Diffusion Filters for Variational Data Assimilation of Sea Surface Temperature in an Intermediate Climate Model

3.1. Sequential Diffusion Filter (SDF)

The sequential diffusion filter (SDF) scheme is similar to the S3DVar method derived by Xie et al. [17]. The SDF scheme uses a sequence of 3DVars to obtain the final estimation to retrieve information from all wavelengths from long- to shortwaves in turn. The matrix B is modeled by applying the diffusion filter sequentially in x and y direction, respectively. SDF begins its sequence with a big value of the diffusion coefficient a; then an initial estimation is obtained through analyzing the observed data. After that, a S3DVar is solved using the diffusion filter with a smaller a than before. For the S3DVars, observations to be assimilated are produced by subtracting the previously analyzed values from the observations assimilated by the previous 3DVar until the diffusion coefficient a is small enough. The final estimation is the summation of all the previous 3Dvar analyses based on the diffusion filter.

From the above description, it is noted that the SDF scheme is a simple extension of the DF, in which information is retrieved step by step from long- to shortwaves. During the process of the SDF, B is changed gradually with the different diffusion coefficient a and thus becomes flow dependent and anisotropic following the multiscale information of the observation.

3.2. Adaptive Diffusion Filter (ADF)

Due to the introduction of the heat diffusion equation, the gradient of the cost function with respect to the state variables can be obtained using the adjoint method with 4DVar. In general, the diffusion coefficients a(x, y) and b(x, y) are not constants but are space dependent. Therefore, it is possible to optimize not only the state variables but also the diffusion coefficients using 4DVar. State variables and diffusion coefficients are used together as control variables, so values of a(x, y) and b(x, y) will change adaptively according to the distribution of observations.

3.3. Gradient Diffusion Filter (GDF)

The algorithm is a variant of the spatial multiscale recursive filter [22]. For small diffusion coefficients a, b, the gradient contains not only all the observational signals from longer to shorter wavelengths, but also a lot of erroneous signals in data sparse regions, which causes lack of coherent longwave structure in space. If this gradient is simply introduced into the minimization algorithm without careful considerations, the analysis departs far from reality. Thus, a prerequisite for the minimization algorithm used in 3DVAR is needed to extract the longwave information from the gradient and at the same time to preserve the valuable shortwave signals.

However, the longwave information implied in the gradient cannot be made best use of to construct a reasonable descent direction in general minimization algorithms. Take the steepest descent algorithm as an example, in which the descent direction is simply chosen as −g(w). Suppose the initial guess of w (i.e., w₀) is equal to zero. Then at the ith iteration, the new solution w_i = w_i−1 + l_i−1∗(−g(w_i−1)) is obtained by using a line search algorithm to find an appropriate step size l_i−1. According to what have been indicated, the gradient g(w₀) actually represents certain scales of observations d, and these scales will be extracted by the line search at the first iteration and incorporated into a new solution w₁. However, if the diffusion coefficients a, b are small, the gradient g(w₀) will lack coherent structure in data sparse regions though it actually carries all observational signals. And since the new solution w₁ is simply obtained along the descent direction, −g(w₀), the same problem will also exist in w₁, which indicates that the longwave information of observations d is not effectively extracted from the gradient g(w₀) at the first iteration. Similarly, at the second iteration, the longwave information of the observation residue after the first iteration will not be extracted from the gradient g(w₁) and incorporated into the new solution w₂, and so on. As a consequence, in data sparse regions, the final analysis will also lose the longwave structure of observations. The same problem also exists for other minimization algorithms such as BFGS and the conjugate gradient method, for the same reason.

If the background term J_b is involved in the cost function J, the same procedure is performed except that g calculated in step (4) is the gradient of cost function J, which includes both J_b and J_o.

4.1. Experiment 1

In this experiment, the number of observations is set to 2000 at first, and the observations are randomly and uniformly distributed in the whole domain, which can be seen from the black dots in Figure 1(b). In the experiment with DF, several values of the diffusion coefficient are used to verify the impacts on the analyzed field. In the experiment with GDF, the processes (1)–(8) described in Section 3.3 are conducted. The diffusion coefficients a, b are set to a small value, of 0.1, which suggests almost all the observational signals, from long to short wavelengths, can be retrieved. However, a large enough value, 1.0, is given to the extra diffusion coefficient β of the gradient at the first step. For the subsequent steps, β is reduced by 0.1 from the previous step. At the last step, β becomes 0.1, which is small enough for the case. The limited memory BFGS quasi-Newton minimization algorithm [23] is used during the minimizing procedure.

