## 1. Introduction

In recent years, multiple instruments capable of measuring sea surface temperature (SST) have been flown on satellites. It has been well established that high-quality global SSTs can be produced by blending SST data from multiple satellites and in situ observations (Reynolds 1988; Reynolds and Smith 1994; Reynolds et al. 2005). Such global blended SSTs have played an important role in a number of climatological analyses and air–sea interaction studies. Motivated by the success of global blended SSTs, it is natural to explore the potential of blended SSTs for coastal applications.

Coastal oceans feature flow systems on spatial scales from a few kilometers to tens of kilometers and temporal scales from several hours to days. For analyzing such fine spatial structures and temporal evolutions, the desired blended SST field must have a spatial resolution of a few kilometers and a temporal resolution of several hours, along with sufficient accuracy. A central question is whether the existing satellite and in situ observations have coverage and quality sufficient for generating the desired blended SST fields.

Currently, there are more than 10 satellites that carry SST sensors. Table 1 summarizes the satellites whose data we use in this study. In Table 1, the sensor types are also listed. The data from the Geostationary Operational Environmental Satellite (GOES) series have a spatial resolution of 6 km and a regular temporal resolution of 30 min. The GOES imager is an infrared (IR) sensor. IR sensors are carried by other satellites, including the Advanced Very High Resolution Radiometer (AVHRR) and the Moderate Resolution Imaging Spectroradiometer (MODIS). The spatial resolutions of AVHRR and MODIS SSTs are as high as 1–2 km. The frequent coverage and high spatial resolution of these IR SSTs are sufficient for generating the desired blended SST fields under clear skies (e.g., He et al. 2003). Unfortunately, SSTs cannot be retrieved from IR sensors in the presence of clouds or even high concentrations of aerosols. However, SSTs can be retrieved under such conditions by microwave (MW) sensors, including the Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) and the Advanced Microwave Scanning Radiometer for Earth Observing System (AMSR-E). In comparison with IR SSTs, MW SSTs have a much lower resolution of about 25 km and are not retrieved near land. Thus, IR and MW SSTs are highly complementary. Further, there are thousands of SST reports available daily from in situ platforms. These SST reports are mainly concentrated near the coast. Wang and Xie (2007) have demonstrated a possibility of resolving diurnal variation by blending all these observations at a moderate resolution of 0.25° × 0.25°. We will demonstrate that the desired blended SST fields may be produced by blending the above mentioned IR and MW SST data and in situ observations using a properly designed quality-control (QC) procedure and an advanced blending algorithm.

Although the importance of QC is self-evident, the requirements on the blended SSTs pose particular requirements on the blending algorithm. We can identify three such requirements on the blending algorithm. The algorithm must 1) efficiently use all available data, 2) accommodate a large volume of observations, and 3) incorporate inhomogeneous and anisotropic characteristics. As noted above, the blended SST needs to be produced at a time interval of several hours to resolve the diurnal cycle. With such a short time interval, the number of observations available can occasionally be quite limited. In these situations, the blending algorithm must be able to fully use all the observations; in particular, the lower-resolution MW SSTs should be fully blended with the high-resolution IR SSTs. On the other hand, at other times, when GOES and AVHRR observations are available, for example, the total number of observations can be huge—as large as 10^{5}–10^{6} over a region no larger than 1000 km × 1000 km. The blending algorithm needs the capability to accommodate such a large volume of observations efficiently and reliably. Another requirement arises from the characteristics of coastal flows. Coastal flows generally display a larger spatial scale offshore and relatively smaller scales near shore, and the variance of the variability is generally larger near shore than offshore. As such, the statistical covariance of flow variability is inhomogeneous and anisotropic. The statistical covariance is the major parameter in statistical interpolation methods on which advanced blending algorithms are based. Thus, an advanced blending algorithm for coastal oceans should have the capability to incorporate inhomogeneous and anisotropic covariances. Most blending algorithms currently employed use the optimal interpolation (OI) algorithm (e.g., Reynolds and Smith 1994; Guan and Kawamura 2004). As discussed in section 2, it is desirable to generalize the OI algorithm mathematically and to improve its computational efficiency to satisfy the requirements outlined above. The two-dimensional variational data assimilation (2DVAR) algorithm developed in this study is a generalization of the OI algorithm, as detailed in section 3.

