The AMSR2 microwave radiometer onboard the GCOM-W1 satellite measures the microwave emission from the Earth, at a nominal incident angle of 55° and a swath width of 1,450 km. The AMSR2 SIC dataset we used here is version 5.4 with a grid resolution of 3.125 km, which utilizes the highest spatially resolving AMSR2 channels at 89 GHz (Melsheimer, 2019). It uses the same ARTIST sea ice (ASI) algorithm, as it was developed for the AMSR-E 89 GHz channel (Spreen et al., 2008). It has a higher spatial resolution than most other AMSR2 SIC datasets, but the atmospheric influence can be higher. The uncertainty is calculated following the same procedure in Spreen et al. (2008), where the overall error sums from three sources: the radiometric error from the bright temperature, the variability of the tie points, and the atmospheric opacity. The uncertainty is expressed in terms of SD. It is noted that this uncertainty does not account for individual, spatially varying atmospheric and surface effects as for example discussed in Rückert et al. (2023) and Rostosky and Spreen (2023).

Due to the large uncertainties in the PMRs for low SIC conditions (Cavalieri, 1994; Kern et al., 2019), we choose the NIS sea ice chart to mitigate the defect. The ice chart is produced based on manual interpretation of satellite data (Dinessen and Hackett, 2018), being a typical manually analyzed product. The ice charting employs a variety of satellite observations to obtain a more realistic SIE and MIZ. The main satellite data used are the weather-independent SAR data from RadarSat-2 and Sentinel-1. The analyst also uses visual and infrared data from METOP, NOAA, and MODIS in cloud-free conditions. These satellite data cover the charting area several times a day and are resampled to 1-km grid spacing. The NIS ice chart includes seven ice categories following the WMO sea ice nomenclature (WMO, 2014): fast ice (SIC = 10/10), very closed drift ice (9−10/10), closed drift ice (7−8/10), open drift ice (4−6/10), very open drift ice (1−3/10), open water (<1/10), and ice free (0). For practical use, a mean value is applied to denote the different ice categories in the ice chart. The uncertainty is approximated as the half of the range of the corresponding ice category, except being 0.01 for the fast ice. Apparently, this uncertainty is a very coarse estimate.

The SMOS SIT is retrieved from brightness temperature measured at the L-band (1.4 GHz) from ESA’s SMOS mission. The retrieval algorithm is based on a thermodynamic sea ice model and a three-layer radiative transfer model, which applies an iterative method to calculate SIT using bulk ice temperature and bulk ice salinity (Tian-Kunze et al., 2014). The SMOS SIT uncertainty is calculated based on the following factors: uncertainty of the measured brightness temperature, uncertainties of the auxiliary data sets (JRA55 reanalysis and sea surface salinity climatology), and the assumptions made for the radiation and thermodynamic models. The uncertainty increases rapidly with increasing SIT, and it is strongly recommended to use only data with a saturation ratio (provided in the dataset) less than 100% (Tian-Kunze et al., 2014). The data is gridded at the 12.5-km grid spacing on polar stereographic projection and is available from mid-October to mid-April in the Arctic. During winter seasons, the data is generated operationally by Alfred Wegener Institute (AWI), Germany, on daily basis with 24-h latency. SMOS SIT is obtained from ESA data collections (ESA, 2023a, version 3.3, accessed on 07.07.2023). As an example, the SMOS SIT and its SD on 16/03/2022 are shown in Figures 3A, B .

The weekly CS2SMOS SIT is also produced by AWI and distributed via the ESA web portal (ESA, 2023b, version 2.05, accessed on 07.07.2023). CS2SMOS provides weekly SIT retrievals from merging daily SMOS thin SIT retrievals (Tian-Kunze et al., 2014) and SIT retrievals from CryoSat-2 (Hendricks and Paul, 2023), using an optimal interpolation approach (Ricker et al., 2017). The uncertainty of the CS2SMOS SIT is a natural part of the optimal interpolation. The data are projected onto the 25-km EASE2 Grid, based on a polar aspect spherical Lambert azimuthal equal-area projection. Figures 3C, D show the weekly CS2SMOS SIT and its SD on 16/03/2022, being an estimate of SIT and its SD during 13–19/03/2022.

Upgridding refers to the process of regridding a source field to a finer-resolution destination field. This applies to both temporal and spatial fields. In this study, regridding of the weekly mean CS2SMOS SIT to daily would require upgridding in the temporal space, whereas both of the SMOS SIT (spatial resolution 12.5 km) and the CS2SMOS SIT (spatial resolution 25 km) would need upgridding of the two-dimensional spatial SIT to the Barents-2.5-km domain. Upgridding is generally performed through the interpolation technique, which is typically composed of nearest neighbor, linear, and cubic interpolations.

