scholarly journals Smoothing error pitfalls

2014 ◽  
Vol 7 (4) ◽  
pp. 3301-3319 ◽  
Author(s):  
T. von Clarmann
Keyword(s):  
Covariance Matrix ◽  
Error Propagation ◽  
A Priori ◽  
A Priori Information ◽  
Atmospheric Variability ◽  
True State ◽  
Statistical Sense ◽  
Atmospheric State ◽  
The Difference ◽  
Priori Information

Abstract. The difference due to the content of a priori information between a constrained retrieval and the true atmospheric state is usually represented by the so-called smoothing error. In this paper it is shown that the concept of the smoothing error is questionable because it is not compliant with Gaussian error propagation. The reason for this is that the smoothing error does not represent the expected deviation of the retrieval from the true state but the expected deviation of the retrieval from the atmospheric state sampled on an arbitrary grid, which is itself a smoothed representation of the true state. The idea of a sufficiently fine sampling of this reference atmospheric state is untenable because atmospheric variability occurs on all scales, implying that there is no limit beyond which the sampling is fine enough. Even the idealization of infinitesimally fine sampling of the reference state does not help because the smoothing error is applied to quantities which are only defined in a statistical sense, which implies that a finite volume of sufficient spatial extent is needed to meaningfully talk about temperature or concentration. Smoothing differences, however, which play a role when measurements are compared, are still a useful quantity if the involved a priori covariance matrix has been evaluated on the comparison grid rather than resulting from interpolation. This is, because the undefined component of the smoothing error, which is the effect of smoothing implied by the finite grid on which the measurements are compared, cancels out when the difference is calculated.

scholarly journals Smoothing error pitfalls

2014 ◽  
Vol 7 (9) ◽  
pp. 3023-3034 ◽  
Author(s):  
T. von Clarmann
Keyword(s):  
Covariance Matrix ◽  
A Priori ◽  
Atmospheric Variability ◽  
Fine Grid ◽  
True State ◽  
Statistical Sense ◽  
Atmospheric State ◽  
The Difference ◽  
Atmospheric Variations ◽  
Priori Information

Abstract. The difference due to the content of a priori information between a constrained retrieval and the true atmospheric state is usually represented by a diagnostic quantity called smoothing error. In this paper it is shown that, regardless of the usefulness of the smoothing error as a diagnostic tool in its own right, the concept of the smoothing error as a component of the retrieval error budget is questionable because it is not compliant with Gaussian error propagation. The reason for this is that the smoothing error does not represent the expected deviation of the retrieval from the true state but the expected deviation of the retrieval from the atmospheric state sampled on an arbitrary grid, which is itself a smoothed representation of the true state; in other words, to characterize the full loss of information with respect to the true atmosphere, the effect of the representation of the atmospheric state on a finite grid also needs to be considered. The idea of a sufficiently fine sampling of this reference atmospheric state is problematic because atmospheric variability occurs on all scales, implying that there is no limit beyond which the sampling is fine enough. Even the idealization of infinitesimally fine sampling of the reference state does not help, because the smoothing error is applied to quantities which are only defined in a statistical sense, which implies that a finite volume of sufficient spatial extent is needed to meaningfully discuss temperature or concentration. Smoothing differences, however, which play a role when measurements are compared, are still a useful quantity if the covariance matrix involved has been evaluated on the comparison grid rather than resulting from interpolation and if the averaging kernel matrices have been evaluated on a grid fine enough to capture all atmospheric variations that the instruments are sensitive to. This is, under the assumptions stated, because the undefined component of the smoothing error, which is the effect of smoothing implied by the finite grid on which the measurements are compared, cancels out when the difference is calculated. If the effect of a retrieval constraint is to be diagnosed on a grid finer than the native grid of the retrieval by means of the smoothing error, the latter must be evaluated directly on the fine grid, using an ensemble covariance matrix which includes all variability on the fine grid. Ideally, the averaging kernels needed should be calculated directly on the finer grid, but if the grid of the original averaging kernels allows for representation of all the structures the instrument is sensitive to, then their interpolation can be an adequate approximation.