The major scales of the truth field are reconstructed by 2000 observations almost fully using the GDF (Figure 2(a)), but not well reconstructed using the DF with different diffusion coefficients (a, b): 1.0 (Figure 2(b)), 0.5 (Figure 2(c)), and 0.1 (Figure 2(d)). The small scale features begin to dominate the analyzed fields when the diffusion coefficients are reduced gradually, while the large scale signals are contaminated dramatically by an abundance of small scale features.

As He et al. [18] indicated, artificial signals can be produced during the data assimilation if the chosen diffusion coefficient cannot represent the actual scale. In contrast, the GDF can handle spatial multiscale analysis pretty well compared to the simple DF with a fixed diffusion coefficient. In addition, the GDF is easy to avoid in empirical selection of the diffusion coefficient.

Figure 3 shows the performance of GDF and DF when the number of observations decreases from 2000 to 500. The GDF (Figure 3(a)) can retrieve large scale information from observations and leave the unresolved scale as errors on top of the resolvable scales. These errors are smaller than those generated by DF with a fixed diffusion coefficient (Figure 3(b)) in the condition of the sparseness of the observations and the lack of information.

4.2. Experiment 2

The second experiment is conducted with removal of observations in the area of 103°E~107°E and 35°N~40°N (Figure 4) to further evaluate the GDF capability in retrieving the multiscale information from observations. The analyzed field of the GDF (Figure 5(a)) performs much better in the data void region than that of the DF with (a, b) = 0.8 (Figure 5(b)) and 0.5 (Figure 5(c)). The GDF can reconstruct the temperature field (Figure 5(a)) reasonably well despite the absence of the observations in the region as shown in Figure 4. The spatial pattern of the whole temperature field can be captured roughly according to the large scale information derived from all the observations in the whole analyzed region. However, the DF fails to reconstruct the temperature field and produces false features especially in the data void region. For example, a strong cold tongue is produced for (a, b) = 0.8 (Figure 5(b)), and large scale temperature field is distorted with displacement of the thermal front in the data void region for (a, b) = 0.5 (Figure 5(c)). Little information of the observations can be extracted from data rich area to the data void region using DF.

Such capabilities make the GDF invaluable to get well represented values for the data void (or insufficiently covered) areas such as a typhoon-affected area during typhoon passage or the Southern Hemisphere Oceans (compared to other ocean basins). The GDF can reconstruct the analyzed field roughly according to the longwave information of the observations beyond the data void area such as typhoon-affected region or the Southern Ocean. On the other hand, both the standard DF and the traditional RF may lead to false results in the data void region, as shown in Figures 5(b)-5(c); an improper analysis is also likely to be produced, which will affect the analysis/forecast accuracy seriously.

In addition, several classical geostatistical tools, such as inverse distance to a power, triangulation with linear interpolation, and Kriging method are used to interpolate such observations (no white noise is imposed on the observations). Compared to the other two geostatistical tools, the Kriging method is able to accurately fill in the hidden information (Figures 6(a) and 6(b) versus 6(c)). However, compared with the variational method, the geostatistical tools have a limited application and cannot handle corrections between different analysis variables or physical balances and other constraints [20].

5.1. Brief Description of an Intermediate Atmosphere-Ocean-Land Coupled Model

All the three model components adopt 64 × 54 Gaussian grid and are forwarded by a leap frog time stepping with a half hour integration step size. There are 2287 and 1169 grid points over the ocean and land, respectively. An Asselin-Robert time filter [30, 31] is introduced to damp spurious computational modes in the leap frog time integration. Default values of all parameters are listed in Table 1 in Wu et al. [24].