The paper is organized as follows: section 2 reviews the OI algorithm and discusses its limitations when used to blend SSTs for coastal oceans. In section 3, the formulation of the 2DVAR algorithm is presented and comparisons with the OI algorithm are described. The specification and estimation of error covariance used by the blending algorithm are also discussed. In section 4, quality-control methods are described. Section 5 presents an evaluation of blended SST fields for the central California coastal region. Finally, section 6 gives a summary.

## 2. An overview of the OI algorithm

Bretherton et al. (1976) formulated an OI algorithm for application to oceanic data analyses. This OI algorithm offers an elegant mathematical formulation and is easy to implement. As such, it is still the algorithm most commonly used for blending SSTs (e.g., He et al. 2003; Guan and Kawamura 2004; Barron and Kara 2006; Kawai et al. 2006) and other quantities, such as sea surface heights (Le Traon et al. 1998). We examine this OI algorithm here and illustrate some of its limitations.

Following Bretherton et al. (1976), an SST value at a general point *x*, *T _{x}* is estimated from observations

*T*at a limited number of data points

_{r}^{o}*r*(

*r*= 1, … ,

*M*). The observation

*T*=

_{r}^{o}*T*+ ɛ

_{r}^{t}*, where the observational errors are white noise that has known variance*

_{r}^{o}*E*. We also introduce a background

*T*, and

_{x}^{b}*T*=

_{x}^{o}*T*+ ɛ

_{x}^{t}*, where ɛ*

_{x}^{b}*is the background error.*

_{x}^{b}*T*is where 𝗔 is the background error covariance between the pair of observing points

_{x}*r*and

*q*, and its entries are given by Here, angle brackets stand for the statistical mean. Note that

*x*and

*r*.

The formulations (1)–(3) are fairly general. The estimate given by (1) can be shown to be the minimum-error variance solution or the maximum-likelihood solution under the assumption of Gaussian distributions, as addressed in the next section. However, this formulation suffers from major limitations when used to blend satellite SSTs for coastal oceans.

The first limitation is related to the different resolutions of the SST observations. It requires that the SSTs are sampled point by point. However, satellite sensors do not measure SSTs point by point but instead measure the average over an area. Depending on the sensor, the size of the averaging area varies. For example, current MW sensors measure averages over an area of 0.25° × 0.25°, whereas a typical IR sensor measures averages over a smaller area. Hence, satellite SSTs have different resolutions. The formulation given by (1) cannot directly blend SSTs of different resolutions. In current uses of (1), an ad hoc method is generally used to circumvent this difficulty. In such a method, a lower-resolution SST is generally used as a point observation, but it is used only at points where no higher-resolution or in situ data are available (e.g., Guan and Kawamura 2004). With this ad hoc method, MW SSTs are not effectively used when other higher-resolution SSTs are available. On the other hand, when no other higher-resolution SST is available, the lower-resolution SST may lead to smoothed blended SSTs at these locations. To use (1) to blend satellite SSTs of different resolutions, a mathematical generalization to (1) is desirable.

The second limitation is related to inhomogeneous features in coastal oceans. Inhomogeneities are ubiquitous and significant in coastal oceans. One major feature is that the correlation has a smaller-length scale near shore than offshore (Li et al. 2008b). Although the formulation (3) allows for an inhomogeneous error covariance *C _{xr}*, it is difficult to implement. The difficulty arises when the number of grid points for blended SSTs is large, because there are not enough data available for estimating the covariance matrices of

*A*and

_{rq}*C*. Also, the large dimensions of

_{xr}*A*and

_{rq}*C*may prevent them from being stored on computers. One method is to parameterize them and construct the error covariance matrices based on analytical functions, such as Gaussian functions. With such a method, however, it is practically complicated to construct inhomogeneous covariance matrices that satisfy the symmetric property of the covariance (Carter and Robinson 1987). In fact, Bretherton et al. (1976) did not give the general form of

_{xr}*A*and

_{rq}*C*as in (4) and (5) but directly formulated them as homogeneous and isotropic.

