The Effect of Linearization Errors on 4DVAR Data Assimilation

Tomislava Vukićević National Center for Atmospheric Research, * Boulder, Colorado

Search for other papers by Tomislava Vukićević in
Current site
Google Scholar
PubMed
Close
and
Jian-Wen Bao National Oceanic and Atmospheric Administration, Boulder, Colorado

Search for other papers by Jian-Wen Bao in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

The authors show that the linear approximation errors in the presence of a discontinuous convective parameterization operator are large for a small number of grid points where the noise produced by the convective parameterization is largest. These errors are much smaller for “smooth convective” points in the integration domain and for the nonconvective regions. Decreasing of the amplitude of initial perturbations does not reduce the errors in noisy points. This result indicates that the tangent linear model solution is erroneous in these points due to the linearization that does not include the linear variations of regime changes (i.e., due to use of standard method).

The authors then show that the quality of local four-dimensional variational (4DVAR) data assimilation results is correlated with the linearization errors: Slower convergence is associated with large errors. Consequently, the 4DVAR assimilation results are different for different convective points in the integration domain. The negative effect of linearization errors is not, however, significant for the cases that are studied. Erroneous points slightly degrade 4DVAR results in the remaining points. This degradation is reflected in decreased monotonicity of the cost function gradient reduction with iterations.

These results suggest that there is a probability for locally bad 4DVAR assimilation results when using standard adjoints of discontinuous parameterizations. In practice, when using for example observations, this is unlikely to cause errors that are larger than errors associated with other approximations and uncertainties in the data assimilation integrations such as the linear approximation errors and the uncertainties associated with the background and model errors statistics. This conclusion is similar to the conclusions of prior 4DVAR assimilation studies that use the standard adjoints but unlike in these studies the results in the current study show that 1) the linearization errors are nonnegligible for small-amplitude initial perturbations and 2) the assimilation results are locally and even globally affected by these errors.

Corresponding author address: Dr. Tomislava Vukicevic, NCAR, P.O. Box 3000, Boulder, CO 80307-3000.

Email: tomi@milena.cgd.ucar.edu

Abstract

The authors show that the linear approximation errors in the presence of a discontinuous convective parameterization operator are large for a small number of grid points where the noise produced by the convective parameterization is largest. These errors are much smaller for “smooth convective” points in the integration domain and for the nonconvective regions. Decreasing of the amplitude of initial perturbations does not reduce the errors in noisy points. This result indicates that the tangent linear model solution is erroneous in these points due to the linearization that does not include the linear variations of regime changes (i.e., due to use of standard method).

The authors then show that the quality of local four-dimensional variational (4DVAR) data assimilation results is correlated with the linearization errors: Slower convergence is associated with large errors. Consequently, the 4DVAR assimilation results are different for different convective points in the integration domain. The negative effect of linearization errors is not, however, significant for the cases that are studied. Erroneous points slightly degrade 4DVAR results in the remaining points. This degradation is reflected in decreased monotonicity of the cost function gradient reduction with iterations.

These results suggest that there is a probability for locally bad 4DVAR assimilation results when using standard adjoints of discontinuous parameterizations. In practice, when using for example observations, this is unlikely to cause errors that are larger than errors associated with other approximations and uncertainties in the data assimilation integrations such as the linear approximation errors and the uncertainties associated with the background and model errors statistics. This conclusion is similar to the conclusions of prior 4DVAR assimilation studies that use the standard adjoints but unlike in these studies the results in the current study show that 1) the linearization errors are nonnegligible for small-amplitude initial perturbations and 2) the assimilation results are locally and even globally affected by these errors.

Corresponding author address: Dr. Tomislava Vukicevic, NCAR, P.O. Box 3000, Boulder, CO 80307-3000.

Email: tomi@milena.cgd.ucar.edu

Save
  • Anthes, R. A., E.-Y. Hsie, and Y.-H. Kuo, 1987: Description of the Penn State/NCAR Mesoscale Model Version 4 (MM4). NCAR Tech. Note NCAR/TN-282+STR, 66 pp. [Available from the National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307.].

  • Brown, J., and K. Campana, 1978: An economical time-differencing system for numerical weather prediction. Mon. Wea. Rev.,106, 1125–1136.

