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Joshua P. Hacker and Lili Lei

, product that reduces the elements of in Eq. (7) and is denoted . When solving the statistical analysis equation with scalar regressions as in Anderson (2003) , the localization is applied as a factor to each regression coefficient. Because the ensemble statistics are derived from an ensemble filter including some form of localization, the analysis increment from a hypothetical observation would be subject to the same localization. Thus, in Eqs. (6) and (7) the same approach would be used

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Sean M. Wile, Joshua P. Hacker, and Kenneth H. Chilcoat

of the analysis perturbation differ in different experiments explained below. By perturbing the analysis at a point x i , the slope of the regression line between x i and J gives the expected forecast change. Further, assimilating an observation at a sensitivity point decreases σ x , reflecting more certainty in the analysis. In this case the uncertainty in forecast J should also reduce as long as the linear approximation is valid, because to a linear approximation each member of the

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Matthew E. Jeglum, Sebastian W. Hoch, Derek D. Jensen, Reneta Dimitrova, and Zachariah Silver

exceeding 100% while the lower slopes consist mainly of alluvial fans whose slopes are much gentler. One particular alluvial fan, on the southeast side of GM (see inset in Fig. 1 ), was the location of intensive observation during the MATERHORN field experiment. Data collected there have been used in multiple previous studies ( Lehner et al. 2015 ; Grachev et al. 2016 ). This area will be referred to here as the east slope and will be the primary focus for analysis in this paper. Fig . 1. Map (UTM

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H. J. S. Fernando, E. R. Pardyjak, S. Di Sabatino, F. K. Chow, S. F. J. De Wekker, S. W. Hoch, J. Hacker, J. C. Pace, T. Pratt, Z. Pu, W. J. Steenburgh, C. D. Whiteman, Y. Wang, D. Zajic, B. Balsley, R. Dimitrova, G. D. Emmitt, C. W. Higgins, J. C. R. Hunt, J. C. Knievel, D. Lawrence, Y. Liu, D. F. Nadeau, E. Kit, B. W. Blomquist, P. Conry, R. S. Coppersmith, E. Creegan, M. Felton, A. Grachev, N. Gunawardena, C. Hang, C. M. Hocut, G. Huynh, M. E. Jeglum, D. Jensen, V. Kulandaivelu, M. Lehner, L. S. Leo, D. Liberzon, J. D. Massey, K. McEnerney, S. Pal, T. Price, M. Sghiatti, Z. Silver, M. Thompson, H. Zhang, and T. Zsedrovits

approximation in ESA ( Hacker and Lei 2015 ). Ensuing results showed that sampling error can be mitigated by reducing regression coefficients according to the expected error in the sensitivities and that the approximation can be easily avoided through a minimum-norm regression. Including full spatial analysis covariance information, and accounting for sampling error, improved the ESA predictions for where observations are most likely to reduce forecast uncertainty. F ig . 11. The 6-h correlation (red) and

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