In contrast to the upgridding, downgridding is the process of regridding a source field to a coarser-resolution destination field. In the present study, the sea ice chart SIC has a spatial resolution of 1 km, and it would require downgridding to the relatively coarser resolution for the Barents-2.5-km domain. While the common interpolation methods, such as the nearest neighbor, linear, and cubic interpolation, are generally applicable for downgridding, conservative interpolation methods are preferred for some special cases which require high accuracy for tiny changes (Pletzer and Fillmore, 2015).

Due to the importance of observation uncertainty on data assimilation, it is essential to accurately determine the uncertainties of satellite observations due to regridding. To the authors’ knowledge, such an effect has not been considered thus far in the data assimilation community.

The effect of downgridding on the uncertainty may be derived as follows. Denote x 1 , x 2 , … , x l as l independent observations with SDs σ 1 , σ 2 , … , σ l , then the total is

which has variance

The mean of these measurements x ¯ is simply given by

The variance of the mean can then be calculated according to the Equations (1–3) such that

From Equation (4) we get the corresponding SD

If the uncertainties of all the l observations are equal, namely, σ k = σ then Equation (5) can be simplified as

Due to different manipulations of data, the effect of upgridding on the uncertainty may differ between temporally and spatially. For temporal upgridding such as the weekly CS2SMOS SIT into daily SIT, an inverse process to the downgridding shall be applied. If we assume that the daily SIT observations are independent on each other and their uncertainties are approximately equal, then the daily uncertainty σ can be estimated from the weekly uncertainty σ x ¯ such that

where l = 7 in this case.

Equations (5–7) indicate the resulting uncertainty tends to decrease during downgridding and increase during upgridding. However, spatial upgridding of satellite observations may need special attention. It is noted that, for satellite products such as the AMSR2 SIC, SMOS SIT, and CS2SMOS SIT, their spatial resolutions may already be at the highest of the products. Therefore, upgridding does not produce more independent observations. As a consequence, the uncertainty would be unlikely to increase as significantly as the temporal upgridding. Further studies are needed to accurately determine the uncertainty variations for this situation. In this study, the uncertainties are assumed unchanged during the spatial upgridding.

Consider a state variable vector x (column vector) such as SIC for a certain spatial domain such as the Barents-2.5-km domain ( Figure 1 ), on a regular grid with the total grid number of n. Suppose we have m observations x k , k =   1 , … , m , for the true state vector x t . These observations are assumed to be taken with different instruments, and their error vector associated with each measurement is ϵ k = x k − x t . We note that the observations are also assumed independent during the temporal regridding process in section 3.1.2. However, those observations are generally obtained using the same instrument but differ in the temporal distributions.

We assume that all the observations have been regridded and that all the error vectors are random, unbiased, and normally distributed. Thus, for the kth observation error vector, we have the mean µ = E ( ϵ k )   =   0 and covariance R k = E ( ϵ k ϵ k T ) , where E denotes expectation operation and the superscript “T” denotes transpose. The probability density function (PDF) of such a error vector can be expressed as,

where | R k | denotes the determinant of R k . If we further assume that the observation error vectors are not mutually correlated, that is, E ( ϵ j ϵ k T )   =   0 , when j ≠ k , the PDF of the joint multivariate normal distribution for all the observation error vectors can be extended from Equation (8) and expressed as (Todling, 1999)

where Π denotes the multiplication operator. It is thus easy to see from Equation (9) that the maximum likelihood estimate of f ( ϵ 1 , ϵ 2 , … , ϵ m ) is obtained by equivalently minimizing the following cost function

where the optimal estimate is considered as an approximate to the true value. Differentiate J(x) in Equation (10) against x and set it as 0,

From Equation (11) we thus have the optimal estimate x o vector

and the optimal observation error vector

Since all the estimates are assumed as unbiased, normally distributed, and not mutually correlated, from Equation (13) we get the optimal observation error covariance

The optimal estimate Equation (12) can be rewritten as

It is noted that the error covariance is symmetric and semi-positive definite. Consider the process for two data sets to be merged with the error covariance being R ₁ and R ₂, respectively. According to Equation (14), the merged data error covariance is

From Equation (16) and consider the properties of positive definite matrix, we see that the trace of R has the following property:

Equation (17) indicates that the sum of the error variance of merged data is no larger than any of the individual observations. For more observations, we can use this analysis successively. With more and more observations, the sum of the merged error variance will become smaller and smaller. Therefore, the MODM process is to combine multiple observations with reducing uncertainty and increasing confidence. In addition, the MODM method can significantly reduce the computational cost and storage for data assimilation, as assimilating the merged multisensor observations is equivalent to assimilating the individual observations concurrently (Wang et al., 2020).

