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Oleg M. Pokrovsky
Main Geophysical Observatory, Karbyshev str.7, St. Petersburg, 194021 Russia
E-mail: pokrov@main.mgo.rssi.ru
The opportunity of a fuzzy set approach for the study of joint diurnal distribution of principal-coupled atmosphere-land variables (temperature of air and soil, humidity of air and soil, atmospheric precipitation, pressure, short-wave radiation and cloudiness) is investigated. Ten years of data sets of 1-hour temporal resolution for several meteorological sites of the Russian northwest region are used. Known and novel interrelationships between various atmospheric and linked soil variables are reviewed. The revealed relationships between solar downward radiation fluxes and soil temperature diurnal patterns allow the simulation of all principal elements of surface energy exchange: long-wave outgoing radiation fluxes in the atmosphere, soil heat fluxes, sensible and turbulent and latent heat fluxes. It was found that the most complicated links take place in fuzzy sets, corresponding to fractional cloudiness diurnal patterns. There is an asymmetrical relationship between solar daily sums of radiance for half cloudy days and mean soil temperature values. That is a main cause for simulation of meteorological and heat balance component diurnal cycles in most ecological models. Introduced stationary and transition modes for main meteorological variable diurnal patterns represent the background for simulation of all known weather phenomena. These modes also are used as a neural network’s nodes for hidden layers. Implementation of neural networks (back propagation algorithm) allows it to perform several modeling experiments. The problem of optimum network configuration (number of nodes in hidden layers) is discussed. For example, it is possible to reconstruct the diurnal cycles of some meteorological variables with sufficient (at the observational error levels) precision for each of them, by using 1, 2, or 3 instantaneous meteorological observations. Adapted non-linear mapping of one group of variables (and its diurnal distributions) on another allow one to investigate the possibility of application of this approach for diagnostic purposes. Potential applications of the developed approach are: regional models; nested models and downscaling in global models; spatial field reconstruction in the case of missing observational data (for some variable(s)); parameterization of energy exchange processes at the surface of the Earth.
Various approaches and methods of simulation of meteorological processes at the surface of the Earth have been actively developed in recent years (e.g., Randall et al., 1991). The principal stimulus for this research is linked with activation of regional and global climate change investigations, and also of complicated processes occurring in ecosystems (Kondratyev et al., 1996, Pokrovsky, 1995, Pokrovsky, 1996).
The headstone for such research is the study and simulation of a diurnal cycle of basic near-surface meteorological parameters (air and soil temperature, humidity, pressure, precipitation, wind, and cloud cover). In connection with necessity of description of energy exchange processes it is essential to consider also radiometric parameters (downward, reflected and net SW radiation, albedo, outgoing LW radiation) (Cherveny et al., 1991, Kubota, 1991). Throughout many years, experts investigated specific features of a diurnal cycle of these parameters on the base of empirical data (Cherveny and Belling, 1992; Williams 1992). Some examples of classification of diurnal or daily cloud cover distributions are represented in the literature (Berlyand, 1961). For many years the given problem for atmospheric temperature and humidity was traditionally considered at a qualitative level. We mention some publications of last time (Mass et al., 1991; Ruschy and Baker 1991). The examples of a classification of radiation characteristics at the surface of the Earth, mainly determined by distribution of a cloud cover, are presented in two works (Flach, 1951; Flach, 1952). In the last few years, activity at study of a diurnal cycle of precipitation amount is exhibited (Haldar et al, 1991). It has allowed the authors to determine basic types of distributions and to make definite qualitative conclusions. Unfortunately, the majority of research in this area is connected with study of a diurnal cycle of separate meteorological parameters; however, for simulation of complicated processes taking place at the surface of the Earth, characterised by natural topography, it is necessary to investigate joint distribution of several parameters simultaneously.
Many model studies (see review: Betts et al., 1996) have shown the importance of the surface boundary condition over land in controlling weather and climate processes of different time-scales. The radiation fluxes over land as well as soil temperature and soil moisture are not routinely observed or are not used as input boundary conditions in global forecast models. Instead, they are evaluated from a full system of parameterized equations at the surface, involving the surface radiation budget, strongly influenced by solar diurnal cycle and poor model representation of the cloud field. The impact of land surface processes in general circulation models has been reviewed by Garratt (1993). The solar terms are dominant in radiation budget, at least during daytime and in summer. Reflection and absorption by clouds and aerosol reduce the incoming radiation at the surface. It is well known that many models underestimate the atmospheric absorption and reflection of solar energy and, as a result, overestimate the incoming short-wave radiation at the surface (Garratt et al. 1993). The excessive net radiation at the surface leads to excessive surface evaporation and dried-out soil moisture. The mentioned study concerns the daily mean values; however, it is of interest to get insight into the deviation statistics for diurnal resolved values because of substantial daily amplitude of main meteorological parameters.