_{xr}The last limitation is the computational difficulty. When (1) is used to blend SSTs, the computational cost depends on the number of observations *M*. This is because the *M* × *M* matrix 𝗔 must be inverted. When *M* becomes large (>10^{5}), the inversion of 𝗔 can be very time consuming. There is possibly a greater difficulty; that is, existing inversion algorithms likely fail to work. There are methods for circumventing these difficulties arising from the large number of observations. One method is to partition a large domain into smaller subregions, and the OI algorithm is then implemented over each subregion. When *M* becomes small enough in a subregion, the inversion of 𝗔 is manageable. However, the blended SSTs of the subregions need to be pieced together, which may produce noises near the partition boundaries. Given these issues, it is fair to say that this OI algorithm is practically impossible to implement with a large number of observations.

We propose next a two-dimensional variational data assimilation algorithm. This algorithm does not suffer from the first and third limitations, and it offers great flexibility in mitigating the second limitation.

## 3. Formulation of the 2DVAR algorithm

This 2DVAR algorithm uses a mathematical principle that is the same as the OI method (Lorenc 1986), but its numerical implementation is different. The major difference is that 2DVAR takes advantage of advanced large-scale numerical optimization methods.

Let the blended SST fields that we are seeking be defined on a given regular grid. The grid can be in a variety of coordinates, such as latitude–longitude or any curvilinear coordinate. Once the coordinate is chosen, we define an *n*-dimensional vector **T** that encompasses the SSTs at all the grid points, where *n* is the total number of the grid points.

**T**: where

**T**

*is the background,*

^{b}**T**

*is the observations,*

_{s}^{o}*N*is the number of types of SST observations, 𝗕 is the error covariance matrix of

**T**

*, and 𝗥*

^{b}*is the observational error covariance of*

_{s}**T**

*. The values of 𝗛*

_{s}^{o}*are known as observational operators, which map the blended SSTs to the observation locations. The cost function can be derived in terms of a maximum-likelihood estimation (Jazwinski 1970); thus, the solution is a maximum-likelihood estimate.*

_{s}*δ*

**T**=

**T**−

**T**

*. Then the cost function becomes where*

_{b}*δ*

**T**

*=*

_{s}^{o}**T**

*− 𝗛*

_{s}^{o}_{s}

**T**

^{b}is known as the innovation vector.

In the cost function (5), 𝗕 and 𝗥* _{s}* are matrices. Because of their large dimensions, simplifications are often needed in practice. In most cases, 𝗥

*can be simplified as a diagonal matrix whose elements are the observational error variances. However, 𝗕 is necessarily dense. To illustrate this, we decompose 𝗕 into 𝗕 =*

_{s}**Σ**𝗖

**Σ**, where

**Σ**is a diagonal matrix, the diagonal elements are the root-mean-square error (RMSE) of

**T**

*, and 𝗖 is the correlation matrix whose elements consist of the spatial correlations among grid points. In our implementation, we use the climatological monthly mean as the background. For such a background, the error correlations are simply the total SST correlations. More importantly, the spatial correlations play the role of spreading information from locations with observations to surrounding areas. Because 𝗖 must be dense, its large dimension may not allow all the elements of 𝗖 to be saved, making its inversion practically impossible. A particular simplification of 𝗖 is necessary to make (5) manageable for performing numerical minimization.*

_{b}Here, we introduce _{χ} ⊗ _{η}, where ⊗ stands for the Kronecker product (Graham 1981) and _{χ} and _{η} are one-dimensional correlations in the cross-shore and alongshore directions, respectively. We introduce a Cholesky factorization, **ΘΘ**^{T}, where **Θ** is the Cholesky triangle matrix. According to the properties of the Kronecker product, **Θ** can be computed as **Θ** = **Θ**_{χ} ⊗ **Θ**_{η}, where **Θ*** _{χ}* and

**Θ**

*are the Cholesky triangle matrices of*

_{η}_{χ}and

_{η}, respectively; that is,

_{χ}=

**Θ**

_{χ}

**Θ**

_{χ}

^{T}and

_{η}=

**Θ**

_{η}

**Θ**

_{η}

^{T}. Because 𝗖

_{χ}and 𝗖

_{η}are one-dimensional matrices, the Cholesky factorization of 𝗖

_{χ}and 𝗖

_{η}can be efficiently performed.

*δ*

**T**=

**Σ**𝗖(

**Θ**

^{−1})

^{T}

*δ*

**x**, we then have the cost function The cost function (6) offers two major merits. One is that inversion of 𝗖 is not required. This merit allows us to construct 𝗖 with great flexibility. The other is that the cost function is preconditioned. The second merit allows us to apply a minimization algorithm to this cost function (6).

