Generalized Adjoint for Physical Processes with Parameterized Discontinuities. Part I: Basic Issues and Heuristic Examples

Xu, Qin

doi:10.1175/1520-0469(1996)053<1123:gafppw>2.0.co;2

Cited by 95 publications

(48 citation statements)

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“…(9) and (12), time evolution of both variables takes the following unified form: (15) where dF (LS is the rate of large-scale advective change, and Ph is compensation due to physical processes. Xu (1996) pointed out quasiequilibrium (F=0, d=0) in a time-averaged sense when dF LS>0, This means that Fl (Fc) satisfies the following 2 quasiequilibrium condition: 1) Fl(Fc) is close to 0 in model stratiform (convective) precipitation areas.…”

Section: Jo Approximationmentioning

confidence: 99%

Direct Assimilation of Multichannel Microwave Brightness Temperatures and Impact on Mesoscale Numerical Weather Prediction over the TOGA COARE Domain

Aonashi¹,

Liu

1999

Journal of the Meteorological Society of Japan

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Direct assimilation using 1-dimensional variational method in the vertical (1D-VAR) was developed to incorporate vertically polarized brightness temperatures (TB's), and rain flag data (index of existence of precipitation) from the special sensor microwave imager (SSM/I), into a mesoscale numerical weather prediction (NWP) model. We used a radiative transfer model (RTM) developed by Liu (1998) to calculate TB's from NWP model variables. Observational residuals of TB's are assumed to be nonlinear functions of precipitation rates in rainy areas, and of other thermodynamic variables in rain-free areas. Quasiequilibrium assumptions on humidity and convective instability in model precipitation areas are used to assimilate TB's in rainy areas and rain flag data, which are functions of precipitation. TB's in rain-free areas are assimilated into total water content (cloud water content + humidity). Smith's method (1990) is used to calculate cloud water content and humidity from total water content. To simplify 1D-VAR, the following approximations are introduced: 1) In the tropics, variations in temperature are much smaller than variations in humidity.2) Variations in divergence dominates the rainy areas, compared to vorticity and mass variables. Based on the above approximation, we assumed that the background error covariance of relative vorticity and the unbalanced component of mass variables is negligible, so we divided 1D-VAR into that for total water content and that for divergence. Newtonian iteration is used to solve these 1D-VAR problems.To study the assimilated variables, assimilation was applied to cases during 19-20 Dec. 1992 over the Tropical Ocean Global Atmosphere Coupled Ocean-Atmosphere Response Experiment (TOGA COARE) domain. Results indicate: 1) Precipitable water content (PWC) obtained by assimilation agreed well with PWC calculated from radiosonde data and PWC retrieved from TB's using Shibata's algorithm (1994).2) Consistency between humidity and cloud water content profiles was attained by Smith's method (1990), although no significant improvement was seen in humidity profile accuracy by assimilation. 3) Assimilation of rain flag and TB's in rainy areas successfully produced model precipitation areas agreeing with radar observation data by Short et al. (1997).To test the impact of assimilation on NWP forecasts, experiments were conducted assimilating TB data for 19 Dec. 1992. Results indicate: 1) Assimilation modified large-scale model humidity distribution, improving large-scale humidity and precipitation forecasts for 48 hours.

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Section: Jo Approximationmentioning

confidence: 99%

Direct Assimilation of Multichannel Microwave Brightness Temperatures and Impact on Mesoscale Numerical Weather Prediction over the TOGA COARE Domain

Aonashi¹,

Liu

1999

Journal of the Meteorological Society of Japan

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“…Unfortunately, due to its strong dependence on the adjoint model of the forecast model (Bormann and Thepaut, 2004;Park and Zou, 2004;Caya et al, 2005;Bauer et al, 2006;Rosmond and Xu, 2006;Gauthier et al, 2007), computer implementation of 4DVar is often expensive and not easy to program because of the need to maintain and update the adjoint model of the forecast model. The situation is even worse when the forecast model is highly non-linear and/or the model physics involves discontinuous parameters (Xu, 1996). Furthermore, the applications of the static, thus flowindependent, background-error covariance in the traditional 4DVar often causes its poor assimilation performance (Beck and Ehrendorfer, 2005;Zhang et al, 2009;Cheng et al, 2010).…”

Section: Introductionmentioning

confidence: 99%

A non-linear least squares enhanced POD-4DVar algorithm for data assimilation

Tian

Feng

2015

Tellus A: Dynamic Meteorology and Oceanography

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A B S T R A C T This paper presents a novel non-linear least squares enhanced proper orthogonal decomposition (POD)-based 4DVar algorithm (referred as NLS-4DVar) for the non-linear ensemble-based 4DVar. In the algorithm, the GaussÁNewton iterative method is employed to handle the non-quadratic non-linearity of the 4DVar cost function while the overall structure of the algorithm still resembles the original POD-4DVar algorithm. It is proved that the original POD-4DVar algorithm is a special case of the proposed NLS-4DVar algorithm under the assumption of the linear relationship between the model perturbations (MPs) and the simulated observation perturbations (OPs). Under the assumption it is also shown that the solution of POD-4DVar algorithm coincides with the solution of the proposed NLS-4DVar algorithm. On the contrary, if the linear relationship assumption is dropped, the solution of the POD-4DVar algorithm is only the first iteration of the proposed NLS-4DVar algorithm. As a result, our analysis provides an explanation for the degraded and inaccurate performance of the POD-4DVar algorithm when the underlying forecast model or (and) the observation operator is strongly non-linear. The potential merits and advantages of the proposed NLS-4DVar are demonstrated by a group of Observing System Simulation Experiments with Advanced Research WRF (ARW) using accumulated rainfall-observations.