The author believes in order to solve this problem it is necessary to refuse from conventional methods of statistical analysis (calculations of moment, correlation functions, regression analysis and etc.), which ensure exposition only of linear relations. At the same time, most meteorological processes and relationships should be referred (by their nature) to non-linear. The final goal is to describe these non-linear processes by modern mathematical methods. The first step on a path should be the classification of basic meteorological parameter diurnal patterns by means of fuzzy logic (Jain and Dubes, 1988; Kohonen, 1989). Their diurnal cycles should be the object of the first phase of research. It is then necessary to proceed to classification of synoptic variability, not only of single meteorological parameters, but also of their joint distribution to study their actual feedback. All mentioned types of patterns are considered as state vectors in multidimensional space. Principal method issues are the revealing of its fuzzy sets. We suggest that most of the weather phenomena could be simulated by sequences of its conservation inside some fuzzy set, and transition from one fuzzy set to another. The more fuzzy sets are used, the more wide set weather phenomena could be described by this approach. Important key points here are the development of the transition rules.
In the first section of the paper, the description of the used cluster analysis method is presented. The problem of classification of diurnal cycles of major meteorological parameters under condition of their statistical sample independence is further considered. In the third section, the most actual problem - the study of joint probability distribution of several meteorological parameters for synchronized periods of observations and appropriate classification of linked diurnal cycles is considered. In the final section, research on classification of joint distribution of meteorological parameters on a time scale equal to 2 days is presented.
In meteorology, much attention is given to the analysis of statistical relations between separate parameters. For study of their space and temporal distributions it is necessary to use methods of a multidimensional statistical analysis. One of the important directions is a combination of cluster and discrimination analysis. In the given research, diurnal distributions of meteorological parameters will be considered; however, the offered approach can be applied to distributions of other temporal scales (monthly, seasonal, annual, etc.).
We assume that it is necessary to investigate a time scale, containing p - references (for example, instants of observations), and we arrange n - realisations of measurements in an indicated time interval. In this case, measurement data can be presented as a matrix:
X = 
= (X1, X2,..., Xn). (1)
Each realisation of the investigated process, represented by the vector - column Xj,(j=1,..., n), can be interpreted as a point in Euclidean multidimensional space of size p. Using notation (1), it is possible to introduce several characteristics, describing scattering of data inside classes and a distance between classes. We define a matrix of size p * p
Sx = 
(Xi - 
) (Xi - ) *, (2)
(* - sign of a transposition)
It is named by a matrix of scattering of an observational set X, and 
= 
Xj / n - is a vector of mean values. The trace of a matrix Sx, equal to a sum of diagonal elements, is named by statistical scattering of a set X.
The cluster analysis is based on partition of an initial set X on subsets, distance between which should be maximum. Now let us assume that there are two data sets X and Y. It is necessary to introduce a measure of a statistical distance between classes, formed by observational data matrixes X and Y:
D = a ( - 
) * (
- ), (3)
where a = n1 n2 / (n1 + n2), n1, n2 - number of realisations in each class.
We shall enter into consideration a combined class Z= (X, Y). Relationship between scattering of observations inside class Z and corresponding values inside classes X and Y:
Sz = Sx + Sy + (( - 
) * ( - )), (4)
is fundamental, as is used in any cluster analysis.
The equation (4) can be interpreted as follows: total scattering of two joint classes of realisations is equal to a sum of scattering inside classes and a distance between these classes(i.e., between its mean values ). There are several fuzzy logic algorithms (Jain and Dubes, 1988). These algorithms are distinguished by that in one case the partition of a set of input objects set into classes begins from one class, contained all objects, and in another, on the contrary, joining of separate objects into classes are performed by a selection of "nearest neighbours." We have used a method concerning the second group. The iterative algorithm implements under conditions of the aforementioned prescribed number of classes. Its essence is reduced to the sequential search of objects from their initial distribution and to trial its transposition from one class to the other to archive both a minimum of inside class scattering and a maximum of centre-to-centre class distance.