NUMERICAL SOLUTION OF TWO-DIMENSIONAL PROBLEMS OF THERMOVISCOELASTICITY

2021 ◽  
Vol 335 (1) ◽  
pp. 60-64
Author(s):  
M. Bukenov ◽  
Ye. Mukhametov
Keyword(s):  
A Priori ◽  
Elastic Collision ◽  
Qualitative Behavior ◽  
Stress Profile ◽  
Two Dimensional ◽  
A Priori Information ◽  
Deformable Solids ◽  
Methods Of Calculation ◽  
The Difference ◽  
Priori Information

This paper considers the numerical implementation of two-dimensional thermoviscoelastic waves. The elastic collision of an aluminum cylinder with a two-layer plate of aluminum and iron is considered. In work [1] the difference schemes and algorithm of their realization are given. The most complete reviews of the main methods of calculation of transients in deformable solids can be found in [2, 3, 4], which also indicates the need and importance of generalized studies on the comparative evaluation of different methods and identification of the areas of their most rational application. In the analysis and physical interpretation of numerical results in this work it is also useful to use a priori information about the qualitative behavior of the solution and all kinds of information about the physics of the phenomena under study. Here is the stage of evolution of contact resistance of collision – plate, stress profile.

scholarly journals Effects of <i>a priori</i> profile shape assumptions oncomparisons between satellite NO<sub>2</sub> columns and model simulations

2020 ◽  
Author(s):  
Matthew J. Cooper ◽  
Randall V. Martin ◽  
Daven K. Henze ◽  
Dylan B. A. Jones
Keyword(s):  
Shape Factor ◽  
A Priori ◽  
Trace Gas ◽  
A Posteriori ◽  
A Priori Information ◽  
Profile Shape ◽  
Satellite Retrievals ◽  
Mass Factor ◽  
The Difference ◽  
Priori Information

Abstract. A critical step in satellite retrievals of trace gas columns is the calculation of the air mass factor (AMF) used to convert observed slant columns to vertical columns. This calculation requires a priori information on the shape of the vertical profile. As a result, comparisons between satellite-retrieved and model-simulated column abundances are influenced by the a priori profile shape. We examine how differences between the shape of the simulated and a priori profile can impact the interpretation of satellite retrievals by performing an adjoint-based 4D-Var assimilation of synthetic NO2 observations for constraining NOx emissions. We use the GEOS-Chem Adjoint model to perform assimilations using a variety of AMFs to examine how a posteriori emission estimates are affected if the AMF is calculated using an a priori shape factor that is inconsistent with the simulated profile. In these tests, an inconsistent a priori shape factor increased errors in a posteriori emissions estimates by up to 80 % over polluted regions. As the difference between the simulated profile shape and the a priori profile shape increases, so do the corresponding assimilated emission errors. This reveals the importance of using simulated profile information for AMF calculations when comparing that simulated output to satellite retrieved columns.

scholarly journals NUMERICAL SOLUTION OF TWO-DIMENSIONAL PROBLEMS OF THERMOVISCOELASTICITY

2021 ◽  
Vol 335 (1) ◽  
pp. 60-64
Author(s):  
M. Bukenov ◽  
Ye. Mukhametov
Keyword(s):  
A Priori ◽  
Elastic Collision ◽  
Qualitative Behavior ◽  
Stress Profile ◽  
Two Dimensional ◽  
A Priori Information ◽  
Deformable Solids ◽  
Methods Of Calculation ◽  
The Difference ◽  
Priori Information

This paper considers the numerical implementation of two-dimensional thermoviscoelastic waves. The elastic collision of an aluminum cylinder with a two-layer plate of aluminum and iron is considered. In work [1] the difference schemes and algorithm of their realization are given. The most complete reviews of the main methods of calculation of transients in deformable solids can be found in [2, 3, 4], which also indicates the need and importance of generalized studies on the comparative evaluation of different methods and identification of the areas of their most rational application. In the analysis and physical interpretation of numerical results in this work it is also useful to use a priori information about the qualitative behavior of the solution and all kinds of information about the physics of the phenomena under study. Here is the stage of evolution of contact resistance of collision – plate, stress profile.

scholarly journals Towards the retrieval of tropospheric ozone with the Ozone Monitoring Instrument (OMI)

2015 ◽  
Vol 8 (2) ◽  
pp. 671-687 ◽  
Author(s):  
T. Mielonen ◽  
J. F. de Haan ◽  
J. C. A. van Peet ◽  
M. Eremenko ◽  
J. P. Veefkind
Keyword(s):  
Radiative Transfer ◽  
Covariance Matrix ◽  
Tropospheric Ozone ◽  
Surface Albedo ◽  
A Priori ◽  
Stray Light ◽  
Retrieval Algorithm ◽  
A Priori Information ◽  
Ozone Profile ◽  
Priori Information

