The GCM Validation

In this chapter the observed and simulated (by the Hamburg GCM) Northern Hemisphere monthly surface air temperatures, averaged within different latitude bands, are statistically analyzed and compared. The objects used for the analysis are the two-dimensional spatial-temporal spectral and correlation characteristics, the multivariate autoregressive and linear regression model parameters, and the diffusion equation coefficients. A comparison shows that, generally, the shapes of the corresponding spectra and correlation functions are quite similar, but their numerical values and some features differ markedly, especially for the tropical regions. The spectra reveal a few randomly distributed maxima (along the frequency axis), the periods of which were not identical for both types of data. A comparative study of the estimates of the diffusion equation coefficients shows a significant distinction between the character of the meridional circulations of the observed and simulated systems. The approach developed gives approximate stochastic models and reasonable descriptions of the temperature processes and fields, thereby providing an opportunity for solving some of the vital problems of theoretical and practical aspects surrounding validation, diagnosis, and application of the GCM. The methodology and results presented make it clear that formalization of the statistical description of the surface air temperature fluctuations can be achieved by applying the standard techniques of multivariate modeling and multidimensional spectral and correlation analysis to the data which have been averaged spatially and temporally. The idea of the statistical approach to the problems of GCM variability validation is contained in the comparison (observed vs. modeled) of the probability distributions of the different atmospheric and ocean processes and fields. At first, such a statement sounds like a standard statistical approach, and its solution would be obvious and simple if the number of climate processes taking place jointly were not huge and if they did not present a tremendously complicated (in its interrelationships and feedbacks) deterministic-stochastic system. As is known, the Stochastic System Identification Theory (see Eikhoff, 1983) deals mostly with the methodology for identifying linear systems, The interdependences of climatic processes and fields are not linear, and the application of this theory can give only highly approximate results.

Random Processes and Fields

In this chapter, the nonparametric methods of estimating the spectra and correlation functions of stationary processes and homogeneous fields are considered. It is assumed that the principal concepts and definitions of the corresponding theory are known (see Anderson, 1971; Box and Jenkins, 1976; Jenkins and Watts, 1968; Kendall and Stuart, 1967; Loeve, 1960; Parzen, 1966; Yaglom, 1986); therefore, only questions connected with the construction of numerical algorithms are studied. The basic results ranged from univariate process to multidimensional field are presented in Tables 3.1 and 3.2. These formulas make it possible to compare and trace the formal character of developing estimation procedures when the dimensionality is increasing. The schemes in these tables, as well as the formulas in the previous chapters, can be used for software development without any rearrangement. In part, this approach presents the application of the methods of Chapters 1 and 2 in evaluating random function characteristics. Of course, the final identification of the algorithm parameters (for example, the spectral window widths) can be made only through trial and error and by taking into account the character of the problem under study, that is, the physical properties of the processes and fields observed. The last section of this chapter presents results of the application of these methods to the analysis of some climatological fields. Here the basic results of the univariate spectral analysis are briefly discussed in order to develop algorithms for a multidimensional case by analogous reasoning. The complete description of the estimation procedures of the spectral and correlation analysis for univariate stationary process can be found, for example, in Jenkins and Watts, 1968.

Averaging and Simple Models

Simple linear procedures for processing correlated observations are considered and interpreted in this chapter. Primarily, they present different schemes for averaging data. These procedures are important because climatology has historically dealt with spatial and temporal averaging of statistically dependent meteorological observations. The accuracy of such averaging is determined by the volume of data arid by its correlation structure. The examples presented in this chapter illustrate the level of accuracy that can be achieved within the framework of some assumptions about such correlation structure. Let us consider the principal relationships of the least squares method for the statistically dependent observations (see Rao, 1973). The basic assumptions are as follows.

Digital Filters

In this chapter, several systems of digital filters are presented. The first system consists of regressive smoothing filters, which are a direct consequence of the least squares polynomial approximation to equally spaced observations. Descriptions of some particular univariate cases of these filters have been published and applied (see, for example, Anderson, 1971; Berezin and Zhidkov, 1965; Kendall and Stuart, 1963; Lanczos, 1956), but the study presented in this chapter is more general, more elaborate in detail, and more fully illustrated. It gives exhaustive information about classical smoothing, differentiating, one- and two-dimensional filtering schemes with their representation in the spaces of time, lags, and frequencies. The results are presented in the form of algorithms, which can be directly used for software development as well as for theoretical analysis of their accuracy in the design of an experiment. The second system consists of harmonic filters, which are a direct consequence of a Fourier approximation of the observations. These filters are widely used in the spectral and correlation analysis of time series. The foundation for developing regressive filters is the least squares polynomial approximation (of equally spaced observations), a principal notion that will be considered briefly.

