In order to successfully predict the weather using a numerical model (a set of algorithms that predict future conditions based upon current and possibly past conditions), one must first provide that model with an accurate portrayal of atmospheric conditions. This process, termed model initialization, provides a starting point from which the relevant equations can be integrated forward in time and thus from which weather predictions based on numerical models are produced. A cornerstone of all model initialization techniques is the utilization of observations. (Obviously, if we want to predict future atmospheric conditions, we should base those predictions on current and past observations.) When utilizing observations, one typically determines what they infer concerning the atmosphere on the model grid. (The model grid is the set of points, in space and time, where prediction values are computed.) Information concerning the atmosphere is generally needed on the model grid in order to start the prediction process.

The techniques that are used to infer atmospheric conditions on the model grid generally depend (at least at some point) upon a process called objective analysis. Objective analysis serves two functions: 1) it provides estimates of fields at locations and times other than those of the observations and 2) it filters the input [i.e., it removes a portion of the input (hopefully noise) and retains other portions of the input (hopefully signal)]. The first function is of practical importance since estimates are required on the model grid, the points of which do not generally correspond with observation points. The second function is also of practical importance. If the second function is performed poorly, then the likelihood of an accurate forecast is significantly diminished. Consequently, it is critically important that filtering properties be taken into consideration when objective analysis techniques are designed.

Until recently, the determination of the exact filtering properties of objective analysis techniques has only been feasible for special situations. Unfortunately, in many sciences (like the atmospheric sciences) these special situations are either rare or nonexistent. Because scientists have had to rely quite heavily on knowledge concerning filtering properties for these special situations, they have often designed filters whose properties not only differ, but often differ greatly, from the desired properties. The result is inferior analyses (owing to lack of techniques, not effort!) that subsequently hinder our ability to both diagnose current atmospheric conditions and to predict future conditions.

The problem that plagues many objective analysis techniques is the determination of their filtering properties in the most general (and common, in the atmospheric sciences) situation of discrete, irregularly-spaced data (data that are collected at specific locations and times and that are not separated by equal distances or time intervals). A fair amount of work has been expended towards this goal. In fact, scientists made significant progress in this area in the early 1970s, but were seemingly distracted away from this problem by other issues. Recently, I have pursued this goal and have discovered a method that provides the exact filtering properties for any objective analysis technique (as long as that technique can be expressed in terms of a generalized framework) and for any data distribution. Not only is this immediately valuable in the diagnostic sense (e.g., for the evaluation of filtering properties of an objective analysis technique), it may be useful in designing adaptive objective analysis techniques that produce, as nearly as possible, desired filtering properties.

An illustration of this new method for determining filtering properties is provided in the Fig. 1. The caption for this figure refers to response functions. These are simply functions that convey the filtering properties of an objective analysis scheme: if one knows a scheme's response function then one knows what that scheme does to the input and, thus, the scheme's filtering properties. The data illustrated in Fig. 1 are from an experiment in which observations were obtained from the dotted line in Fig. 1a at the locations marked with the small arrows. [As is apparent from Fig. 1a, the observations are both discrete (at particular locations instead of everywhere) and irregularly-distributed.] Then, analysis values were produced using two methods: 1) via an objective analysis scheme (denoted by the solid line in Fig. 1a), and 2) via the response function (denoted by the plus symbols in Fig. 1a). If the generalized response function that I obtained is correct, then the actual and response-function predicted values should coincide, which they do. This test verifies the veracity of the generalized response function.

The amplitude and phase modulations of the objective analysis scheme, as determined using the generalized response function, are illustrated in Fig. 1b. These values indicate the degree to which the input wave was damped (amplitude modulation) and shifted (phase modulation). When filtering data, one typically wishes to diminish the intensity of some waves while retaining their phase. If their phases are changed (which is indicated by a nonzero phase modulation), then in effect the input waves have been moved! If an accurate portrayal of current atmospheric conditions is desired, then one certainly would not want to move the observations around. Consequently, the information provided in Fig. 1b can be used to evaluate the quality of an objective analysis scheme.

With the development of the generalized response function, other research avenues have now opened. One important avenue is the utilization of the generalized response function in improving objective analysis techniques. It is expected that this will help to improve the initial conditions that are provided to numerical weather prediction models and will thus ultimately assist in enhancing weather forecasts.

 

 

 

 

 

 

 

 

 

FIG. 1. (a) The input field (dotted line), observations (diamond symbols), and actual (solid line) and response-function predicted (plus-sign symbols) analysis fields for an objective analysis of one-dimensional, discrete, irregularly-distributed data. (b) The amplitude (solid line, left axis) and phase (thin-dashed line, right axis) modulation functions and the ideal response function (dotted line, left axis; assumes infinite and continuous data) for the test shown in (a). In both (a) and (b) the thick-dashed lines indicate the limits of possible observation locations and the arrows denote actual observation locations. The observational field is given by , with , and .