In recent years, more and more sea ice remote sensing observations begin to provide local variance or SD as a measure of uncertainty (Tian-Kunze et al., 2014; Ricker et al., 2017; Tonboe et al., 2016; Lavergne et al., 2019). Accordingly, the MODM method may be simplified by further assuming that each observation error vector is spatially uncorrelated. In this case, the kth error covariance, R _k (Equation 14), becomes

where σ j , k is the SD of the kth observation at the jth grid, where k =   1 , … , m and j =   1 , … , n . In this case, the error covariance (Equation 14) and the optimal estimate (Equation 15) of the multisensor observations can be expressed on individual grid (Equation 18),

where j =   1 , … , n is the grid ordinal number. Equations (19) and (20) are used in this study for MODM of SIC and SIT.

Sea ice volume (SIV) is not directly observed, and there has been no studies to estimate the SIV uncertainty. Here, we deduce the formulation for SIV (V) based on the observed SIC (a) and SIT (h) and their SDs σ a and σ h . It is generally reasonable to assume that a and h are two independent random variables; thus, V can be simply expressed as

and its SD can be calculated according to the variance of the products (Goodman, 1960)

where the subscript “mean” denotes the mean values of SIC and SIT. It is noted that Equation (25) is the exact variance of products, whereas the Gaussian propagation of uncertainty is an approximate solution after neglecting high-order derivatives and cross-correlated terms.

According to the World Meteorological Organization (WMO, 2014), SIE is defined as the demarcation at any given time between open sea and sea ice of any kind. It can generally be separated into two types: compacted and diffuse. The compacted SIE refers to the close and clear-cut SIE, which is compacted by wind or current, usually on the windward side of an area of drift ice. The diffuse SIE refers to the poorly defined SIE, which has an area of dispersed ice, usually on the leeward side of an area of drift ice. In practical usages, SIE is often defined as the demarcation where SIC = 0.15 in the sea ice and climate modeling communities. By contrast, it is often defined as the demarcation where SIC = 0.1 in the sea ice charting community, such as the NIS ice chart (https://cryo.met.no ) and US National Ice Center (NIC) ice chart (https://usicecenter.gov/Products ). Wang et al. (2023) argue that choosing SIC = 0.1 as the demarcation for SIE has several benefits. Most importantly, it has a clear physical representation that distinguishes open water (SIC<1/10) and very open drift ice (SIC in 1–3/10). In addition, it provides a consistent definition for the joint sea ice modeling and charting community. In this study, we also use SIC = 0.1 as the demarcation for SIE.

MIZ is generally referred to the transition region from open water to dense pack ice that is affected by open ocean processes (Wadhams, 1986; Johannessen et al., 1987), although its accurate definition is still under intensive discussion from different viewpoints and concerns. Typical MIZ conditions are found along the southern edges of the ice pack in the Bering, Greenland, and Barents seas, in the Baffin Bay, and along the complete northern edge of the Antarctic ice cover (Røed and O’Brien, 1983). There have been several definitions for the MIZ. The most widely used one is solely based on SIC, commonly defined as the region where SIC ∈ [0.15,0.8]. In order for a consistent definition in both ice charting and sea ice modeling, the MIZ is here defined as follows

where a is the SIC, and the subscript “t” denotes traditional. This traditional definition has been applied in a variety of applications, such as sea ice charting (e.g., the NIC ice chart), satellite observations (e.g., Strong, 2012; Liu et al., 2019a), sea ice modeling (e.g., Wang et al., 2023), primary productions (e.g., Barber et al., 2015), marine ecosystems (e.g., Wassmann, 2011; Arrigo, 2014), and ship navigation (e.g., Palma et al., 2019).

The above traditional definition of MIZ provides a reasonable quantification of the MIZ extent. However, it is often inadequate for a detailed description of MIZ dynamics (see Bennetts et al., 2022 and references therein). In such cases, the effect of waves must be taken into consideration. To observe the dynamical MIZ, Dumont (2022) suggests three approaches, in which sea ice displays vortical motions, wavy motion, or a dominant floe size less than an upper value (in the order of 200 m–500 m). We comment that the vortical and wavy motions are generally unstable features, so they are not proper for a consistent determination of the dynamical MIZ. For example, the ice eddies or waves in the ice could be temporally diminished in the MIZ whereas the ice floes remain unchanged. In such cases, the extent of the MIZ should remain according to the floe size method, rather than vanished according the other two methods. Therefore, the floe size method appears to be the most appropriate method for observing the dynamical MIZ.

The sea ice floe size is not operationally observed for the sea areas around Svalbard. As an alternative, we approximate the dynamical MIZ using combined SIC and SIT based on the model results from a coupled wave and ice model (Dumont et al., 2011),

where a and h denote SIC and SIT and the subscript “d” denotes dynamical. The lower SIC bound of 0.1 is here used to be consistent with the SIE. Compared with the traditional MIZ t the dynamical MIZ d also includes part of the very close drift ice, although the SIT tends to be thinner with increasing SIC.