Starting from initial conditions $$, where ψ⁰, ϕ⁰, $$, and $$ are zonal mean values of corresponding climatological fields, the coupled model is run for 60 years to generate the model states $$). The last 10 years’ model states (Z1) are used as the “truth” fields. Figure 7 shows the annual mean of ψ (Figure 7(a)), ϕ (Figure 7(b)), T_o and T_l (Figure 7(c)), where the associated wave trains in the ψ field are observed. For ϕ, one can see the distinct pattern of the western boundary currents, gyre systems and the Antarctic Circumpolar Current (ACC). For T_o and T_l, reasonable temperature gradients are also produced. Note that the low temperature in tropical lands can be attributed to the linear damping of K_T in the solar forcing. The above model configuration is called the “truth” model, which has reasonable but rough representation for the basic climate characteristics of the atmosphere, land, and ocean.

5.2. Model “Bias” Arising from the Initial States

However, starting also from the same initial conditions Z0, the Gaussian random numbers are added to ψ⁰ and ϕ⁰, with standard deviations of 10⁷ m²s⁻¹ (for ψ⁰) and 10⁵ m²s⁻¹ (for ϕ⁰), respectively. The coupled model is also run for 60 years to generate the model states $$. The last 10 years’ model states are used for analysis. This model configuration is called the biased model.

The model “biases” induced by perturbed initial fields are examined. Figure 8 shows time series of the spatial averaged root mean square errors (RMSEs) of ψ, ϕ, T_o, and T_l for the assimilation model, which are calculated according to the difference in the assimilation model and the “truth” model. The obvious difference about all the four components can be seen from Figure 8. The RMSE of ψ reaches about 1.6 × 10⁷ m²s⁻¹ with a high frequent oscillation (see Figure 8(a)). In contrast, the RMSE of ϕ performs smoothly and rapidly decreases within the first year and gradually reaches a low and stable value about 10⁴ m²s⁻¹ (see Figure 8(b)). The RMSEs of T_o (Figure 8(c)) and T_l (Figure 8(d)) increase rapidly in the first year, which are generated by the initially perturbed ψ⁰ and ϕ⁰ through the coupling. High frequency oscillation is noted in the time series of the RMSE of T_l, which indicates that the land surface temperature T_l is dominated by the atmospheric motion (ψ). However, time series of the RMSE of T_o is much smoother than that of T_l, indicating that T_o is modulated by the oceanic motion (ϕ).

The spatial distribution of RMSE of T_o for the biased model (Figure 9) shows a notable bias in the ACC region of the Southern Ocean with a maximum value over 15 K near the southern tip of Africa. Besides, obvious biases also exist in the west boundaries of the ocean in the subtropical regions.

5.3. Twin Experiment Design

In this section, with the intermediate model and DF/GDF data assimilation scheme described above, a perfect twin experiment framework is designed with the assumption that the errors of initial model states are the only source of assimilation model biases. Starting from the model states, Z1, described in Section 5.1, the “truth” model is run for 1 year to generate the time series of the “truth” states. Only synthetic observations of T_o are produced through sampling the “truth” states at specific observational frequencies. A Gaussian white noise is added for simulating observational errors. The standard deviations of observational errors are 0.5°C for T_o. The sampling period is 24 hours. The “observation” locations of T_o are global randomly distributed with the same density of the ocean model grid points.

The biased model uses the biased initial fields depicted in Section 5.2. Starting from the biased model states, Z2, the experiment E_GDF consists on assimilating observations into model states using the GDF scheme. In comparison, the experiment E_DF is carried out, where the standard DF scheme is used with the diffusion coefficients a, b = 0.5. In addition, a control run without any observational constraint, called CTRL, serves as a reference for the evaluation of assimilation experiments.

5.4. Impact of the GDF on the Estimate of the States

The performance of GDF is investigated. Figures 10(a)–10(d) show time series of RMSEs of ψ, ϕ, T_o, and T_l for the CTRL (solid line) and the GDF (dash line). Compared to the CTRL (solid line in Figure 10(c)), T_o of the GDF has significant improvement (dash line in Figure 10(c)), in which the RMSE decreases to approximately 0.5 K. Figure 11 presents the spatial distributions of RMSEs of T_o using GDF. The RMSE of T_o over ocean is obviously reduced compared with that of the CTRL (see Figure 9), especially in the Southern Ocean, the subtropical and the subpolar regions. In particular, the reduction of RMSE is much significant in the ACC region, in which the RMSE decreases from above 15 K to below 3 K.

