AnalySeries 1.0 has just been released. It is available free of charge on the Internet, for European users at ftp://ftp-lmce.cea.fr/incoming/paillard/AnalySeries, and for American users at location: www.ngdc.noaa.gov/paleo/softlib.html (or ftp://ftp.ngdc.noaa.gov/paleo/softlib).
Previous versions of the software are already used by many scientists throughout the world, particularly in the field of paleoclimatology, for which it was designed. However, many tools available in AnalySeries are more generally useful, such as the spectral analysis methods and most of the other time-series mathematical treatments. Similarly, the methods of stratigraphic correlation are widely used in Earth sciences.
AnalySeries is available only for Macintoshes (For spectral analysis on UNIX workstations, you may use the "user-friendly" SSAToolkit package [ Dettinger et al. , 1995], also available on the Internet). AnalySeries was designed to provide powerful mathematical tools to a broad range of potential users, and most programming efforts were aimed at defining an intuitive graphical user interface. The software requires at least about 1 MB of available RAM and about 1.5 MB of disk space. The number of data points in series is limited only by available RAM and computing speed. Depending on the computer, a few tens or hundreds of thousands of points is probably a practical limit.
AnalySeries is a Macintosh application that follows the general requirements of graphically oriented, menu-driven applications. Input and output of data are possible either through file opening and saving, or more simply by copying and pasting columns from a spreadsheet software. Therefore, it is easy to work on AnalySeries and a spreadsheet at the same time, while exchanging data between the two applications. All operations performed during an AnalySeries session are listed on its main window, which can be saved independently of the data series to provide a permanent record.
Data series appear on the AnalySeries window as rectangles that can be selected and edited by mouse clicks. Once selected, any series or group of series can be manipulated by one of the commands available in the menu bar, eventually creating new result series. Series or groups of series can be plotted on screen, with several possible options (zooming, x- and y-scaling, for example). Output series include a small header explaining how they were obtained. AnalySeries offers fast and easy access to sophisticated and powerful mathematical time-series analysis methods, as well as to other useful tools. A simple on-line help is also included in the application.
AnalySeries was designed specially to facilitate the study of paleoclimatic records using the approach and some of the methods defined by the SPECMAP group [ Martinson et al. , 1987; Imbrie et al. , 1984 , 1989 ]. It addresses two main problems: transforming "data versus depth" records into "data versus age" records; and spectral analysis of the paleoclimatic records for studying their relationships with insolation, ice volume, and other climatic parameters in the frequency domain.
A classical method for establishing an age-scale on a sedimentary record is to use a comparable well-dated signal as a reference signal and then to optimize some measurement of the similarity between the two series, while changing the depth scale of the first one to the age-scale of the second. Two main questions come into mind then. First, what is a good measure of the similarity between the two time-series? And second, what kind of an age-depth relation is desirable? Very often, when many quantities are measured on the same sedimentary record, a third problem is then to use as much as possible of the available information to construct a final consistent age-depth relation.
With regard to the first question, two classical approaches are commonly used. The simplest
is to "put together the corresponding remarkable features of both signals" [e.g., Prell et al. , 1986]. Though its simplicity is appealing, this method may
give subjective "user-dependent" results; the identification of "remarkable features" may
sometimes be somewhat arbitrary. The second method is to use a mathematical measure of the
similarity between both signals--for example, a correlation coefficient--and then optimize this
measure when adjusting the age-depth relation [ Martinson et
al. , 1987]. This procedure is likely to give a more objective result. Unfortunately, the
"fit" is not always as good as with the simple visual correlation. A mathematical measure such as
a correlation coefficient will indeed give more weight to the large timescale signal fluctuations
(low-frequency variations) where much of the variance is located than to the rapid ones that
usually account for little of the variance. As a result, the rapid transitions or spikes are not
exactly in phase, as they should be according to the underlying simultaneity hypothesis ( Figure 1 ). This second approach is, therefore, more objective, but often less
precise.