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“…But whether discontinuous "on-off" switches impact the existence of the CF gradient has been investigated only in the case of single grid point model characterized by an ordinary differential equation (Xu, 1996(Xu, , 1997Zou, 1997;Mu and Wang, 2003). In the case of partial differential equations, the numerical experiments performed by Mu and Zheng (2005) show that traditional numerical treatment at the switches would result in terrible CF zigzags (see Fig.…”

Section: The Existence Of the Gradient Of The Cf With Respect To Icmentioning

confidence: 99%

“…Zou et al, 1993;Verlinde and Cotton, 1993;Zupanski, 1993;Bao and Warner, 1993;Zupanski and Mesinger, 1995;Zou, 1997;Xu, 1996Xu, , 1997Xu, , 1998Xu, , 1999Mu and Wang, 2003;Mu and Zheng, 2005). Due to the difficulties of the involved problems, attentions are mostly paid to simplified theoretical "on-off" models: from an ordinary differential equation describing the evolution of specific humidity on one grid point to a partial differential equation describing the evolution of specific humidity on a vertical or horizontal interval (Xu, 1996(Xu, , 1997(Xu, , 1998Zou, 1997;Mu and Wang, 2003;Mu and Zheng, 2005). The researches on the idealized simple models demonstrate that when discontinuous "on-off" switches occur in the governing equation, the effects of the VDA depend upon two facts: one is the simple discretization at the switches of the forward model, which usually induces the associated discrete cost function (CF) to depend discontinuously on the initial condition (IC) and the numerical solution of the forward model to contain the zigzag oscillations (Xu, 1997;Mu and Zheng, 2005).…”

Section: Introductionmentioning

confidence: 99%

The effects of the model errors generated by discretization of "on-off'' processes on VDA

2006

Nonlin. Processes Geophys.

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Abstract. Through an idealized model of a partial differential equation with discontinuous "on-off" switches in the forcing term, we investigate the effect of the model error generated by the traditional discretization of discontinuous physical "on-off" processes on the variational data assimilation (VDA) in detail. Meanwhile, the validity of the adjoint approach in the VDA with "on-off" switches is also examined. The theoretical analyses illustrate that in the analytic case, the gradient of the associated cost function (CF) with respect to an initial condition (IC) exists provided that the IC does not trigger the threshold condition. But in the discrete case, if the on switches (or off switches) in the forward model are straightforwardly assigned the nearest time level after the threshold condition is (or is not) exceeded as the usual treatment, the discrete CF gradients (even the one-sided gradient of CF) with respect to some ICs do not exist due to the model error, which is the difference between the analytic and numerical solutions to the governing equation. Besides, the solution of the corresponding tangent linear model (TLM) obtained by the conventional approach would not be a good first-order linear approximation to the nonlinear perturbation solution of the governing equation. Consequently, the validity of the adjoint approach in VDA with parameterized physical processes could not be guaranteed. Identical twin numerical experiments are conducted to illustrate the influences of these problems on VDA when using adjoint method. The results show that the VDA outcome is quite sensitive to the first guess of the IC, and the minimization processes in the optimization algorithm often fail to converge and poor optimization retrievals would be generated as well. Furthermore, the intermediate interpolation treatment at the switch times of the forward model, which reduces greatly the model error brought by the traditional discretization of "onoff" processes, is employed in this study to demonstrate that Correspondence to: Q. Zheng (qinzheng@mail.iap.ac.cn) when the "on-off" switches in governing equations are properly numerically treated, the validity of the adjoint approach in VDA with discontinuous physical "on-off" processes can still be guaranteed.

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Generalized Adjoint for Physical Processes with Parameterized Discontinuities. Part I: Basic Issues and Heuristic Examples

Cited by 95 publications

References 0 publications

Direct Assimilation of Multichannel Microwave Brightness Temperatures and Impact on Mesoscale Numerical Weather Prediction over the TOGA COARE Domain

Direct Assimilation of Multichannel Microwave Brightness Temperatures and Impact on Mesoscale Numerical Weather Prediction over the TOGA COARE Domain

A non-linear least squares enhanced POD-4DVar algorithm for data assimilation

The effects of the model errors generated by discretization of "on-off'' processes on VDA

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Generalized Adjoint for Physical Processes with Parameterized Discontinuities. Part I: Basic Issues and Heuristic Examples

Cited by 95 publications

References 0 publications

Direct Assimilation of Multichannel Microwave Brightness Temperatures and Impact on Mesoscale Numerical Weather Prediction over the TOGA COARE Domain

Direct Assimilation of Multichannel Microwave Brightness Temperatures and Impact on Mesoscale Numerical Weather Prediction over the TOGA COARE Domain

A non-linear least squares enhanced POD-4DVar algorithm for data assimilation

The effects of the model errors generated by discretization of &quot;on-off'' processes on VDA

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The effects of the model errors generated by discretization of "on-off'' processes on VDA