Unlike the RMSEs of T_o, there is no direct observations constraint for ψ, ϕ, and T_l; therefore, their RMSEs decrease gradually owing to the effect of the coupling. The RMSEs of ψ for the GDF are reduced significantly from about 1.6 × 10⁷ m²s⁻¹ to about 1.1 × 10⁷ m²s⁻¹ with a high frequent oscillation (see Figure 10(a)). The ϕ in GDF is also improved significantly comparing to CTRL (solid versus dash lines in Figure 10(b)), whose RMSE decreases gradually and smoothly, but it does not reach a stable value within the experimental period, indicating that the low frequency signal needs a much longer time to reach equilibrium compared to the high frequency signal. For T_l, the GDF reduces the error by approximately 60%. Note that T_o has no direct effect on T_l, which can be realized according to the framework of the coupling model (see (18)–(20)). Instead, T_o affects T_l indirectly via ψ. The improved T_o by the observational constraint increases the quality of ψ over land through the dynamical constraint. Then, the improved ψ ameliorates T_l through the process of the external forcing.

5.5. Removal of Observational Data in the Southern Ocean

In the real ocean, the observations are scarce in the southern polar region. Therefore, another set of data assimilation experiment is carried out, which is the same as the experiment in Section 5.3, but in which the observations, south of 50°S and 50°E~300°E, are removed completely.

Figures 12(a)–12(d) show the time series of RMSEs of ψ, ϕ, T_o, and T_l with the GDF (black line) and the standard DF with a, b = 0.5 (red line). The RMSE of T_o for the DF increases persistently during the experimental period, while the RMSE for the GDF begins to descend after 0.2 years and converges after 0.6 years (Figure 12(c)). When the diffusion coefficients are set to different values in the DF experiment (e.g., a, b = 0.2,0.8,1.0), similar results as the ones presented in Figure 12 are obtained. Results indicate that DF cannot correct the model bias in the data void region. However, the GDF is able to mitigate the model bias to some degree through extracting the spatial multiscale information from the available observations to the data void region. Figure 13 presents the spatial distributions of RMSEs of T_o for the GDF and the standard DF with a, b = 0.5. Compared to the DF, the GDF produces a significant improvement within the data void region in the Southern Ocean (compare Figures 13(a) and 13(b)).

The RMSE of ψ in the GDF is not always smaller than the DF owing to the strong nonlinear nature of the high frequent atmosphere (red line versus black line in Figure 12(a)). In contrast, the evolution of T_l in the model (see (20)) is rather simple (i.e., linear); the RMSE in the GDF is almost always smaller than that for the DF (red line versus black line in Figure 12(d)). For the low frequent component ϕ (see Figure 12(b)), the RMSEs of both the GDF and the DF decrease gradually, indicating that the effect of the data void region on the low frequent signal is small in the given time scale.

5.6. Impact of the GDF on the Forecast

From a more practical point of view, the role of the GDF should be judged from the model forecast. In this section, two forecast experiments without any observational constraint are integrated for 1 year, respectively, starting from the final analyzed states of the above two assimilation experiments (the GDF and the DF).

Figures 14(a)–14(d) show the forecasted time series of RMSEs of ψ, ϕ, T_o, and T_l for the GDF (black line) and the DF with a, b = 0.5 (red line). The GDF performs much better than the DF in 1 year’s forecast lead time of all the state variables such as the high frequent component ψ and the low frequent component ϕ (black versus red curves in Figures 14(a) and 14(b)). It is interesting that the forecasted RMSE of ϕ still decreases inertially owing to the longer adjustment time of the low frequent signal, but whose trend becomes mildly with the increase of the forecasted lead time. For T_o, because of the absence of the observational constraint, the forecasted RMSE has an obvious positive trend (see Figure 14(c)), indicating that the forecasted state is gradually drifting away from the truth. Anyway, the GDF retains its superiority relative to the DF during the entire forecasted lead time. The forecasted RMSEs of T_l have similar patterns to those of T_o (see Figure 14(d)).