In addition to offering both classical methods, AnalySeries also provides a trade-off by allowing the simultaneous use of both methods, associating remarkable features identified on both signals, and optimizing a correlation coefficient in other regions of the record where no such clear features have been identified.
Furthermore, the age-depth relationship cannot be fully arbitrary, for the number of available constraints on the age-scale is always much smaller than the number of data points. Most commonly, when using tie-points to constrain the age of a record, a constant sedimentation rate is assumed between the tie-points (piecewise-linear, age-depth relation). This is probably the method of choice in AnalySeries (command Linage). The only clear drawback of this method is the introduction of discontinuities in the sedimentation rate. When computing fluxes to the sediment, this results in undesirable artificial transitions. The simplest alternative is probably to use a spline interpolation between tie-points instead of a piecewise-linear one. But one must be careful that the corresponding age-depth relation continues to increase, as required by basic stratigraphy. This is automatically enforced in AnalySeries by using a special class of spline, where the continuity of the second derivative is eventually relaxed to enforce monotony. A third possibility offered by AnalySeries is to use a linear + sinusoidal relation for reasons of compatibility with the Martinson et al. [1987] algorithm.
But one main feature of AnalySeries is the possibility of performing this kind of stratigraphic adjustment simultaneously using several proxies of the same sedimentary record. One example of such a situation is when comparable records are obtained from two nearby sites. It is then desirable to build a common stratigraphic framework, using not one, but eventually many proxies in both records. AnalySeries allows the user to put tie-points on one pair of signals, while interactively showing the effect on all other pairs. By changing pairs appearing on screen and adding and removing tie-points while controlling the result, it is possible to quickly establish a consistent, common stratigraphic scale.
AnalySeries calculates the astronomical and daily insolation time series following Berger [1978]. The results are accurate up to about 1 million years [ Berger and Loutre , 1991]. The astronomical series are the eccentricity of the Earth orbit, the obliquity (the tilt) of the Earth axis, and the precessional parameter, classically defined as e sin w, where e is the eccentricity and w is (please make sure that, if possible, w is the greek letter 'omega') the longitude of the perihelion. These three parameters govern insolation, the amount of solar energy received by the Earth at the top of the atmosphere.
AnalySeries offers two different possible calculations for the insolation. First, the "daily insolation," which is defined as the instantaneous amount of energy received at a given latitude and a given orbital position (that is, a given pseudocalendar date). It is averaged over one Earth rotation. The second option is to average this previous value between two orbital positions (two pseudocalendar dates), taking into account the varying speed of the Earth on its orbit. It is worth mentioning that the present-day calendar cannot be applied directly to define the Earth position on its orbit, for the number of days between each astronomical season (solstices and equinoxes) varies at different geological time periods. For example, winters are approximately 90 days and summers 92 days long in our present calendar, but the opposite was true 11 kyr ago. Therefore only the true orbital position (the angular sector defined between the Earth and the vernal point) is significant. The pseudocalendar provided in AnalySeries is an averaged calendar, composed of 360 days (12 months of 30 days) corresponding to the 360 angular degrees on the Earth orbit.
The days thus defined are not 24-hour time lapses, but just a convenient way to define the Earth's position on its orbit.
Given a time series, one of the first concerns is often to identify recurrent features or periodicities, and spectral analysis is then the tool of choice. Many methods have been suggested for identifying periodicities in time series, and each has its advantages and drawbacks. AnalySeries provides a set of classical spectral analysis methods that are often complementary in terms of robustness versus resolution.
The periodogramme method is the most straightforward and unstable method for doing spectral analysis. The data (eventually multiplied by a window, so as to minimize side effects, or leakage) are just Fourier transformed, and the spectrum is calculated directly from the Fourier coefficients. This method should not be used for real (noisy) data, because results are not stable with respect to small changes in the input signal. It is present in AnalySeries mainly for pedagogical reasons, as the simplest possible method.
