In variational assimilation, the most probable state of a dynamical system under Gaussian assumptions for the prior and likelihood can be found by solving a least-squares minimization problem . In recent years, we have seen the popularity of hybrid variational data assimilation methods for Numerical Weather Prediction. In these methods, the prior error covariance matrix is a weighted sum of a climatological part and a flow-dependent ensemble part, the latter being rank deficient. The nonlinear least squares problem of variational data assimilation is solved using iterative numerical methods, and the condition number of the Hessian is a good proxy for the convergence behavior of such methods. In this paper, we study the conditioning of the least squares problem in a hybrid four-dimensional variational data assimilation (Hybrid 4D-Var) scheme by establishing bounds on the condition number of the Hessian. In particular, we consider the effect of the ensemble component of the prior covariance on the conditioning of the system. Numerical experiments show that the bounds obtained can be useful in predicting the behavior of the true condition number and the convergence speed of an iterative algorithm

翻译：在变分同化中，通过解最小二乘最小化问题，可以找到在假设先验和似然符合高斯分布条件下动力系统的最可能状态。近年来，混合变分数据同化方法在数值天气预报中日益流行。这些方法中，先验误差协方差矩阵是气候学部分和流依赖集合部分的加权和，后者存在秩亏。变分数据同化的非线性最小二乘问题通过迭代数值方法求解，而Hessian矩阵的条件数是衡量此类方法收敛性的良好指标。本文通过建立Hessian矩阵条件数的界，研究了混合四维变分数据同化（Hybrid 4D-Var）方案中最小二乘问题的条件性。我们特别考虑了先验协方差矩阵的集合分量对系统条件性的影响。数值实验表明，所得界可用于预测真实条件数的行为及迭代算法的收敛速度。