It is noted that the dynamical MIZ formulation Equation (27) is solely based on the simulation results at the Fram Strait of a 1D coupled wave-ice model (Dumont et al., 2011). Its accuracy and validity for other sea areas needs further verification. The upper SIC bound of 0.85 is used here to be consistent with the simulation results (Dumont et al., 2011), which is slightly larger than the traditional upper bound of 0.8 (see Equation 26). This difference can partly be explained by the constraint h ≤ 2.0 in Equation (27). For h > 2.0 m, the upper SIC bound is supposed to become lower than 0.85 and approach to 0.8.

The MODM of SIT follows much of the same procedure for the SIC. The original SIT and SD for the daily SMOS and weekly CS2SMOS are shown in Figure 3 , both covering the whole northern hemisphere, with the spatial resolutions of 12.5 km and 25 km, respectively. These two SIT products are firstly upgridded to the Barents-2.5-km domain using the nearest-neighbor interpolation ( Figures 5A–D ). For clarity purposes, the uncertainties have both remained unchanged during the regridding. It can be seen that there are considerable differences in these two products ( Figure 5 ). The SMOS SIT has a relatively large data hole around the North Pole. The uncertainty increases rapidly when the observed SIT is over 1 m, as can also be seen in Figures 5A, B . By contrast, the weekly CS2SMOS SIT has a full coverage of the whole domain, with the SIT generally up to 3 m ( Figure 5C ). The CryoSat-2 SIT has large uncertainties when it is thinner than 1 m (Ricker et al., 2017), and the weekly CS2SMOS SIT effectively reduces the overall uncertainty by combining the CryoSat-2 SIT and the SMOS SIT.

It is noteworthy that the weekly CS2SMOS SIT is a weekly mean; therefore, it can be biased when used for daily purposes. One such case can be seen in the north Greenland Sea, west of Svalbard (cf. Figures 5A, C ). We can see that the thin SMOS SIT there is generally approximately 0.5 m ( Figure 5A ), whereas the weekly CS2SMOS SIT is mostly over 1 m ( Figure 5C ). This discrepancy is due mainly to the weekly average of the CryoSat-2 SIT and SMOS SIT over the whole week, in which both thin and thick ice drifting before and after the day are accounted for. Therefore, for daily usage, a more accurate estimate of the thin ice should be closer to the daily SMOS SIT. It is apparent that assimilation of such weekly SIT as a daily data would introduce considerable systematic bias.

The merged daily CS2SMOS SIT and SD with and without the temporal upgridding effect are shown in Figures 5E–H , in which the subscript “0” denotes no temporal upgridding effect. It is seen that the merged SD is noticeably larger than SD₀ (cf. Figures 5F, H ), particularly for the larger SIT areas in the Arctic Ocean and Greenland Sea. By contrast, the merged SIT is slightly lower than SIT₀ (cf. Figures 5E, G ). On the whole, the merged daily CS2SMOS SIT is closer to the SMOS SIT, whereas the SIT₀ is closer to the weekly CS2SMOS SIT, although the thick ice in both SIT and SIT₀ are close to the weekly CS2SMOS SIT.

The merged thin SIT to the west of Svalbard is very close to that in the SMOS SIT (cf. 5e and 5a), showing a successful merging as discussed above. However, the overall distributions of the merged SIT and SIT₀ in the Kara Sea (east of Novaya Zemlya) appear much closer to the weekly CS2SMOS ST than the SMOS SIT, although more similarities are seen to the SMOS SIT when considering the temporal upgridding effect (cf. Figures 5A, C, E, G ). Since the SIT there is generally approximately 0.7 m–0.9 m, a reasonable result would be that the merged SIT is closer to the SMOS SIT rather than the weekly CS2SMOS SIT. The most probable reason for the discrepancy is that the uncertainty there in the weekly CS2SMOS SIT ( Figure 5D ) is underestimated. This is confirmed by the weekly CS2SMOS SIT SD, which is generally less than 0.06 m in this area ( Figure 5D ). It is much lower than the SMOS SIT SD (generally approximately 0.7 m as shown in Figure 5B ), even after the temporal downgridding from daily to weekly. A further refinement of the weekly CS2SMOS SIT uncertainty would be highly desirable.

The fused SIT and its SD are generally similar to the merged ones in much of the domain (cf. Figures 5I, J, E, F ). Their differences are calculated according to Equations (22) and (23), and shown in Figures 5K, L . The differences are mainly located near the SIE, with the additional thin SIT in several cm and additional SD below 1 cm. Such a supplementary effectively overcomes the shortcoming of the PMRs, thus generating a more consistent and accurate observation of the SIE and MIZ compared with the merged SIT and SD. The application of the ice chart also removes some coastal sea ice along the mainland Norway ( Figures 5K, L ).