A basis for our research was archival hourly observational data for several sites of northwest region of Russia during 1984-93. The input data arrays, concerning various meteorological parameters, were synchronised on time and presented separately for each of the months. The calculations were carried out for several stations. The results of the classification have appeared similar; therefore, results for only one characteristic site Voejkovo ( 60 N, 30 E) is presented below.
Originally diurnal distributions, obtained by an independent way for atmospheric temperature and humidity, downward total (scattered plus direct) SW irradiance, atmospheric pressure, precipitation, direction, and velocity of wind, were considered. Then rows of X-type matrix were performed. Each row contains diurnal distributions for a certain day; thus, for each parameter and each month, a matrix was formed and the calculations on a partition of initial sample on a number of classes by means of cluster analysis, described above, were carried out. Calculations allowed us to allocate each class as a set of distributions, with the minimum deviating in a Euclidean distance sense from the average diurnal cycle (in a given class). As mentioned above, the applied algorithm is aimed at maximizing a Euclidean distance between mean value vectors, appropriate to diurnal pattern distributions of different classes. Despite a formal character of this algorithm, it has appeared that the obtained outcomes have reasonable physical interpretation.
We shall begin by reviewing the diurnal cycle of total solar downward radiation. Its diurnal cycle in the summer is the most significant due to amplitudes. Average (for 6 classes) distributions are shown in Figure 1. The obtained results suppose the simple interpretation. The upper curve corresponds to conditions of clear sky, and the lower one to overcast cloud cover. Intermediate curves correspond to conditions of a fractional cloud cover, which can vary during the day (Table 1). For example, at forth class in the first half of a day, the fractional cloud cover is of .3, and in the afternoon is of .7, and at fifth class, the diurnal pattern has an opposite structure. It is necessary to note that results of empirical classification of a daytime cycle of cloud cover, presented in works (Flach, 1951; Flach, 1952), are closely interlaced with data from Table 1. To substantiate
the interpretation of classification results presented in Table 1, parallel calculations of the daytime distribution of downward SW radiation for the latitude 60° and height of the Sun, adequate to the appropriate month of the year, were performed. The calculations were carried out on the base of the Delta-Eddington approximation technique for homogeneous cloud model, with optical thickness equal to 30, the albedo of single scattering equal to 0.99, and asymmetry factor of the phase function g=0.85. The version of this technique for conditions of a fractional cloud cover, described in (Schmetz, 1984), was used. The values of fractional cloud cover given in Table 1 were selected by minimisation of a divergence between calculated and empirical distributions of a daytime cycle of radiation fluxes. It is interesting to note that cloud diurnal pattern classification results obtained in previous studies (see review at Berlyand, 1961) are closely linked to data given in Table 1. The distributions show that net radiation diurnal patterns have the structure analogous to that obtained for surface solar irradiation. This fact explains rather strong links between diurnal distributions of surface solar irradiation and surface air temperature and other parameters, which will be discussed below.
Table 1. Classification of cloudiness and solar radiation diurnal patterns
|
Cluster Number |
Cloudiness distribution |
|
1 |
Clear sky - during a day |
|
2 |
Clear sky -in the morning, fractional cloud cover of 0.3 - at noon and in the afternoon |
|
3 |
Fractional cloud cover of 0.3 - in the morning and at noon, clear sky in the afternoon |
|
4 |
Fractional cloud cover of 0.3 - in the morning, fractional cloud cover of 0.7 - at noon and in the afternoon |
|
5 |
Fractional cloud cover of 0.7 - in the morning and at noon, fractional cloud cover of 0.3 - in the afternoon |
|
6 |
Overcast cloudiness - during the day |
Since a mathematical classification method is used, the question concerning the structure of each class can be risen. Figure 2 shows the bands including the diurnal radiation distributions for two (from six) clusters (their "centres' " are given in Figure 1). It is clear that classification can be considered representative, not only if the class elements have minimal deviations from the centre diurnal distribution (here it is mean values), but also if they have the same structure: coordinates of maximum and time gradient values. The given results demonstrate the fact that the proposed classification method guarantees precisely this similarity for most elements from the same class.
