The discretization of time (values of t1, t2, ..., tN, Eq. (18)) is an input of the problem. On contrary to the ramp/step method, no explicit constraint is imposed on the surface temperature history. However, excessive number of time intervals in the trial model can result in the instability of the inversion process. On the other hand, we must find the smallest time intervals that can be determined from the given temperature log. The resolution matrix plays an important role in the parametrization of time. In the case of sharp cutoff, the resolution matrix can be defined as R = VVT. The jth column of this N X N matrix is the least squares solution for maximizing the jth parameter. At the proper choice of the discretization of time the resolution matrix exhibits delta-like behavior (compact resolution); the column with the best resolving power is nearly always the column with the maximum diagonal element (Bodri and Cermak, 1995, 1997a). Thus, the diagonal elements of the resolution matrix can be used as the measures of the resolving power. When choosing the preliminary discretization scheme, one has to take into account the "thermal" memory of the ground for the events on its surface (see Section 2.2 and Figure 23). One can choose a preliminary timescale with relatively long time intervals gradually increasing toward the past. Subsequently this scale should be refined with the help of the resolution matrix. For such operation a semi-empirical rule that proved to be very effective in the designing of structural models of the Earth from surface-wave and free oscillation data (Wiggins, 1972) can be applied. If, after computing the resolution matrix, we find that a single time interval is nearly perfectly resolved, this means that we have not selected the time interval short enough in the vicinity of the given instant of time to determine the exact shape of the resolution. In such a case the problem should be recomputed for a shorter time interval.
In practice the GST reconstruction problem cannot be solved uniquely unless a priori information is incorporated into the analysis from the very beginning (Jackson, 1979). As a priori information the covariance matrices for both measured data and unknown GST are generally used in the SVD approach. For example, one can assume that the measured data are not statistically independent, but are characterized by a positive definite covari-ance matrix S. The covariance matrix for the data is symmetric (M X M) matrix, the elements of which are
where a2 is the standard deviation of the data, r(Az) the autocorrelation function, and Az the depth lag. The autocorrelation function characterizes the range (intensity) of the interdependence of the measured signal. Dependence can be either short-range or longrange. The short-range dependence is characterized by correlations that decrease exponentially fast, while long-range dependence occurs when the correlations decrease like a power function. Our calculations have shown that measured borehole temperatures are characterized by a short-range dependence, thus, by the correlation that decreases exponentially fast r(Az)~exp(-Az/D) (Bodri and Cermak, 1995; Bodri et al., 2001). The decorrelation parameter, D, corresponds to the depth lag at which the autocorrelation falls to (1/e), i.e. defines the distance at which the individual temperature values can be considered statistically independent. This quantity represents an individual characteristic of the given hole. According to the field experience, for the majority of the boreholes the D-value ranges from 50 to 300 m. For boreholes with the fast decorrelation and the high number of measured points an advantageous technique of the data thinning ("scarcing") can be used for inversion. In this case, the original data set of measured temperatures can be divided into subsets, and different parameters of the time discretization of the GST history can be obtained from different sections of the T-z profile. This procedure significantly enhances the reliability of estimated parameters, but requires that the data sets used be statistically independent (Twomey, 1977). Details of application of the data thinning technique are presented in Section 3.1 (Chapter 3) for Finnish boreholes.
