), KC (Fig. 1c) and TP (Fig. 1d) remain at about −3 at the scale of 50−200 km. This indicates that the mesoscale motions can be resolved by the OSPO in these three regions. In the GS (Fig. 1b) area, the spectrum of the OSPO decreases rapidly at 100 km, corresponding to the relatively low capability of the OSPO in resolving mesoscale eddies in the GS. Note that a mountain-like pattern occurs from 35 km to the finer scale, which results from the ineffective interpolation of low-resolution data onto the high-resolution grid. This reflects the fact that the grid resolution of the OSPO (0.05°) is “wasted”. The PSD of the REMSS can decay naturally from the large scale to the mesoscale in all four regions except for the GS. A reasonable spectrum for the slope of between −2 and −3 occurs in the SIO (Fig. 1a), the KC (Fig. 1c) and the TP (Fig. 1d), except for the narrow flattening trend near the Nyquist frequency. This demonstrates that the effective resolution of the REMSS is around 30−40 km, which is very close to the grid resolution (0.1°, about 20 km). The spectral behavior is supportive of the high quality of the REMSS for research on mesoscale motions. It must be noted that the product is not perfect, considering the anomalous spectrum in the GS (Fig. 1b) at about 50 km. Nevertheless, the REMSS is the only product that can effectively resolve motions with a scale of 50 km among the four SST products.

Based on the above analysis, the OISST and the OSTIA cannot effectively resolve the mesoscale features within 100 km, while the grid resolution of the OISST is roughly comparable with the effective resolution. Considering that the potential application of ocean salinity to climatological analysis, the OISST is considered to be suitable to play the role of a low-resolution template. For the two 0.05° SST products, i.e., the OSPO and the OSTIA, there is a significant mismatch between the nominal resolution and the feature resolution. Taking the OSPO or the OSTIA as the template would not only fail to synthesize high-resolution SSS products, but also increase the burden of computation and storage. Conversely, the REMSS is verified to be a truly eddy-resolving product and its grid resolution is comparable to its feature resolution. Consequently, the REMSS is selected as the high-resolution template in the SSS fusion.

After selection of the SST templates, four combinations of MFF fusion were considered for this study, namely, the OISST and REMSS as the templates along with the BEC and CCI as the source data. To highlight the merits of MFF, the original product and the 50-km Gaussian-filtered products were also validated. By subtracting the salinity of the satellite data from the NSS of Argo, the differences between the salinity products for each satellite and Argo were compared, as shown in Fig. 2. There are many large differences in the original data of the BEC (Fig. 2a), which are mainly located in the low-latitude sea area of the Pacific Ocean (corresponding to a heavy precipitation area in the intertropical convergence zone), the strong current area in the western boundary, the diluted water areas (Bay of Bengal, Amazon River Estuary, etc.) and the sea areas with large evaporation (Arabian Sea, Mediterranean Sea, etc.), etc. The largest difference of salinity was more than 0.5. The CCI data (Fig. 2e) effectively eliminated the salinity errors in these corresponding regions, and in the Southern Hemisphere, a large area with a reduction in the difference was evident. This indicates that the CCI data can effectively reduce both the random errors and the systematic bias of the SSS. There are still clear differences between the CCI data in those strong flow areas and those diluted water areas, but this does not necessarily demonstrate the errors of the CCI data. These large values are in good agreement with the inconsistent observations revealed in the study by Yan et al. (2021), indicating that most of these differences are likely be attributed to the inherent differences between the in situ observations and the remotely sensed observations.

Through comparison of the different panels in the same column, it can be noted that the differences between the fused products and the original products are not visually significant. Although a seemingly small reduction of salinity error (e.g., 0.05−0.10) cannot be highlighted easily in the figure, the improvement of data quality is quite significant considering the magnitude of the error (about 0.2−0.3) of the SSS products. However, a large number of extreme value pixels in the BEC products are effectively removed in the CCI products, e.g., in the tropical Pacific and the Amazon plume. This suggests that the calibration and correction processes of the CCI can effectively mitigate the errors, while the fusion algorithm provides a complementary means for improving the quality of the SSS products only.