Second Moments of Rain

The first part of this chapter presents a description of the GATE rain rate data (Polyak and North, 1995), its two-dimensional spectral and correlation characteristics, and multivariate models. Such descriptions have made it possible to show the concentration of significant power along the frequency axis in the spatial-temporal spectra; to detect a diurnal cycle (a range of variation of which is about 3.4 to 5.4 mm/hr); to study the anisotropy (as the result of the distinction between the north-south and east-west transport of rain) of spatial rain rate fields; to evaluate the scales of the distinction between second-moment estimates associated with ground and satellite samples; to determine the appropriate spatial and temporal scales of the simple linear stochastic models fitted to averaged rain rate fields; and to evaluate the mean advection velocity of the rain rate fluctuations. The second part of this chapter (adapted from Polyak et al., 1994) is mainly devoted to the diffusion of rainfall (from PRE-STORM experiment) by associating the multivariate autoregressive model parameters and the diffusion equation coefficients. This analysis led to the use of rain data to estimate rain advection velocity as well as other coefficients of the diffusion equation of the corresponding field. The results obtained can be used in the ground truth problem for TRMM (Tropical Rainfall Measuring Mission) satellite observations, for comparison with corresponding estimates of other sources of data (TOGA-COARE, or simulated by physical, models), for generating multiple rain samples of any size, and in some other areas of rain data analysis and modeling. For many years, the GATE data base has served as the richest and most accurate source of rain observations. Dozens of articles presenting the results of the GATE rain rate data analysis and modeling have been published, and more continue to be released. Recently, a new, valuable set of rain data was produced as a result of the TOGA-COARE experiment. In a few years, it will be possible to obtain satellite (TRMM) rain information, and a rain statistical description will be needed in the analysis of the observations obtained on an irregular spatial and temporal grid.

Historical Records

In this chapter, the historical records of annual surface air temperature, pressure, and precipitation with the longest observational time series will be studied. The analysis of the statistically significant systematic variations, as well as random fluctuations of such records, provides important empirical information for climate change studies or for statistical modeling and long-range climate forecasting. Of course, compared with the possible temporal scales of climatic variations, the interval of instrumental observations of meteorological elements proves to be very small. For this reason, in spite of the great value of such records, they basically characterize the climatic features of a particular interval of instrumental observations, and only some statistics, obtained with their aid, can have more general meaning. Because each annual or monthly value of such records is obtained by averaging a large number of daily observations, the corresponding central limit theorem of the probability theory can guarantee their approximate normality. In spite of this, we computed the sample distribution functions for each time series analyzed below and evaluated their closeness to the normal distribution by the Kolmogorov- Smirnov criterion. As expected, the probability of the hypothesis that each of the climatic time series (annual or monthly) has a normal distribution is equal to one with three or four zeros after the decimal point. As seen in this section, the straight line least squares approximation of the climatic time series enables us to obtain very simple and easy-to-interpret information about the power of the long period climate variability. Carrying out such an approximation, we assume that the fluctuation with a period several times greater than the observational interval will become apparent as a gradual increase or decrease of the observed values. Using only a small sample, it is impossible to determine accurately the amplitude and frequency of such long-period climate fluctuation. Consequently, the straight-line model is the simplest approach in this case. Let us begin with an analysis of the annual surface air temperature time series, the observations of which are published in Bider et al., (1959), Bider and Schiiepp, (1961), Lebrijn (1954), Manlcy (1974), and in the World Weather Records (1975).

Multivariate AR Processes

This chapter is devoted to different computational aspects of multivariate modeling. An algorithm for fitting such models with sequential increasing of the order is given. Several examples of approximation of multiple climatological time series by first-order AR models are considered. It is emphasized that such modeling can be used not only for forecasting, but also for the analysis of the parameter matrix as a matrix of the interactions and feedbacks of the observed processes. Some problems in identifying stochastic climate models are also discussed. It is shown that formulation of climatology problems within the strict frameworks of fundamental theory will facilitate natural progress along with the development of these methods.

Variability of ARMA Processes

In this chapter, the numerical and pictorial interpretation of the dependence of the standard deviation of the forecast error for the different types and orders of univariate autoregressive-moving average (ARMA) processes on the lead time and on the autocorrelations (in the domains of the permissible autocorrelations) are given. While the convenience of fitting a stochastic model enables us to estimate its accuracy for the only time series under consideration, the graphs in this chapter demonstrate such accuracy for all possible models of the first and second order. Such a study can help in evaluating the appropriateness of the presupposed model, in earring out the model identification procedure, in designing an experiment, and in optimally organizing computations (or electing not to do so). A priori knowledge of the theoretical values of a forecast’s accuracy indicates the reasonable limits of complicating the model and facilitates evaluation of the consequences of certain preliminary decisions concerning its application. The approach applied is similar to the methodology developed in Chapters 1 and 2. Because the linear process theory has been thoroughly discussed in the statistical literature (see, for example, Box and Jenkins, 1976; Kashyap and Rao, 1976; and so on), its principal concepts are presented in recipe form with the minimum of details necessary for understanding the computational aspects of the subject. Consider a discrete stationary random process zt with null expected value [E(zt) = 0] and autocovariance function . . . M(T) = σ2 ρ(T), (4.1) . . . where σ2 is the variance and ρ(T) is the autocorrelation function of zt. Let at be a discrete white noise process with a zero mean and a variance σ2a. Let us assume that processes zt and at are normally distributed and that their cross-covariance function Mza(T) = 0 if T > 0.