# Using Linear Inverse Modeling

## Using Linear Inverse Modeling to Systematize Model Improvement

Deterministic prediction is limited by error growth but statistical prediction can actually exploit error growth to obtain statistical quantities. This is so because highly nonlinear terms in the dynamical equations scatter energy rapidly among system structures producing rapid decorrelation and a broad continuous temporal spectrum of fluctuations. It follows that the statistics of the climate can be captured to a good approximation by linear stochastic analysis. The dynamics of the climate state, φ, are approximately governed by the linear forward model (LFM):

/dt = Αφ + Fε(t)

where Α is the linearized operator and Fε(t) is a stochastic forcing accounting for the fast nonlinear processes (Farrell and Ioannou, 1996). Climate statistics are predicted with surprising accuracy by this linear stochastic dynamics (see for example Farrell and Ioannou, 1995, and Zhang and Held, 1999); see also Whitaker and Sardeshmukh (1998). An important advantage proceeding from the fact that the climate is well modeled by a linear stochastic system is that it is accessible to understanding using powerful linear analysis methods. A recent application of this theory led to a comprehensive assessment of the sensitivity of perturbation and the structural stability of the midlatitude jet (Farrell and Ioannou, 2002). The accurate approximation to statistical dynamics provided by such linear stochastic systems motivates the attempt to obtain approximations to the above equation from observations.

Related to linear stochastic modeling is Linear Inverse Modeling (LIM); a method for extracting the intrinsic linear dynamics that govern the climatology of a complex system directly from observations of the system. Practically, LIM provides an accurate approximate dynamics that becomes more exact as the system becomes more chaotic whether this chaos arises internally from turbulence or externally from unknown disturbances with short spatial and temporal correlation in comparison with the system dynamics. That the climatology of complex and turbulent atmospheric flows is amenable to such a reduction was examined by DelSole and Farrell (1996), who were able to extract from nonlinear simulations of a two-layer baroclinic channel flow the effective linear operator that was controlling the climatology of the channel flow. DelSole and Hou (1999) extended these methods to demonstrate the accuracy of LIM in inverting for the climatology of a full hemispheric GCM. Additionally, successful modeling of the intraseasonal variability of the midlatitude atmosphere in the traditional variables of velocity and temperature and of ENSO dynamics in the SST have been achieved with these methods (Farrell and Ioannou, 1995; Penland and Magorian, 1993). Of particular interest to us is that the LIM method is not confined to traditional variables such as velocity and temperature; it can be applied to obtain intrinsic dynamics of a system in any variable. The advantage of this approach is that it provides information on the primary structures of the system response and their temporal variability together with a link among the structures in the form of a dynamical system. We believe that analysis of the intrinsic dynamics of the radiance spectrum captures the maximum practically obtainable information on the spectra and their temporal variability. Comparison among models and between models and data can be sensitively made by contrasting their spectra both in structure and dynamics using this method.

Identifying model error and improving model parameterizations is a very difficult task to accomplish by appealing to physical arguments and first principle reasoning alone, as the variety of cumulus parameterizations testifies. Continued progress in model refinement requires developing methods to systematize parameterization improvement. The success of LIM reveals that inverting observations for dynamics is a powerful methodology compared with approximating dynamics of complex processes from first principles. This is particularly true when, as in the case of clouds and precipitation, the first principles are not themselves well known.