It is noteworthy that the cost function (6) can be applied to problems with large dimensions. In the cost function (6), the dimension of the problem is apparently limited by the size of two matrices, **Θ** and 𝗖. However, the dimension of **Θ** is not a limiting factor because it is computed by using two smaller-dimension matrices, **Θ*** _{χ}* and

**Θ**

*. For 𝗖, we do not need to save it explicitly but only compute its entries (correlations at grid points) when they are used.*

_{η}*δx*. The final blended SST is where the second term

^{a}*δ*

**T**

^{a}=

**Σ**𝗖(

**Θ**)

^{−1}

*δ*

**x**

^{a}is known as the analysis increment.

Now that we have described the 2DVAR algorithm, we next examine its relation with the OI algorithm described in section 2. The estimator given by (1) can be derived from the minimization of (5) as a special case (see appendix). Thus, this 2VDAR algorithm is a generalization of the OI algorithm. One obvious generalization is the use of the observational operators, 𝗛* _{s}*, which map the SSTs to be estimated to observation locations. When higher-resolution SSTs are used, so-called superobservations (superobs) are formed, and the superob grid has the same resolution as blended SSTs. The SSTs that are to be estimated are mapped by 𝗛

*to the superobservation grid, and the superob grid is a simple diagonal matrix with its diagonal elements of zero or one, determined by the existence of superobservations. When lower-resolution MW SSTs are used, an operator 𝗛*

_{s}*includes two functions: averaging the SST to be estimated and then mapping the averaged values to the MW SST location that is specified in the data. Because MW SSTs used have a resolution of 0.25° × 0.25° but the blended SSTs have a resolution of 6 km (as explained later), the average in 𝗛*

_{s}*is the average applied to the SSTs located in the range of 0.125° from the MW SST location. In this way, the MW SSTs are fully treated as area averages and can be used anywhere in either the presence or absence of other high-resolution and in situ SSTs.*

_{s}As addressed in section 2, one limitation of the OI algorithm is the rapid increase of computational load as the number of observations increases. In this 2DVAR algorithm, an increase in the number of observations results in a very small increase in computational load. We can come to this conclusion by examining the cost function in (6). The computational load arises primarily from the calculation of 𝗖(**Θ**^{−1})^{T}*δ***x**, which is independent of the number of observations. This computational property allows us to blend a large volume of observations in real time. As such, we do not need to perform any data selection as the OI algorithm does. For example, the OI method generally allows only a single observation at a grid point. The 2DVAR algorithm described here allows multiple observations at a grid point, which offers some benefits. First, when different observations are independent by nature, the use of multiple observations is statistically equivalent to reducing observational errors. Second, the observational term can dominate the cost function; thus, the weight of the background SSTs is relatively reduced. Third, all data are used.

We apply the algorithm to blend GOES, AVHRR, MODIS, AMSR, and TMI SSTs and in situ observations. As such, we have *N* = 6 in the cost function (1) (the computing region is shown in Fig. 2). A curvilinear coordinate is used so that one of the coordinates approximately follows the coastline. The blended SSTs are produced every 6 h at a resolution of 6 km. The spatial resolution of 6 km is chosen because it is the resolution of the GOES SST.

In the subsequent calculation, we will use the cost function (8) because of the relatively straight coastline in the region. Thus, the background error covariance is given by *C̃*, and we need to specify the RMSE, as well as the one-dimensional correlations in the cross-shore and alongshore directions. The RMSE in this case is the variance of SST variability in the region. We specify it as 3.5°C. The correlations can be constructed analytically or estimated from model output as has been done by the National Meteorological Center [NMC; now the National Centers for Environmental Prediction (NCEP)] method (Parrish and Derber 1992). In this study, we follow the NMC method and use model output to estimate the decorrelation length. The model output used is described in Li et al. (2008b). Then, the decorrelation length is used to construct the one-dimensional correlations, assuming they have a Gaussian character. Because the Kronecker product is used, we are allowed to consider decorrelation lengths that vary cross-shore.

For brevity, we do not describe the resulting correlation structures but instead illustrate them using the single observation experiments shown in Fig. 1. The correlation has a larger scale offshore and a smaller scale near shore, which represents a typical inhomogeneity of correlations in coastal oceans.

