Test Simulation Result
(2014-2023 Test Dataset)
RMSE Improvement & Analysis Increment
The data assimilation simulations over a 10-year historic typhoon dataset (Fig. 1) clearly demonstrate that modulating the quadrant-specific observation error covariance (R) significantly enhances the accuracy of the analysis field (x_a).
Fig. 1. Comparison of quadrant-specific SST RMSE improvement rates (left) and mean analysis increments (right) for 2014–2023. The introduction of Adaptive R increases the overall error reduction rate, demonstrating that the highly uncertain RR quadrant suppresses analysis increments via strict weight control, while increments in the more reliable sectors (FR, FL, RL) are actively enhanced.
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RMSE Improvement: Compared to the baseline configuration (Fixed R), the proposed Adaptive R framework achieved a general increase in the RMSE improvement rate across the quadrants, demonstrating the effectiveness of the system. Here, the improvement rate is quantified as the percentage reduction in RMSE achieved by the analysis field (x_a) relative to the initial background field (x_b).
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Analysis Increment vs. Kalman Gain: In the Rear-Right (RR) quadrant, where representation errors are large, the Fixed R approach produced a relatively strong mean increment of 0.085 K. In contrast, the Adaptive R framework reduced the mean increment to 0.072 K by assigning a higher observation error covariance (0.68^2). This quantitatively confirms that an increased observation error covariance (R) effectively reduces the Kalman gain. Such a conservative adjustment helps prevent over-fitting to noisy microwave satellite observations in the cold wake region.
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Enhanced Increments in the Remaining Quadrants (FR, FL, RL): In contrast, quadrants with higher observational reliability showed larger analysis increments under the Adaptive R configuration compared to the Fixed R baseline: FR (0.089 ⇒ 0.101 K), FL (0.075 ⇒ 0.095 K), and RL (0.070 ⇒ 0.077 K). By increasing the Kalman gain in these more reliable regions, the system more strongly incorporated observational information, leading to improved overall error reduction.
The Adaptive R framework improves RMSE by reducing increments in high-uncertainty regions(RR) and increasing them in reliable quadrants through spatially varying Kalman gains, leading to more balanced and effective data assimilation than the Fixed R baseline.
Representative Error Case Study Analysis: TC 1517
To evaluate the physical feasibility of the proposed quadrant-based data assimilation system, we conducted a spatial distribution analysis of Typhoon 1517 (2015-09-08 12:00 UTC), which showed substantial initial errors and pronounced correction effects within the 10-year test period (2014–2023).
Spatial Distribution Analysis in Geographic Coordinates
The geographic coordinate maps show the data assimilation process in the context of the full typhoon track and sea surface temperature (SST) distribution over the Northwest Pacific.
Fig. 2.Spatial distribution of data assimilation fields for TC 1517 in geographical coordinates (2015-09-08 12:00:00). (a) The 4-quadrant mask centered on the typhoon track, (b) the initial ERA5 background bias (x_b - x_t) showing a severe warm bias in the right hemisphere, (c) the resulting DA increment (x_a - x_b) introducing strong negative corrections, and (d) the final adaptive analysis bias (x_a - x_t) illustrating a significant reduction in overall prediction errors.
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Background Error Distribution (b): As Typhoon 1517 propagated northwestward, a pronounced warm bias (red regions) exceeding 1.5 K in the ERA5 reanalysis was observed across a wide area to the right of the typhoon center, primarily over the ocean south of Japan.
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Data Assimilation Increment (c): Upon assimilation of negative innovations (y - x_b) from the Microwave Satellite SST (MW SST), the negative increments (dark purple regions) were generated. These adjustments were concentrated in the right-hand sector, where the warm bias was most pronounced, effectively cooling the model background field. Although the Kalman gain (K) was conservatively reduced by the adaptive R formulation, the largest increments still appeared in the rear-right sector of the storm due to the substantial initial mismatch between the ERA5 warm bias and the observed cold wake.
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Final Analysis Bias (d): Post-assimilation, the spatial extent and intensity of the severe biases shown in the initial error map (b) were visibly reduced, demonstrating a significant convergence toward the truth proxy (OISST).
Across the background error, assimilation increment, and final analysis fields, the system effectively corrected a large ERA5 warm bias associated with Typhoon 1517 by applying MW SST–driven negative innovations, producing coherent cooling increments and improving agreement with OISST.
Quantitative Analysis in Typhoon-Centric Local Coordinates
The local coordinate maps standardize the storm domain into a 500-km rectangular grid by aligning the typhoon's heading with the positive y-axis (along-track direction), thereby isolating the error control mechanisms relative to the storm's asymmetric physical structure.
Fig. 3. Comparison of quadrant-specific SST RMSE improvement rates (left) and mean analysis increments (right) for 2014–2023. (a) The normalized quadrant layout aligned with the storm's heading, (b) the concentrated ERA5 background bias in the rear right hemisphere (RR sectors), (c) the DA increment demonstrating a conservative, smooth correction in the highly uncertain RR quadrant, and (d) the optimized final analysis bias field showing stabilized error reduction across the entire local domain.
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Quadrant Alignment and Error Concentration (a, b): Rotating the coordinate system based on the storm motion vector highlight the ERA5 warm bias (red), revealing its strong concentration in the rear-right (RR) quadrant.
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Quadrant-Specific Adaptive Weight Control (c): To reduce the large errors in the right hemisphere, strong negative increments (purple) were generated in RR quadrants(bottom-right). Notably, the RR quadrant, where a larger observation error covariance (R) was assigned to account for greater observational uncertainty, exhibited a smoothly constrained and conservative correction through reduced Kalman gains (K). As a result, the large magnitude of the initial innovation in the RR quadrant effectively counterbalanced the reduced gain, consistent with the analysis increment formulation.
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Optimization of Analysis Bias (d): Owing to the adaptive mechanism that prevents localized over-correction from observational noise, the final analysis bias map exhibits a well-structured reduction of errors across the entire domain.
By using a storm-relative coordinate system and quadrant-specific adaptive error weighting, the framework effectively controls Kalman gains and reduces ERA5 warm-bias errors, producing a spatially consistent improvement across the domain.