Shay Gilpin

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• Gilpin, S., Matsuo, T., & Cohn, S. (2023). A generalized, compactly supported correlation function for data assimilation applications. Quarterly Journal of the Royal Meteorological Society, 149(754). doi:10.1002/qj.4490
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  This work introduces a new, compactly supported correlation function that can be inhomogeneous over Euclidean three-space, anisotropic when restricted to the sphere, and compactly supported on regions other than spheres of fixed radius. This function, which we call the Generalized Gaspari–Cohn (GenGC) correlation function, is a generalization of the compactly supported, piecewise rational approximation to a Gaussian introduced by Gaspari and Cohn in 1999 and its subsequent extension by Gaspari et al in 2006. The GenGC correlation function is a parametric correlation function that allows two parameters (Figure presented.) and (Figure presented.) to vary, as functions, over space, whereas the earlier formulations either keep both (Figure presented.) and (Figure presented.) fixed or only allow (Figure presented.) to vary. Like these earlier formulations, GenGC is a sixth-order piecewise rational function (fifth-order near the origin), while the coefficients now depend explicitly on the values of both (Figure presented.) and (Figure presented.) at each pair of points being correlated. We show that, by allowing both (Figure presented.) and (Figure presented.) to vary, the correlation length of GenGC also varies over space and introduces inhomogeneous and anisotropic features that may be useful in data assimilation applications. Covariances produced using GenGC are computationally tractable due to their compact support and have the added flexibility of generating compact support regions that adapt to the input (Figure presented.) field. These features can be useful for covariance modeling and covariance tapering applications in data assimilation. We derive the GenGC correlation function using convolutions, discuss continuity properties relating to (Figure presented.) and (Figure presented.) and its correlation length, and provide one- and two-dimensional examples that highlight its anisotropy and variable regions of compact support.
• Gilpin, S., Matsuo, T., & Cohn, S. (2022). Continuum Covariance Propagation for Understanding Variance Loss in Advective Systems. SIAM-ASA Journal on Uncertainty Quantification, 10(3). doi:10.1137/21m1442449
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  Motivated by the spurious variance loss encountered during covariance propagation in atmospheric and other large-scale data assimilation systems, we consider the problem for state dynamics governed by the continuity and related hyperbolic partial differential equations. This loss of variance has been attributed to reduced-rank representations of the covariance matrix, as in ensemble methods for example, or else to the use of dissipative numerical methods. Through a combination of analytical work and numerical experiments, we demonstrate that significant variance loss, as well as gain, typically occurs during covariance propagation, even at full rank. The cause of this unusual behavior is a discontinuous change in the continuum covariance dynamics as correlation lengths become small, for instance in the vicinity of sharp gradients in the velocity field. This discontinuity in the covariance dynamics arises from hyperbolicity: the diagonal of the kernel of the covariance operator is a characteristic surface for advective dynamics. Our numerical experiments demonstrate that standard numerical methods for evolving the state are not adequate for propagating the covariance, because they do not capture the discontinuity in the continuum covariance dynamics as correlations lengths tend to zero. Our analytical and numerical results show that this leads to significant, spurious variance loss in certain regions and gain in others. The results suggest that developing local covariance propagation methods designed specifically to capture covariance evolution near the diagonal may prove a useful alternative to current methods of covariance propagation.
• Gilpin, S., Anthes, R., & Sokolovskiy, S. (2019). Sensitivity of forward-modeled bending angles to vertical interpolation of refractivity for radio occultation data assimilation. Monthly Weather Review, 147(1). doi:10.1175/mwr-d-18-0223.1
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  Assimilation of radio occultation (RO) observations into numerical weather prediction (NWP) models has improved forecasts, where RO is typically one of the top five observational systems contributing to forecast accuracy. By measuring the phase delay of radio waves traversing Earth's atmosphere between global positioning system (GPS) and low-Earth orbiting satellites, RO obtains quasi-vertical profiles of bending angles (BA) of the radio waves' trajectories. BA are the RO observation most often assimilated into NWP models, but since they are not computed or analyzed in the models, they must be computed from model data using a forward model. First, model refractivity N is computed from variables specified on the model's vertical levels, then using the Abel integral, BA are computed from N. The forward model requires vertical differentiation of N, and accurate differentiation requires vertical interpolation of N between model levels. The interpolation results in errors, which then propagate through the forward model to produce BA errors. In this study, we investigate the sensitivity of forward-modeled BA to five different methods of vertical interpolation of N between model levels to determine a method that minimizes interpolation errors. We use RO-observed N to isolate the interpolation errors and determine an accurate method that can be applied to any NWP model. Of the five methods investigated, the log-spline interpolation reduces N and BA errors the most, regardless of the vertical resolution of the model grid.
• Gilpin, S., Rieckh, T., & Anthes, R. (2018). Reducing representativeness and sampling errors in radio occultation-radiosonde comparisons. Atmospheric Measurement Techniques, 11(5). doi:10.5194/amt-11-2567-2018
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  Radio occultation (RO) and radiosonde (RS) comparisons provide a means of analyzing errors associated with both observational systems. Since RO and RS observations are not taken at the exact same time or location, temporal and spatial sampling errors resulting from atmospheric variability can be significant and inhibit error analysis of the observational systems. In addition, the vertical resolutions of RO and RS profiles vary and vertical representativeness errors may also affect the comparison. In RO-RS comparisons, RO observations are co-located with RS profiles within a fixed time window and distance, i.e. within 3-6h and circles of radii ranging between 100 and 500km. In this study, we first show that vertical filtering of RO and RS profiles to a common vertical resolution reduces representativeness errors. We then test two methods of reducing horizontal sampling errors during RO-RS comparisons: Restricting co-location pairs to within ellipses oriented along the direction of wind flow rather than circles and applying a spatial-temporal sampling correction based on model data. Using data from 2011 to 2014, we compare RO and RS differences at four GCOS Reference Upper-Air Network (GRUAN) RS stations in different climatic locations, in which co-location pairs were constrained to a large circle ( ∼ 666km radius), small circle ( ∼ 300km radius), and ellipse parallel to the wind direction ( ∼ 666km semi-major axis, ∼ 133km semi-minor axis). We also apply a spatial-temporal sampling correction using European Centre for Medium-Range Weather Forecasts Interim Reanalysis (ERA-Interim) gridded data. Restricting co-locations to within the ellipse reduces root mean square (RMS) refractivity, temperature, and water vapor pressure differences relative to RMS differences within the large circle and produces differences that are comparable to or less than the RMS differences within circles of similar area. Applying the sampling correction shows the most significant reduction in RMS differences, such that RMS differences are nearly identical to the sampling correction regardless of the geometric constraints. We conclude that implementing the spatial-temporal sampling correction using a reliable model will most effectively reduce sampling errors during RO-RS comparisons; however, if a reliable model is not available, restricting spatial comparisons to within an ellipse parallel to the wind flow will reduce sampling errors caused by horizontal atmospheric variability.