To further quantify the differences between satellite-based salinity and Argo salinity, the bias, the STD and the correlation coefficients (Corr) were computed and the values are shown in Fig. 3. In addition to the regions in Fig. 1, four additional regions were considered, including the global ocean (Glo; 60°S−60°N, 180°W−180°E), the Agulhas Current (AC; 20°−40°S, 20°−60°E), the Bay of Bengal (BoB; 10°−30°N, 80°−100°E) and the Amazon plume (Amazon; 10°S−10°N, 30°−60°W,). It can be seen in Fig. 3a that both the BEC and the CCI are unbiased globally. Meanwhile, the Corrs for all products were >0.9. The STD for the original data of the BEC was 0.39, which was much larger than that of the CCI (0.22). Both the Gaussian filter and the MFF can effectively reduce the STD for the BEC, especially for the MFF fusion product with the template of the OISST (STD about 0.28). The denoising performance was not significant in terms of the CCI product, indicating that most of errors were eliminated in the original data. For the SIO (Fig. 3b) and the TP (Fig. 3e), the STDs for the CCI-related products were 0.15 or lower, and the Corrs were around 0.98. These statistics demonstrate the high quality of the CCI in the open ocean. The STDs for the BEC-based products in these two regions were still significantly inferior to that of the CCI_ori product, even though the Gaussian filter and MFF can lower the STD.

In the GS region (Fig. 3c), the original BEC product showed a significant negative bias of −0.14 and a high STD of 0.6, while the original CCI product was slightly biased (0.02) and had a much lower STD (0.42). Instead of improving the quality of the SSS product, the Gaussian filter actually increased the STD of the CCI and decreased the Corrs of both the BEC and the CCI. Nevertheless, the MFF can still take effects in the GS, especially for the BEC-fused product with the template of the OISST. The statistics for the KC (Fig. 3d) were dissimilar to those for the GS, but more similar to those for the SIO. Both the MFF and the Gaussian filter can effectively reduce the STD and improve the Corr, especially for the BEC products. A similar pattern of statistics can also be seen for the AC (Fig. 3f), except for the extremely low correlation value, i.e., lower than 0.8. Especially, the Corrs of the BEC-based product can be as low as 0.66, albeit the correlation has been significantly improved by the MFF. This demonstrates that the remotely sensed SSS product cannot perform satisfactorily in depicting the structure within the AC.

In the Bay of Bengal (BoB) (Fig. 3g) and the Amazon plume (Fig. 3h), neither the Gaussian filter nor the MFF can effectively improve the statistics of the SSS products. Note that the Gaussian filter is an effective tool for mitigating the random errors, while the performance of the MFF depends on the template of the SST. The poor performance of the Gaussian filter and the MFF highlights the special distribution of errors in these two regions. The variation of the salinity in the BoB and the Amazon is dominated by the motion of freshwater. The variation of the freshwater is independent of the variation in the temperature. Meanwhile, the low sensitivity of the remote sensor to freshwater can import large systematic errors into the SSS products. Consequently, the MFF and the Gaussian filter are not applicable. In this case, the quality of the SSS products is determined by a series of algorithms concerned with data retrieval, calibration, and correction, resulting in divergent statistics for the two products. In the BoB, the STDs/Corrs of the CCI-based products were significantly lower/higher than those of the BEC-based products, while the positive biases of the BEC-based products turned negative for the CCI-based products. For the Amazon, the STDs/Corrs for the BEC-based products were comparable to that of the CCI-based products. Meanwhile, the biases were amplified in terms of the CCI-based products.

The SSS products were further validated by the 1-m salinity of the TAO buoys. Only the data whose quality flag was 1 were considered. As revealed in Tang et al. (2017), several buoys on the date line were erroneous and thus were excluded. The results are shown in Fig. 4, where the color bar is consistent with Fig. 3, i.e., the lighter color corresponds to a better agreement with TAO. The statistics for the different products at each mooring buoy are plotted in terms of 8 pixels, the distribution of which is shown in the bottom diagram in Fig. 4.

