text: 209,85,1319688341118455809,967806578425516032,Dr Jemima Tabeart,"We (@DrSarahDance, @amoslawless, @J_A_Waller and Nancy Nichols) have a new paper on arxiv this week! <LINK>  As is my new tradition, here's an as-non-technical-as-possible overview (this time with GIFs!) 🧵: <LINK> In my PhD I studied how introducing correlated observation error covariance matrices (OECs) alters mathematical properties of variational data assimilation (DA) problems. DA appears everywhere, but is most well known for its use in weather forecasting. <LINK> In this new paper we consider the preconditioned problem - this is where we solve a different but related problem that is computationally cheaper. The standard preconditioner for DA terms puts the OEC matrix in the same term as the background error covariance (BEC) matrix. <LINK> Some maths: we study how cheap/expensive our problem is using a condition number. Large condition number = expensive! Eigenvalues are properties of a matrix and relate to their condition number - if all the eigenvalues are close the condition no. is small, spread out = large <LINK> Previously we found for the UNpreconditioned system (BEC + OEC) that the smallest eigenvalue of the OEC matrix is important theoretically. If the eigenvalue is small, the DA problem is expensive! Numerics show that if we increase this eigenvalue, we can make the DA problem faster <LINK> For the preconditioned problem (OECxBEC) we found that although the smallest eigenvalue is still important, numerically the DA problem is solved fastest when BEC and OEC have similar properties (e.g. related eigenvalues). <LINK> So if all we care about is a fast solution, we should make BEC and OEC as similar as possible, right? Usually we don't get to choose BEC/OEC - they come from the underlying physics. However, in practice we often have to adapt OEC matrices before we use them. <LINK> Our new theory and results could help us choose how to do this modification - and mean even faster convergence! This is great: OEC information is really important, but can be expensive to use. Better OEC matrices = better weather forecasts (NB I'm not promising better weather!) <LINK> Thanks for making it this far! Finishing this paper during COVID-19 times has been difficult. Collaborating virtually has taken some adjustment, and I've found focusing hard (not ideal for editing manuscripts!). But I'm going to celebrate this small win and keep plodding on! <LINK>",https://arxiv.org/abs/2010.08416,"Data assimilation algorithms combine prior and observational information, weighted by their respective uncertainties, to obtain the most likely posterior of a dynamical system. In variational data assimilation the posterior is computed by solving a nonlinear least squares problem. Many numerical weather prediction (NWP) centres use full observation error covariance (OEC) weighting matrices, which can slow convergence of the data assimilation procedure. Previous work revealed the importance of the minimum eigenvalue of the OEC matrix for conditioning and convergence of the unpreconditioned data assimilation problem. In this paper we examine the use of correlated OEC matrices in the preconditioned data assimilation problem for the first time. We consider the case where there are more state variables than observations, which is typical for applications with sparse measurements e.g. NWP and remote sensing. We find that similarly to the unpreconditioned problem, the minimum eigenvalue of the OEC matrix appears in new bounds on the condition number of the Hessian of the preconditioned objective function. Numerical experiments reveal that the condition number of the Hessian is minimised when the background and observation lengthscales are equal. This contrasts with the unpreconditioned case, where decreasing the observation error lengthscale always improves conditioning. Conjugate gradient experiments show that in this framework the condition number of the Hessian is a good proxy for convergence. Eigenvalue clustering explains cases where convergence is faster than expected. ","New bounds on the condition number of the Hessian of the preconditioned