Abstract. We have assessed the sensitivity of the operational Ozone Monitoring Instrument (OMI) ozone profile retrieval algorithm to a number of a priori and radiative transfer assumptions. We studied the effect of stray light correction, surface albedo assumptions and a priori ozone profiles on the retrieved ozone profile. Then, we studied how to modify the algorithm to improve the retrieval of tropospheric ozone. We found that stray light corrections have a significant effect on the retrieved ozone profile but mainly at high altitudes. Surface albedo assumptions, on the other hand, have the largest impact at the lowest layers. Choice of an ozone profile climatology which is used as a priori information has small effects on the retrievals at all altitudes. However, the usage of climatological a priori covariance matrix has a significant effect. Based on these sensitivity tests, we made several modifications to the retrieval algorithm: the a priori ozone climatology was replaced with a new tropopause-dependent climatology, the a priori covariance matrix was calculated from the climatological ozone variability values, and the surface albedo was assumed to be linearly dependent on wavelength in the 311.5–330 nm channel. As expected, we found that the a priori covariance matrix basically defines the vertical distribution of degrees of freedom for a retrieval. Moreover, our case study over Europe showed that the modified version produced over 10% smaller ozone abundances in the troposphere which reduced the systematic overestimation of ozone in the retrieval algorithm and improved correspondence with Infrared Atmospheric Sounding Instrument (IASI) retrievals. The comparison with ozonesonde measurements over North America showed that the operational retrieval performed better in the upper troposphere/lower stratosphere (UTLS), whereas the modified version improved the retrievals in the lower troposphere and upper stratosphere. These comparisons showed that the systematic biases in the OMI ozone profile retrievals are not caused by the a priori information but by some still unidentified problem in the radiative transfer modelling. Instead, the a priori information pushes the systematically wrong ozone profiles towards the true values. The smaller weight of the a priori information in the modified retrieval leads to better visibility of tropospheric ozone structures, because it has a smaller tendency to damp the variability of the retrievals in the troposphere. In summary, the modified retrieval unmasks systematic problems in the radiative transfer/instrument model and is more sensitive to tropospheric ozone variation; that is, it is able to capture the tropospheric ozone morphology better.

scholarly journals A method to derive the Site Atmospheric State Best Estimate (SASBE) of ozone profiles from radiosonde and passive microwave data

2015 ◽  
Vol 8 (3) ◽  
pp. 3399-3422 ◽  
Author(s):  
E. Maillard Barras ◽  
A. Haefele ◽  
R. Stübi ◽  
D. Ruffieux
Keyword(s):  
Brightness Temperature ◽  
A Priori ◽  
A Priori Information ◽  
Data Set ◽  
Ozone Data ◽  
Best Estimate ◽  
Vertical Range ◽  
Ozone Profile ◽  
Atmospheric State ◽  
Priori Information

Abstract. We present a method to derive the site atmospheric state best estimate (SASBE) of the ozone profile combining brightness temperature spectra around the 142 GHz absorption line of ozone measured by the microwave radiometer SOMORA and ozone profiles measured by the radiosonde (RS). The SASBE ozone profile is obtained using the radiosonde ozone profile as a priori information in an optimal estimation retrieval of the SOMORA radiometer. The resulting ozone profile ranges from ground up to 65 km altitude and makes optimal use of the available information at each altitude. The high vertical resolution of the radiosonde profile can be conserved and the uncertainty of the SASBE is well defined at each altitude. A SASBE ozone profile dataset has been generated for Payerne, Switzerland, with a temporal resolution of 3 profiles a week for the time period ranging from 2011 to 2013. The relative difference of the SASBE ozone profiles to the AURA/MLS ozone profiles lies between −3 to 6% over the vertical range of 20–65 km. Above 20 km, the agreement between the SASBE and AURA/MLS ozone profiles is better than the agreement between the operational SOMORA ozone data set and AURA/MLS. Below 20 km the SASBE ozone data are identical to the radiosonde measurements. The same method has been applied to ECWMF-ERA interim ozone profiles and SOMORA data to generate a SASBE dataset with a time resolution of 4 profiles per day. These SASBE ozone profiles agree between −4 and +8% with AURA/MLS. The improved agreement of the SASBE datasets with AURA/MLS above 20 km demonstrates the benefit of better a priori information in the retrieval of ozone from brightness temperature data.

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