Let’s consider a diurnal cycle of other meteorological parameters. For simplicity we will discuss the case of partition of initial sample onto 4 classes. Results of calculations (Figures 3 and 4) have shown, that in this case the obtained classes for temperature and humidity, as well as for downward solar irradiance, can be referred to two different groups. The first group includes distributions for which both parameters correspond to a stationary condition of atmosphere (i.e., during the day a considerable decrease or increase of parameter values deviations from multiannual average diurnal cycle is not observed). Proceeding from these it is possible to name the appropriate average diurnal distributions as stationary modes. They are two and they correspond to keeping warm or cold weather patterns during the day. The second group includes classes, the average distribution of which, on the contrary, changes during the day compared to corresponding average diurnal cycles. Such distributions are possible to name as transient modes, as they follow the modifications of temperature and humidity in the course of the day. These modifications are connected with increasing or decreasing values of considered parameters. Curves, adequate to indicated types of modes, are showed on Figures 3 and 4. Four classes considered above allow the simulation of closed loop variations of meteorological parameters during several sequential days, for example, synoptic variability. It is necessary to note that transient modes for solar radiation are mainly due to cloud cover varying in the course of the day.
Typical diurnal patterns for other meteorological parameters shown in Figures 5 and 6 demonstrate an unlike characteristic type of relationships.
For example, in the case of low or high pressure (clusters 1 and 4), diurnal oscillations (Figure 5) are more often observed. A study of precipitation has shown that rainfall before dawn, after midday, and in the evening, is the most probable; moreover, the rain intensity increases in the indicated sequence and reaches a maximum at 1900 local sun time (Figure 6).
Direction of the wind is one of the most changeable and informative for the meteorological parameter forecasting. Our study shows that the direction of wind is seldom saved during a day. The most probable modification time intervals are before dawn, in the morning, after midday and in the evening. In addition, the most intensive and quick are the changes from northwest wind to northeast, east, and southern. Changes of southern wind to western are less intensive.
For a long time the problem of joint probability distribution of meteorological parameters has attracted the attention of meteorologists; however, until now, in a majority of studies, authors limited themselves by only revealing the cross-correlation and construction of regression models. Taking into account that feedback in the atmosphere - Earth surface, atmosphere - cloud cover and other systems, determined by energy exchange processes have a non-linear character, the above-mentioned traditional statistical approaches, which involve only linear relations, give rather approximate conception on the essence of occurring processes.
The diurnal pattern of one or several meteorological parameters, corresponding to the given day, can be represented as a point in a phase space of appropriate size. Our research has shown that the points of such multidimensional phase space can be classified rather good, i.e., the input sample of points, adequate to empirical data, are split into an easily separated group. It means that in the course of time, a point belonged to one cluster (adequate to one type of weather), and is transferred to another cluster. For some time between these transpositions, the point stays in the former cluster. It means that at this period, the weather processes were stabilised. It is clear that the mentioned processes could not be accurately described by linear models.
Let’s consider outcomes of performed calculations. Results of joint diurnal patterns of three meteorological parameters: solar radiation, air surface temperature, and humidity, are similar to patterns, presented at Figures 1, 3, and 4. The chosen average distributions were divided rather conditionally into two groups of modes: stationary and transient. The sense of introduced terminology is obvious. Stationary modes correspond to the points in phase space that are staying in the same cluster. Transient modes correspond to a transitional position of these points. The minimum number of clusters for this case is equal to 4. Stationary modes in a diurnal temperature, and humidity cycles correspond to distribution of the clear sky radiation: maximum temperatures and minimum humidity, and also maximum diurnal cycle amplitudes. Similarly other types of stationary modes of temperature and humidity correspond to radiation under conditions of continuous cloud cover. The difference is connected with minimum values of diurnal cycle amplitude. A varying fractional cloud cover corresponds to transient modes of temperature and humidity distribution. The indicated transient modes describe passages from large to small values of meteorological parameters (temperature and humidity) and to the contrary.
The joint distribution of total solar radiation, precipitation, and atmospheric pressure demonstrates their coherence. With clear sky conditions, high pressure and temperature, low air humidity, and the absence of precipitation are observed.
In this case one finds two distributions corresponding to conditions of overcast cloud cover; thus, curve 2 corresponds to a distribution associated with the decrease of pressure (Figure 5). On the contrary, continuous cloud cover (Figure 5, the curve 4) at low pressure corresponds to the curve 4 on Figure 6, revealing the presence of precipitation. Pattern, describing increase of pressure and absence of precipitation, corresponds to distribution of radiation at a fractional cloud cover. Practically it is possible to speak about two classes of distributions for precipitation: days with rainfall and without rainfall.