## 4. Quality control

Following Donlon et al. (2002), SSTs can be classified into four types: interface SST (infinitely thin layer at the exact air–sea interface), skin SST (a thin layer of about 50 *μ*m), subskin SST [at the bottom of the skin SST temperature gradient (layer), where molecular and viscous heat transfer processes begin to dominate], and subsurface (or bulk) SST. Here, the bulk SST represents the temperature at a depth of about 1.0 m, which is the depth of buoy-measured SSTs. Different satellite SSTs belong to different types, depending on the sensor used, calibration method, and retrieval algorithms. SSTs retrieved from MW sensors are subskin SSTs. SSTs retrieved from IR sensors are skin SSTs. However, AVHRR SSTs are bulk SSTs because they are statistically regressed to buoy SSTs, although the AVHRR sensor is an IR sensor. MODIS SSTs are skin SSTs. MODIS retrievals are based on algorithm coefficients derived from comparisons with buoy SSTs but are rendered into skin SSTs based on at-sea measurements. The differences between the skin/subskin and bulk SSTs can be significant in some weather conditions. For example, when solar insolation is strong and winds are weak, the difference between the skin/subskin and bulk SSTs can be as large as 1.0°–2.0°C (e.g., Donlon et al. 2002). When blending satellite SSTs, such differences need to be accounted for.

To account for the differences, we first need to decide the type of the blended SSTs. When the blended SST product is targeted for atmospheric applications, the blended SSTs should be interface SSTs because interface SSTs determine air–sea heat flux exchange. Because the blended SSTs here are targeted for oceanic applications, we produce blended bulk SSTs; this is because skin SSTs may not have a close connection to oceanic flow systems. When blended SSTs are bulk SSTs, skin/subskin SSTs need to be adjusted before they are blended.

Because the purpose here is to demonstrate the blending algorithm, we use a simple method to address the difference. During nighttime and under conditions when wind speeds are larger than several meters per second, the difference is a fraction of 1.0°C (e.g., Donlon et al. 2002) and we ignore it. The adjustment is then needed only during daytime and under weak wind conditions. However, this adjustment is not straightforward because it depends in a complex way on weather conditions. To circumvent this difficulty, we exclude skin and subskin SSTs that are during the daytime in areas with very weak winds. Specifically, we do not use skin/subskin SSTs in areas where wind speeds are less than 4 m s^{−1}, based on winds from a blended wind product (Chao et al. 2003) during daytime.

Satellite SST data are generally provided along with quality flags. When quality flags are available, we use the most restrictive flags. Even so, it is still crucial to use additional QC tests to insure the quality of the blended SSTs. We apply an additional gross check. The gross check consists of comparing the observed SSTs with regional climatological monthly means. Observations are not used if the difference is larger than 6°C in magnitude, which is close to 2.0 times the SST standard deviation (3°C) in the region. Further, a time evolution check is applied. The observed SSTs are compared with the blended SST produced 6 h previously. It is empirically decided that the observed SSTs are not used when the difference is larger than 3.0°C in magnitude.

Because of the multiple satellite data available, a very powerful QC method is the cross-check. The cross-check consists of a comparison between different types of observed SSTs. This cross-check is applied to three satellite SST observations: the GOES, AVHRR, and MODIS SSTs, which are IR SSTs and suffer from cloud contaminations. The objective of this cross-check is to identify and remove cloud-contaminated SSTs. Because TMI and AMSR-E SSTs do not suffer from cloud contamination, the GOES, AVHRR, or MODIS SSTs are abandoned when they are 3.0°C or more lower than TMI and AMSR-E SSTs at collocated grid points and within a 12-h window. Figure 2 shows an example of this. Although we have used the most restrictive flags, the GOES SSTs still suffer from cloud contamination, as can be seen near the western boundary. The cross-check procedure effectively removes these cloud-contaminated SSTs. This cross-check turned out to be crucial in improving the quality of the blended SSTs.

After all these QC procedures are applied, we consider that the observation errors are mainly instrument and algorithm errors. As such, the observational errors are specified as follows: 0.7°C for GOES SSTs, 0.5°C for both AVHRR and MODIS SSTs (this value is smaller than the instrumental error because they have a spatial resolution 3 times greater than the grid of the blended SSTs and spatial averaging is performed), 0.7°C for both AMSR and TMI SSTs, and 0.35°C for the in situ SSTs.

