Exploring Ultra Rapid Data Assimilation Based on Ensemble Transform Kalman Filter with the Lorenz 96 Model

TL;DR

The paper investigates ultra-rapid data assimilation (URDA) within nonlinear dynamics and identifies stability issues when preemptive forecasts are repeatedly updated. It analytically links URDA preemptive forecasts to EnKF analyses in nonlinear settings and demonstrates deterioration in RMSE and ensemble spread with conventional inflation/localization schemes using the Lorenz 96 model. To address these challenges, the authors introduce relaxation-to-baseline perturbations (RTBP) and relaxation-to-baseline forecast (RTBF), showing that URDA with RTBP/RTBF yields superior short-term accuracy while preventing long-lead deterioration. The work suggests URDA, augmented with RTBP and RTBF, as a promising approach for high-frequency data assimilation in practical NWP, pending validation on more realistic models and observations.

Abstract

To explore the effectiveness of ultra-rapid data assimilation (URDA) for numerical weather prediction (NWP), this study investigates the properties of URDA in nonlinear models and proposes technical treatments to enhance its performance. URDA rapidly updates preemptive forecasts derived from observations without integrating a dynamical model each time additional observations become available. First, we analytically demonstrate that the preemptive forecast obtained by URDA in nonlinear models is approximately equivalent to the forecast integrated from the analysis. Furthermore, numerical experiments are conducted with the 40-variable Lorenz 96 model. The results show that URDA in nonlinear models tends to exhibit deterioration of forecast accuracy and collapse of ensemble spread when preemptive forecasts are repeatedly updated or when the forecasts are extended over longer periods. Furthermore, the roles of inflation and localization, both essential technical treatments in NWP, are examined in the context of URDA. It is shown that although inflation and localization are essential to URDA, conventional inflation techniques are not suitable for it. Therefore, this study proposes new technical treatments for URDA, namely relaxation to baseline perturbations (RTBP) and relaxation to baseline forecast (RTBF). Applying RTBP and RTBF mitigates the difficulties associated with URDA and yields preemptive forecasts with higher accuracy than the baseline forecast. Consequently, URDA, particularly when combined with RTBP and RTBF, would stand as a step toward practical application in NWP.

Figures (10)

• Figure 1: The concept of URDA. A situation is assumed in which the baseline forecast already exists from time $0$ to time $T$, and additional observations become available at each subsequent time step. URDA sequentially updates the preemptive forecasts using the existing forecast ensembles and the additional observations.
• Figure 2: The RMSE and the spread of the conventional URDA (multiplicative inflation factor $\delta=1.05$ and localization scale $\sigma=1.0$). In both figures, each solid line represents the preemptive forecast $\mathbf{X}_{k \mid j}^{f}$ for each forecast lead time $k$ at each forecast reference time $j$. The dashed line represents the baseline forecast $\mathbf{X}_{k \mid 0}^{f}$, and the dash-dotted line represents the initial forecast $\mathbf{X}_{j+1 \mid j}^{f}$ for each forecast reference time. (a) Dark and light red solid lines show the RMSE for the early and latter parts of the forecast reference time. (b) Dark and light blue solid lines show the spread for the early and latter parts of the forecast reference time.
• Figure 3: The ensemble transform matrix $\widecheck{\mathbf{W}}_{j}^{prod,loc}$ of the conventional URDA (multiplicative inflation factor $\delta=1.05$ and localization scale $\sigma=1.0$ ) for each forecast reference time from (a) $j=0$ days to (i) $j=30$ days at grid point $g=0$.
• Figure 4: The RMSE of the initial forecast $\mathbf{X}_{j+1 \mid j}^{f}$ at each forecast reference time from (a) 2 days to (f) 30 days for the multiplicative inflation factor $\delta$ and localization scale $\sigma$. The black dots indicate the minimum RMSE at each forecast reference time.
• Figure 5: Similar to Figure \ref{['fig:sensitivity_multiplicative_inflation']}, but for the inflation factor $\alpha^{RTPP}$.

Theorems & Definitions (18)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Proposition A.1
• proof