The cluster determination of joint distribution of main meteorological parameters - pressure, precipitation, and wind direction, also gives characteristic correlation of a diurnal cycle. High pressure is accompanied by the lack of precipitation and western direction of a wind with passage to southern and southeastern direction. At lower pressure (curve 3 at Figure 5) the precipitation also is absent, and the wind has a mainly western direction. At a further drop of pressure (the curve 2 at Figure 5) wind has mainly a southeastern direction. At low pressure (curve 4 at Figure 5), precipitation is observed, and wind changes from a southern to a western direction. In July, the increasing pressure curve corresponds to "no" precipitation, and to a probable change of wind from west to south and southwest direction. Providing lower pressure, precipitation is absent too, but wind has prevalent southeast or northeast direction. The decreasing pressure curve corresponds to higher probability of precipitation and wind changes from a south to a west direction.
The mentioned results allow the following conclusions. The typical diurnal cycle of basic meteorological parameters is characterised by consistent distribution of either stationary or transient modes. Fundamental influence of solar radiation on atmospheric temperature and humidity becomes clearer. Characteristic pressure distributions and appropriate distributions of precipitation and wind direction specify certain weather conditions and their modifications. Analogue conclusions are obtained for relations between cloud cover and precipitation. The number of clusters could be enlarged for a more detailed description of other weather conditions. The obtained classification could form a basis for a solution of both fundamental and applied problems. To the first group it is possible to refer to the problem of modeling a diurnal cycle of meteorological parameters, and the problem of the short-term weather forecasting, or automatic monitoring of observational data, could be referred to the second group.
Modifications of weather conditions occur not only during day period, but also consecutive days on a synoptic temporal scale; therefore, the problem of cluster determination of two and three-diurnal joint distributions of basic meteorological parameters was considered. Here, as well as above, we are interested in studying the stationary and transient modes. Only 4 classes for joint distribution of solar radiation, temperature, humidity, and precipitation were selected for a minimum model. The major interest represents a role of joint distribution of parameters. Figures 7 and 8 demonstrate the cluster discrimination results of two-diurnal patterns of joint radiation, temperature, and humidity distributions. As was mentioned above, the stationary modes illustrate preservation of two types of distributions at passage from the first day to the second. There are two modes of such kind: keeping cold or warm weather types (curve 1 and 3).
On the contrary, the transient modes describe modifications of weather conditions. Passage from the clear sky conditions (to the overcast cloud cover) results in the decreasing of temperature (curve 2, Figure 7) and increasing relative humidity (curve 2, Figure 8). Passage from the continuous cloud cover to the clear sky leads to the inverse variations of the rest of the parameters, i.e., rise of temperature and drop of humidity (curve 4, Figure 7 and 8 ). Once again it proves the determining role of radiation processes, controlled by cloud cover.
Independent distribution of basic meteorological parameters on an interval equal to 2 days looks similar to the case considered above of joint distribution of parameters; however, obtained modes are not completely logically matched. For example, transient modes of one parameter can fit similar modes of another parameter, more appropriate by physical reasons to other modes.
The obtained results principally differ from conventional studies on statistical features of meteorological parameters widely represented in scientific literature. The conventional approaches are usually based on application of correlation or regression analysis techniques, which are linear by nature. These methods usually deal with norms of meteorological parameters and statistical estimates of deviations from these norms. In this case it is possible to get estimates of deviations of some parameters as a linear function of the other. It concerns both diagnostic, and prognostic problems. Determination of an area of applicability of linear methods and emerging new techniques of non-linear process modeling, developing in parallel to significant progress of computer facilities, give stimulus for application of non-linear methods for statistical description of meteorological values and fields. Here we mean the "neural network" techniques, which originally emerged from the problems of an artificial intelligence and obtained broad spreading at a solution of many applied problems in various branches of science and engineering. Applying "neural network" methods (Allen and Le Marshall, 1994; Bankert and Aha, 1996) it is necessary first of all to define cells or nodes of such network. The purpose of the given work consists not only in study of non-stationary diurnal and multi-diurnal oscillations (modes), but also in the definition of elementary neural network for modeling diurnal cycles and multi-diurnal patterns of main meteorological parameters; thus, it is possible to consider the present research as the introduction to the application of new computing techniques for simulation of non-linear meteorological processes observable at the surface of the Earth.
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