Another kind of additional information is connected to the interdependence between climatic signals. Highly fluctuating climatic time series exhibit scale invariance or scaling behavior over the wide range of timescales; that is, climatic variations at small scales are related to longer ones by the same scaling law without showing any preferred mode. The investigations of the scaling properties of climatologic data received considerable impetus from the paper by Lovejoy (1982) on the fractal dimension of clouds and rain. By the analysis of meteorological and climatic data using further refined methods, it was concluded that climatic processes exhibit a much more complex structure than previously assumed, where statistical properties at various scales are related through different intensity-dependent dimensions, rather than through a single fractal exponent (Tessier et al., 1993; Lovejoy et al., 2001). The introduction of the multiscaling behavior that may be interpreted as the outcome of a so-called multiplicative cascade process is common to all recent analyses. Generally, such models can be characterized by the range (intensity) of their interdependence, the heaviness of the probability tails, and the degree of nonlin-earity. As about the interdependence of the meteorological/climatic signals, the so-called persistence of weather is a well-known phenomenon. If, e.g. given day is sunny and warm, there is high probability that the next day will be similar. Such tendency appears also on the longer scales. Early attempts to quantify this behavior were made in the works by Kutzbach and Bryson (1974) and Hasselmann (1976). Further these investigations were continued in the works by Lovejoy and Schertzer (1986), Ladoy et al. (1991), Bodri (1993), Beran (1994), and Rangarajan and Sant (2004). The common consequence of all research works was the quantitative establishment of the long-range dependence of correlations within different meteorological/climatologic time series that occurs when the correlation decreases like a power function in such a way that the spectrum diverges at low frequencies. Thus, the climate persistence, characterized by the correlation C(At) of temperature variations separated by the interval At, follows a power law, C(At) ~ At_y. Long-term persistence appears to characterize most climatic phenomena and exists over the spectral range of 1-106 year. Most recent investigations of the weather persistence were performed in the works by Koscielny-Bunde et al. (1998) and Talkner and Weber (2000) using long meteorological temperature records from various climatologic zones in Europe, North America, and Australia. The daily and annual cycles were removed from the data. Investigations with modern detrended fluctuation analysis (DFA) and wavelet techniques that can systematically overcome possible non-stationarities in the data revealed power law correlation decay with roughly the same exponent y = 0.7 (~2/3) in the range of time lags between 10 days to at least 25 years. The range of persistence law is limited by the total length of the time series considered. The authors cannot exclude the possibility that it may exceed detected limit. The persistence of the climatic variations can be taken into account by means of the covariance matrix for the unknown parameters. The covariance matrix is a symmetric (NX N ) matrix, the elements of which are
where s is the a priori standard deviation, At the shift in time, and t the characteristic correlation time. Similarly to the introduced above characteristic distance D, the characteristic correlation time corresponds to the practical vanishing of the correlation between climatic events. The values of s and t are the input parameters of the problem. As will be shown below, generally, smaller s results in the correspondingly smaller disturbances of the GST, and the longer the time t the smoother the obtained solution is. Including an additional information modifies the SVD procedure in such a way that it minimizes both £tS-1£ and VTW-1V0.
Further limits may be imposed on the unknown parameters. In most of the inversion problems it is customary to pose some bounds (so-called hard limits) on the values of parameters imposed by the physics of the problem. For example, when determining the Earth's structure, the densities of the lithospheric rocks may be assumed to vary between 3000 and 4000kglm3, and their shear velocities between 4 and 6kmls. One could treat these statements as pairs of inequality constraints c1 < bTV0(t) < c2, (25)
where vector b is the moment of solution V0(t) and explores the limits of the solution space.
In the case of climatic changes, however, these hard limits tend to be so large that they become irrelevant for practical purposes. The probability distribution of climatic changes may be used to put so-called "soft limits" on the solution. As in the above example with the lithospheric structure, two constants c1 and c2 may be chosen to represent limits on V0(t), but there will be some non-zero probability that V0(t) will violate these bounds. As shown by numerous investigators (e.g. Ladoy et al., 1991; Fraedrich and Lardner, 1993; Olsson, 1995) the cumulative probability distribution (probability that random fluctuation dT exceeds a fixed value T) of climatic time series generally has a nearly Gaussian shape in the center and a tail (probability of the extreme events) that is "heavier" than would be expected for a normal distribution. The "fat-tailed" probability distributions are general characteristics of climatic time series.
When the fluctuations are of this type, the phenomenon is so intermittent that the return times of extreme events are much shorter than those for Gaussian process. We illustrated the difference between Gaussian and the real probability distribution with the use of the two-millennia long homogeneous temperature anomaly time series for Northern Hemisphere developed by Mann and Jones (2003) by using different proxy indicators of climate (Figure 3). The cumulative probability of this data is presented in Figure 24. The difference of the probability tails is clearly visible. For example, the temperature fluctuations corresponding to more than three standard deviations (anomalies of more than -0.016K or less than -0.504K) for Gaussian process would have a probability level of 0.0013 that gives the return period for such anomalies of near 770 years.
for Northern Hemisphere 200-1980 A.D.
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