As can be seen from Fig. 4a, the BEC-based SSS products (i.e., the upper four pixels at each mooring buoy) were biased by >0.1 at multiple sites, and the biases cannot be removed by the Gaussian filter or the MFF, e.g., the data for the four buoys at 170°W. The CCI-based products (i.e., the lower four pixels at each mooring buoy) effectively calibrated the bias at most buoys, whereas there were still some buoys with heavy bias, such as the three buoys at 147°E. In Fig. 4b, the STDs for all products at all buoys were within 0.3, which is consistent with the results in Fig. 3e. It is easy to check that the STDs for the CCI-based products was lower than that for the BEC-based products at most buoys. Meanwhile, the contribution of the Gaussian filter and the MFF to reducing the STD (i.e., the color of pixel gradually becoming lighter from left to right) can be seen, e.g., at several buoys at 137°E. Focusing on Table 1, the STD of salinity can be reduced from 0.25 to about 0.2 for BEC and from 0.14 to 0.11 for CCI. There was no significant difference between the STDs of the Gaussian-filter products and those of the MFF products, indicating that the comparable denoising capability of the two algorithms. Similar characteristics can also be found in Fig. 4c, namely, the better performance of the CCI-based products versus the BEC-based products and the comparable performance between the Gaussian filter and the MFF, which can be highlighted from the correlation coefficients in Table 1. On the one hand, the Gaussian filter and MFF were carried out only from the spatial dimension perspective, while the time series of the mooring buoys were not effectively denoised. On the other hand, the eddy motions in the tropical ocean are not quite active. Consequently, the performance of the MFF was not significant compared with the Gaussian filter for the validation with respect to the mooring buoys. Nevertheless, the evidences are supportive of a denoising capability for the MFF at least at large scales.

The resolution of the different SSS products can be assessed qualitatively from inspection of the images shown in Fig. 5. The images of the two original SSS products were noisy, where hardly any organized motions such as front and eddies can be seen. The contours of the BEC (Fig. 5) were much more congested than those of the CCI (Fig. 5), however, it cannot be determined whether mesoscale signals have been totally masked by the noise. Although the Gaussian filter can improve the quality of the BEC (Fig. 5), the map for the SSS was still dominated by noise, which corresponded to the spatially dependent noise that cannot be mitigated by the Gaussian filter. Nevertheless, the noise has been effectively eliminated in terms of the CCI’s Gaussian filter product (Fig. 5), where the frontal structure and large eddies can be clearly observed. The better application of the Gaussian filter demonstrates that the calibration and correction processes of the CCI can take effect in removing the systematic errors. Taking the OISST as the template, the MFF products can eliminate the noises of the BEC product (Figs 5c and g), even when these noises cannot be effectively denoised by the Gaussian filter. This indicates that the MFF can not only remove the white noise, but also mitigate the spatially-dependent noises. In the case of the CCI data, compared with the Gaussian filter product, the MFF product presented a stronger and narrower salinity front. The rationale for the frontal structure will be further validated by the wavenumber spectra. The MFF products (Figs 5d and h) with the high-resolution REMSS template presented detailed structures, such as the small eddies. The BEC_REMSS contained more small-scale structures than the CCI_REMSS, however, it cannot be demonstrated whether these structures corresponded to signals or noises given the lack of mesoscale observations.

The singularity exponent (SE) is an exponent representing the fractal structure, i.e., the geometric structure (Hoareau et al., 2018). The isoline of the SE extracted from the ocean scalers (e.g., SST, SSS, chlorophyll) should be consistent with the streamline (Nieves et al., 2007; Turiel et al., 2008; Pont et al., 2009). The more detailed and clearer SE isolines correspond to higher resolution, while the negative values correspond to strong currents. The computation of the SE relies on the horizontal gradient and can be obtained via the graphical user interface (GUI) provided by the BEC singularity analysis service (https://bec.icm.csic.es/data/singularity-analysis-service/). The SE maps corresponding to Fig. 5 are shown in Fig. 6. Hardly any stream-like SE structures can be seen from the products of BEC_ori, BEC_Gauss and CCI_ori (Figs 6c, b, and c), e.g., the main axis of the Kuroshio extension was masked by noise. The Gaussian filter product of CCI (Fig. 6f) showed a smoothed SE structure in the region. However, the SE map was extremely fuzzy, indicating that the detailed fractal structure was removed by the Gaussian filter. As for the two products fused by the template of the OISST (Figs 6c and g), the SE maps showed clear frontal structures. Comparing panel Figs 6f and g, the maps demonstrates that the fractal structure was blurred as a result of applying the Gaussian filter but the structure can be restored by the MFF. However, the fractal structure of the BEC_OISST product was notably different from that of the CCI_OISST product, which suggests that the horizontal gradient of at least one product was distorted. In Figs 6d and h, fine streamlines were shown and a number of mesoscale eddies were effectively depicted. Nevertheless, the Kuroshio extension was obscure in the BEC_REMSS product. It would appear that the gradient of the BEC product is problematic, and this needs to be confirmed in future work.