## 5. Evaluation of blended SSTs

We assess the accuracy of the blended SSTs by analyzing the blended SSTs for August 2003. During this month, the Autonomous Ocean Sampling Network field experiment was implemented in the Monterey Bay region of California (Chao et al. 2009). We first make a comparison with independent observations from two moorings operated by the Monterey Bay Aquarium Research Institute (MBARI). One mooring (M2) is located in the open water at 36.75°N, 122.03°W; the other mooring (M1) is located within Monterey Bay at 36.70°N, 122.38°W. The observations from these two moorings consist of 248 data points, as shown in Fig. 3. The bias of the blended SSTs is 0.17°C and the root-mean-square of the difference between the mooring and blended SSTs is 0.81°C, which is close to the specified error of the GOES SST.

Because the comparison with the two moorings reflects only the local quality of the blended SSTs, we seek comparisons with more global independent observations. To do so, we withhold AVHRR SSTs from the blended SSTs. Then, AVHRR SSTs can be used as the independent observations for the evaluation of the blended SSTs produced using the other five sets of SST observations. We withhold AVHRR SSTs because the AVHRR sensor and its retrieval algorithm are known to be well calibrated. Figure 4 presents a snapshot of the differences between the blended SST and the independent AVHRR SSTs. The differences are smaller than 0.5°C over most areas, with a maximum difference of about 1.5°C. For the entire month of August 2003, the root-mean-square difference between the blended SSTs and the AVHRR SSTs is 0.56°C. This number is comparable to the accuracy of the AVHRR SSTs, which is estimated to be 0.5°–0.7°C. It should be noted that the AVHRR SSTs are averaged over a bin of 6 km × 6 km and that this averaged AVHRR SST has a somewhat higher accuracy.

In addition to the accuracy, another interesting question is how the blended SST “optimally” extracts information from each individual dataset. To illustrate this, Fig. 5 shows the blended SSTs and those used in the blending for 2100 UTC 16 August 2003. The GOES SSTs show low values near the shore, which are the consequence of a strong upwelling event (Chao et al. 2007). These cold SSTs can also be seen in the in situ observations. In the GOES SST, we can infer that there is a cold SST filament off Point Arena, California, but this filament is not fully captured by the GOES measurements because of a data gap. In the AMSR-E SSTs, we can see a weak signal of the filament. In the blended SSTs, both the cold SSTs alongshore and the filament appear with a relatively complete structure.

In the introduction, we stated that the spatial resolution of our blended SSTs is high enough to allow analysis of the evolution of finescale structures in coastal flow systems. As an example, Fig. 6 shows a sequence of blended SSTs from 12 to 22 August 2003. Figure 6a (for 2100 UTC 12 August 2003) shows a pattern typical of summertime in this region with relatively uniform and warm temperatures offshore and colder temperatures with more spatial variability in a band along the coast. This pattern is a reflection of strong coastal upwelling of cooler waters. This particular period of upwelling lasted through 18 August (Fig. 6d). After this, the upwelling favorable winds decreased substantially and warmer offshore surface waters replaced the colder waters near the coast. During the upwelling period, we can see a clearly revealed filament of colder nearshore water form near Point Arena (Figs. 6b,c), develop and extend well into the otherwise much warmer offshore region (Figs. 6d,e), and then weaken as the relaxation in the upwelling took hold (Fig. 6f). This filament is an example of the type of finescale structure that can be resolved and studied using our blended SSTs.

In addition to the high spatial resolution, the temporal resolution is also designed to show diurnal variations, which we examine briefly next. For verification, we compare the diurnal variations in the M1 mooring SST data with our collocated blended SSTs. To fully describe the diurnal variation, we need to account for both the amplitude and shape. Because they are generated for every 6 h, the blended SSTs are not adequate to describe the shape of the diurnal variation. We can discuss only the amplitude. Here the amplitude is represented by the difference between the 2100 UTC [1300 local standard time (LST)] and 0900 UTC (0100 LST) SSTs. The mooring data used are averaged from hourly data over a time window of 6 h centered at 2100 and 0900 UTC. Figure 7 shows the daily amplitude during the month. There are two weak diurnal variation events during mid-August and before the end of the month, and there are two strong diurnal variation events in early and late August. It turns out that the weak diurnal variation events correspond to upwelling events, whereas the strong diurnal variation events correspond to relaxation events (Chao et al. 2009).