The Blackman-Tukey method [ Blackman and Tukey , 1958] is the classical method for doing spectral analysis. The algorithm computes first the autocovariance of the data, then applies a window, and finally Fourier-transforms it to compute the spectrum. It is a very robust method, unlikely to present spurious spectral features. The main drawback is its poor resolution in the spectral domain: most of the time sharp features are considerably smoothed. This method requires you to choose a resolution versus confidence parameter: the length of the autocovariance series. You can choose it in terms of a number of lags (number of autocovariance values calculated), percentage of the series length (number of lags divided by the length of the series), or some predefined levels. It is best to perform several computations with different resolution/confidence levels. Different types of window are also available, though this should not considerably affect the results for typical (short and noisy) geophysical time series (except for the square window, which may cause considerable aliasing). This method also provides an error bar on the spectrum, as well as a band width (error bar on the frequency). The confidence level associated with the error bar on the spectrum can be adjusted as desired. Cross-spectral analysis is also provided.
The maximum entropy method [e.g., Haykin , 1983] is useful for its high resolution. Its main drawback is the lack of any statistical confidence estimate. As in the Blackman-Tukey method, you need to choose a resolution/confidence parameter. The conventional wisdom is to make several calculations, increasing this parameter to increase the resolution, but to stop the procedure when too many spectral lines pop out of the background spectrum. This method should not be used alone, but in conjunction with a more robust method, like Blackman-Tukey.
The multitaper method [ Thomson , 1982] is relatively new. It offers some very interesting features: a high resolution and statistical estimates that are independent of the spectral power (small amplitude oscillations may have a high significance level). Some caution is nevertheless advised: the statistical confidence levels given by this method are often (much) more optimistic than the ones given by more classical methods. This is due to different statistical null-hypothesis, therefore the confidence levels are not directly comparable to those of the other methods.
The singular spectral analysis method [ Vautard and Ghil , 1989] is not truly a spectral analysis method. It performs an empirical orthogonal function (EOF) analysis in the time domain, and thus represents the signal as a sum of components that are not necessarily oscillations, but more general, data adaptive functions. It can thus not only be used to identify spectral lines (which appears as a pair of nearly sine and cosine functions in the signal decomposition), but also as a very powerful noise filter, through its ability to separate self-coherent features from random ones.
Several other commonly used tools are also included in AnalySeries. First, a powerful "resampling" command allows resampling of input signals using any other x-scale (a user-defined evenly spaced one, or a combination of scales in other series). The resampling can be done by interpolation (using a piecewise-constant, piecewise-linear, or spline function), by integral-sampling (new samples are averages of previous ones), or by fitting to some model function (piecewise-constant, piecewise-linear, or spline). It is therefore extremely easy to put several records on the same scale, while undersampling or oversampling them. Smoothing and filtering are also provided. Filtering is performed using a band-pass Gaussian filter, whose center and width are chosen by the user.
For more information, or to submit comments or suggestions about AnalySeries, contact Didier Paillard at paillar@asterix.saclay.cea.fr .
Acknowledgments: The AnalySeries software is derived from a previous Macintosh program written on MPW/Fortran format by E. Chol, L. Jodet and F. Lecoat under the direction of L. Labeyrie, and financial support from the French Programme National d Etude de la Dynamique du Climat. We thank J. Overpeck for helpful suggestions on the manuscript, as well as the numerous users of AnalySeries who have made comments and reported bugs. This is LMCE contribution number 00375 and CFR contribution number 1882.-- Didier Paillard, Laboratoire de Modelisation du Climat et de l'environnement, Gif-sur-Yvette, France, and NCAR, Boulder, Colo.; Laurent Labeyrie, CNRS, Gif-sur-Yvette, France, and Universite d'Orsay, Orsay, France; and Pascal Yiou, Laboratoire de Modelisation du Climat et de l'environnement, Gif-sur-Yvette, France
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