For the deep convective storm studied here, the tangent linear solutions, which describe the evolution of perturbations along trajectories of a time-dependent nonlinear base state, represent the corresponding nonlinear perturbation fields very accurately up to about 50 min for a 1% moisture perturbation. Considering the highly nonlinear and discontinuous properties of solutions in a full-physics nonhydrostatic cloud model such as ARPS, these results are encouraging for future studies of storm predictability, data assimilation, Doppler radar retrieval, and ensemble forecasting, all of which require derivative or sensitivity information.
In the supercell simulation, bias-type errors in the water vapor in different regions of the model exert influences on storm dynamics in different ways. Perturbations introduced inside the rain region above cloud base mostly affect the secondary cells, while those outside the rain region mainly influence the main storm. When the perturbations are introduced in the subcloud layer, both the main and secondary cells are affected. Among the vapor perturbations in different regions, the perturbations inside the rain region have the largest influence on storm dynamics.
These results imply that we may need high-quality vapor data from either observations or retrievals in order to obtain accurate predictions of storm behavior. The required accuracy of water vapor can be estimated once the criteria on the forecast accuracy is determined. For example, suppose that a relative sensitivity of the forecast error [15] to water vapor is 20, which implies that the forecast error changes by 20% as a result of a 1% error in water vapor. If one wishes a forecast with only a 10% error, the observation for water vapor should have an error smaller than 0.5%.
For perturbations inside the rain region, the cost function showed the largest sensitivity with respect to temperature, followed by pressure and then water vapor. For perturbations in ambient environment, the cost function showed the largest sensitivity to temperature, followed by water vapor and then pressure. All other variables have almost negligible effect on the cost function. This result is also demonstrated in our 1-D experiments [16].
When applied to variational data assimilation, sensitivity information, especially derivatives of the cost function with respect to all initial fields, can indicate which initial field must be modified by a large amount and which may be altered by only a small amount to change a specific amount of cost function on the way to its minimum state. With this information, the minimization algorithm can be appled in a selective way to save computing times: that is, a variable that exerts little influence on the cost function may be put in the minimization process in a larger iteration step, while a variable with strong effect (especially temperature) may be applied in every step.
Even though ADIFOR does not produce the adjoint, it gives more information than handcoded tangent linear or adjoint models. In our experience, an AD tool dispenses with much labor and time in handcoding the adjoint model, yet provides a great amount of gradient information needed for sensitivity analysis and data assimilation. Compared with the divided-difference approach, AD avoids the difficulty of choosing an optimal perturbation size, to which the solutions of cloud model are extremely sensitive, and also saves a great amount of computing time by avoiding numerous runs with full numerical model. The ADIFOR-generated code is especially efficient and useful for investigating how a perturbation inserted at any given intermediate time propagates through the model variables at later times. Furthermore, it is demonstrated that automatic differentiation can be applied with no problem to a compressible model using mode-splitting time integration techniques.
In the context of data
assimilation especially for 3-D models, however, we note that it is
computationally impractical to compute sensitivities with respect to
all model grid variables through the ADIFOR-generated code, mainly
because of
memory limitations. For example, the nonlinear ARPS with 
grids requires about 9.5 MWords on a Cray-C90, while that machine in
the Pittsburgh Supercomputing Center has a maximum memory of 512
MWords. Therefore, the maximum number of IVs that can be computed
through the SE-ARPS is only about 50. In data assimilation and Doppler
radar retrieval, we usually require the gradient information of the
cost function with respect to all model grid variables, which
constitutes 98315 IVs for only one forecast aspect in our case.
Furthermore, for the purpose of data assimilation, the ADIFOR-generated code is computationally very expensive compared with the pure adjoint model. The reason is that the former is basically a forward model and thus repeats the sensitivity computation implicitly for the number of IVs. Although we may save computing time by applying the sparse matrix option in generating the derivative codes and by using the pseudo-adjoint technique [6], a comparative study has not been performed yet for a 3-D model.