  • Courtier, P., and O. Talagrand, 1987: Variational assimilation of meteorological observations with adjoint vorticity equation: Part II. Numerical results. Quart. J. Roy. Meteor. Soc.,113, 1329–1368.

  • Cressman, G., 1959: An operational objective analysis system. Mon. Wea. Rev.,87, 367–374.

  • Errico, R. M., K. Raeder, and T. Vukićević, 1994: Description of the NCAR Mesoscale Adjoint Modeling System version 1 (MAMS1). NCAR Tech. Note NCAR/TN-410+IA, 214 pp. [Available from the National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000.].

  • Grell, G. A., 1993: Prognostic evaluation of assumptions used by cumulus parameterizations. Mon. Wea. Rev.,121, 764–787.

  • Hack, J. J., B. A. Boville, B. P. Briegleb, J. T. Kiehl, P. J. Rasch, and D. L. Williamson, 1993. Description of the NCAR Community Climate Model (CCM2). NCAR Tech. Note NCAR/TN-382+STR, 107 pp. [Available from the National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000.].

  • Kuo, Y. H., X. Zou, and Y. R. Guo, 1996: Variational assimilation of precipitable water using nonhydrostatic mesoscale adjoint model. Part I: Moisture retrievals and sensitivity experiments. Mon. Wea. Rev.,124, 122–147.

  • Le Dimet, F. X., 1982: A general formalism of variational analysis. CIMMS Rep. 22, Cooperative Institute for Mesoscale Meteorological Studies, 34 pp. [Available from CIMMS, 815 Jenkins St., Norman, OK 73019.].

  • Lewis, J. M., and J. C. Derber, 1985: The use of adjoint equations to solve a variational adjustment problem with advective constraints. Tellus,37A, 309–322.

  • Madala, R. V., 1981: Efficient time integration schemes for atmosphere and ocean models. Finite-Difference Technique for Vectorized Fluid Dynamics Calculations, D. L. Book, Ed., Springer, 56–74.

  • Thepaut, J.-N., and P. Courtier, 1991: Four-dimensional data assimilation using the adjoint of a multilevel primitive equation model. Quart. J. Roy. Meteor. Soc.,117, 1225–1254.

  • Trenberth, K. E., 1992: Global analyses from ECMWF and atlas of 1000 to 10 mb circulation statistics. NCAR Tech. Note NCAR/TN-373+STR, 191 pp. [Available from the National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000.].

  • Tsuyuki, T., 1996: Variational data assimilation in the Tropics using precipitation data. Part II: 3D model. Mon. Wea. Rev.,124, 2545–2661.

  • Tziperman, E., W. C. Thacker, and R. B. Long, 1992: Oceanic data analysis using a general circulation model. Part I: Simulations. J. Phys. Oceanogr.,22, 1434–1457.

  • Vukićević, T., and R. M., Errico, 1993: Linearization and adjoint of parameterized moist diabatic processes. Tellus,45A, 493–510.

  • Xu, Q., 1996a: Generalized adjoint for for physical processes with parameterized discontinuities. Part I: Basic issues and heuristic examples. J. Atmos. Sci.,53, 1123–1142.

  • ——, 1996b: Generalized adjoint for for physical processes with parameterized discontinuities. Part II: Vector formulation and matching conditions. J. Atmos. Sci.,53, 1143–1155.

  • Zou, X., 1997: Tangent linear and adjoint of “on-off” processes and their feasibility for use in 4-dimensional variational data assimilation. Tellus, 49A, 3–31.

  • ——, I. M. Navon, and F. X. Le Dimet, 1993: Variational data assimilation with moist threshold processes. Tellus,45A, 370–387.

  • ——, Y. N. Kuo, and Y. R. Guo, 1995: Assimilation of atmospheric radio refractivity using a nonhydrostatic adjoint model. Mon. Wea. Rev.,123, 2229–2249.

  • Zupanski, D., 1993: The effect of discontinuities in the Betts-Miller cumulus convection scheme on four dimensional data assimilation. Tellus,45A, 511–524.

  • ——, and F. Mesinger, 1995: Four-dimensional variational assimilation of precipitation data. Mon. Wea. Rev.,123, 1113–1127.

  • Zupanski, M., 1993: A preconditioning algorithm for large scale minimization problems. Tellus,45A, 478–492.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 131 39 3
PDF Downloads 76 25 2