# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## 1 Introduction  
Launched in 2016, the Cyclone Global Navigation Satellite System (CYGNSS) mission provides a suite of land and ocean surface sensing products through a technique known as Global Navigation Satellite System reflectometry (GNSS-R). With GNSS-R, the eight CYGNSS satellites act as receivers in a bistatic radar system, where GPS satellites are the transmitters, and the Earth's surface is the target of interest. The CYGNSS mission aims to improve tropical cyclone prediction by measuring ocean surface wind speeds over the tropics [1]. Other satellite observations are impaired by the difficulty in collecting wind speeds underneath thick cloud layers in tropical cyclones, and in situ measurements are often challenging and dangerous to collect due to extreme conditions within tropical cyclones. By exploiting GPS broadcasts in the L-band, where there is limited attenuation from thick rain and cloud layers, CYGNSS can collect observations through mature tropical cyclones. With eight satellites in equatorial low-Earth orbit, CYGNSS can collect global observations with a median revisit time of 3.8 h. However, the sampling characteristics of CYGNSS differ from many other space-based weather observations. Instead of imaging, CYGNSS data are comprised of tracks of specular points that streak across the Earth's surface. Images are reconstructed from composites of multiple satellites. This unique sampling mechanism poses challenges for assimilation into numerical weather prediction (NWP) models. Data assimilation (DA) is the component of an NWP system that incorporates observations with evolving forecast models. A common DA technique poses the problem as a variational equation that minimizes a cost function [2]:  
\[\underset{\mathbf{x}}{\text{argmin}}\mathbf{J}(\mathbf{x})=(\mathbf{x}-\mathbf{ x_{b}})^{T}\mathbf{B}^{-1}(\mathbf{x}-\mathbf{x_{b}})+(\mathbf{y}-H[\mathbf{x}])^{T} \mathbf{R}^{-1}(\mathbf{y}-H[\mathbf{x}]), \tag{1}\]  
where \(\mathbf{J}(\mathbf{x})\) represents the cost function, \(\mathbf{x}\) is the variational argument, \(\mathbf{x_{b}}\) is the background model state of dimension \(n\), \(\mathbf{B}\) is the \(n\)-by-\(n\) background error covariance matrix, \(\mathbf{y}\) is theobservation vector of dimension \(p\), \(H\) is the forward model operator for transforming from model space to observation space, and \(\mathbf{R}\) is the \(p\)-by-\(p\) observation error covariance matrix. In practice, a great deal of focus is placed on estimating \(\mathbf{B}\)[3; 4] because incorrectly specified \(\mathbf{B}\) can have significant impacts on the performance of the model. Famously, introducing perfect observations to a perfect model can still lead to forecast degradations if \(\mathbf{B}\) is poorly constructed [5]. On the other hand, \(\mathbf{R}\) is comparatively overlooked. NWP data assimilation (DA) schemes often assume that each observation error is completely uncorrelated [6; 7; 8] and, as such, treat \(\mathbf{R}\) operationally as a diagonal matrix. This is performed for three main reasons: to simplify the assumptions needed for minimizing the cost function in Equation (1) and reduce the computational resources required to solve the forecast problem; to reduce the data handling and storage requirements for observation data at low latencies; and because frequently the information required to specify correlated observation errors was unavailable or difficult to ascertain. Regardless of the rationale, the under-specification of correlated observation error in \(\mathbf{R}\) leads to the suboptimal use of the information content in observations. There are some major recent advancements on this front. Discussions of correlated instrument errors are generally limited to explorations for interchannel correlated errors in infrared [9; 10] and microwave [11] sounders, with some physical models of correlation proposed [12; 13]. Spatial observation error correlation is discussed in [11; 14] but is generally restricted to estimates of total observation error correlation as estimated from a weather model [15].