## 6. Summary

We propose here and produce blended SST fields with high spatial and temporal resolutions. The high spatial resolution allows analyses of the structure of finescale structures in coastal flow systems, whereas the high temporal resolution resolves the diurnal cycle. A two-dimensional variational data assimilation (2DVAR) method was used. The method offers a number of statistical advantages over currently used OI methods. A major advantage is its global solution characteristics that yield spatially coherent SSTs. The algorithm allows specification of inhomogeneous and anisotropic correlations of the background error, which are common features of coastal ocean regions. In particular, this 2DVAR is computationally efficient with a large number of observations. In addition to the advanced blending algorithm, we use a specially designed QC procedure. In this QC procedure, in addition to the use of the QC flags provided with the satellite SSTs, we perform cross-checks between IR and MW SSTs to remove remaining cloud-contaminated IR SSTs.

The blended SSTs off the central California coast were examined. Satellite IR and MW SSTs and in situ observations were blended (Table 1). The blended SST fields were produced at a spatial resolution of 6 km and a temporal resolution of 6 h. The spatial resolution allows us to resolve flow systems as small as a few tens of kilometers. The temporal resolution is high enough to show diurnal variations, a first among existing blended SST products. An evaluation using independent observations showed RMS errors of less than 1°C. The blended SSTs showed the spatial structures and temporal evolution of filaments associated with coastal upwelling and relaxation, demonstrating their utility in the analysis of fine structures in the coastal ocean. A preliminary look at the diurnal cycle in blended SSTs showed a capability of reproducing the diurnal variation day-to-day changes near shore.

In this study, we tried to specify the major parameters used in the algorithm objectively, but some of them are still somewhat arbitrary, including the background error variances and observational error covariance. For example, it has been noted that microwave SSTs may have a warm bias under rough ocean surface conditions in some areas near shore and that the bias in GOES SSTs may significantly affect both the amplitude and shape of estimates of the diurnal cycle in SSTs (Wick et al. 2002). The observational errors used in our 2DVAR method should reflect such geographical dependencies. The bias of observed SSTs is another important issue that was not addressed in this study. This issue has been emphasized by Reynolds and Smith (1994). The most important issue is the QC. We expect that the QC and the adjustment between skin/subskin and bulk SSTs can be improved by using improved and comprehensive QC flags and other auxiliary datasets developed by the Global Ocean Data Assimilation Experiment (GODAE) High Resolution SST Pilot Project (GHRSST-PP). The GHRSST-PP datasets will particularly facilitate and enhance the cross-check QC method that we have emphasized. By accounting for these biases and improving the QC procedure, it is expected that the high-resolution blended SST fields can be further improved. Application of this 2DVAR algorithm in other coastal regions is currently being investigated.

## Acknowledgments

The research described in this publication was carried out, in part, by the Jet Propulsion Laboratory (JPL), California Institute of Technology, under a contract with the National Aeronautics and Space Administration (NASA). The AMSR-E and TMI SSTs were provided by Remote Sensing Systems (available online at http://www.ssmi.com). The GOES and MC SSTs were provided by PO.DAAC (available online at http://podaac.jpl.nasa.gov). The in situ observations were provided by the Global Ocean Data Assimilation Experiment (GODAE; available online at http://www.usgodae.org). Support from the Office of Naval Research (ONR) through a subcontract from the Monterey Bay Aquarium Research Institute (MBARI) is acknowledged. We thank Drs. J. Vazquez and E. Armstrong for useful discussion. The comments of two anonymous reviewers contributed to improve the original manuscript.

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## APPENDIX

### Relationship between the OI and 2DVAR Algorithms

**T**

*is the observational vector that encompasses*

^{o}**T**

*and 𝗛 is the block diagonal matrix [i.e., 𝗛 = diag(*

_{s}^{o}*H*

_{1}, … ,

*H*)].

_{M}*δJ*(

**T**)/

*δ*

**T**= 0. We then obtain With the equality (𝗕

^{−1}+ 𝗛

^{T}𝗥

^{−1}𝗛)

^{−1}𝗛

^{T}𝗥

^{−1}= 𝗕𝗛

^{T}(𝗛𝗕𝗛

^{T}+ 𝗥)

^{−1}, we immediately have If we consider the case where all the observations are at grid points, 𝗛 becomes a diagonal matrix and its diagonal entries are 0 or 1 and (A2) is degenerated to (1). Thus, OI is equivalent to our 2DVAR method in this idealized example.

Satellite SST data used.