To analyze quantitatively the feature resolution of the different products, the wavenumber spectra of the four regions were examined. The spectra are presented in Fig. 7. In the SIO (Fig. 7a), the PSD of the original BEC data was the highest and the slope is quite flat, which demonstrated the dominance of white noises. The PSD of the original CCI data was relatively low and the slope was reasonable, indicating that the quality of the CCI data is higher than that of BEC. The PSD of the two Gaussian filter products dropped rapidly at 200 km, corresponding to a decrease in the capability to resolve the motions within 200 km. The spectra begin to oscillate around 120 km, which means that signals finer than this scale can no longer be resolved by the Gaussian filter products. The Gaussian filter is based on the average of the moving window, which results in the pseudo mixing across isohalines and consequently a degradation in the effective resolution. Two fusion products based on the OISST template showed lower PSD values. 150 km was the node where the slope drops sharply, i.e., features finer than 150 km were blurred in the two OISST-based products. The spectral behavior of the two fusion products was basically consistent with the SST template. Similar to the OISST, the spectra levelled off at about 75 km, corresponding to the dominance of white noise at a scale finer than 75 km.

In the SIO (Fig. 7a), the spectra of the two fusion products based on the REMSS template were similar. The PSD decayed from the large scale to 30−40 km following a reasonable slope of between −2 and −3. Then the spectra become flatter and the PSD was dominated by white noises. The situation of the TP in Fig. 7d was similar to that of Fig. 7a. It can be judged that in these two open oceanic regions, the effective resolution of the two salinity products based on the OISST template was approximately 75 km, and that based on the REMSS template was approximately 40 km.

For the two strong current regions in Figs 7b and c, the PSD for the BEC_ori was significantly higher than that for any other product. Besides, the flat slope of the BEC_ori indicated the high level of noise in this product, which has been revealed qualitatively in the salinity and fractal maps of Figs 2 and 3. The PSD of CCI_ori was consistent with that of other products at large scale, however, the slope was flatter than −2 and the PSD was significantly higher than the other products, corresponding to a relatively low signal-to-noise ratio at the mesoscale. As for the products synergized by the Gaussian filter, the common problem of over-smoothing was also reflected in the data, while the PSD of the CCI_Gauss product was extremely low, highlighting the loss of resolution as a result of denoising. The slope of the fusion products based on the OISST template was close to −3, which was more reasonable than that in Fig. 7a, indicating that the slightly higher feature resolution of the fusion products in the strong currents. Judging from the inflection points of the spectra, the effective resolution of the MFF_OISST product in the two strong current regions was about 75 km. The spectral slope of the fusion products based on REMSS was about −3, and the spectrum levelled off at a scale of about 40 km in the GS. In the KC, a similar behavior in the spectra can be observed except that the levelling-off point was at about 30 km. Therefore, the sharp drop of the spectra never appears for the REMSS-based products, demonstrating the considerably high feature resolution in the two strong current regions.

Comparing the fusion data based on the two different templates, it can be seen that the high-resolution template can indeed improve the feature resolution of the SSS products, and the shape of the spectra of the MFF fusion data were consistent with the SST template in Fig. 1. Besides, the MFF can counteract the differences between the CCI and the BEC in spectral behaviors. Taking Figs 1a and d as an example, the spectrum of the BEC_REMSS and that of the CCI_ori were almost coincident despite the significant differences between the spectra of BEC_ori and that of CCI_ori. Nonetheless, the spectra of BEC_REMSS and CCI_REMSS can be significantly divergent when their original products are widely different, e.g., in Figs 1b and c.

Comparing Figs 1 and 7, the MFF could make the spectral behavior of the source data (SSS) approach that of the template (SST). The successful application of MFF relies on the similarity in the spectral behaviors of SST and SSS given that both of them exhibit the characteristics of the passive tracer to some degree. Consequently, not all ocean scalars are suitable to play the role of the template. For example, the spectrum for the SSH has been demonstrated to have a slope between −5/3 (predicted by the surface quasi-geostrophic turbulence) and −5 (predicted by the quasi-geostrophic turbulence) (Vergara et al., 2019), which means that the SSH cannot be an effective template to synthesize the SSS due to the much steeper slope.