**1. Introduction**

Extensive climatic data for the past four ice ages and earlier, the period in which *Homo sapiens* evolved, is available in various scientific reports analyzing ice cores, commencing from the mid-1950s. There are various sites on Antarctica and

Greenland where intensive ice core drilling has occurred since 1956, with several countries supporting more than a score of different drilling projects. Currently, intensive ice core drilling is being conducted in other areas as well, so an even larger data set is anticipated. In previous papers [1–4] the history and limitations of ice core drilling are described in detail. Our purpose in this paper is to employ Vostok ice core data for time and frequency related analyses to set a basis for improving prediction of climate variations. The data sets include derivations of relative temperature, carbon dioxide (CO2), methane (CH4), oxygen, dust and solar variation (insolation), during the past 420,000 years. Because isotopic fractionation of oxygen-18 and deuterium in snowfall is temperature dependent and a strong spatial correlation exists between mean annual temperature and mean isotopic ratios it is possible to derive ice-core climate records. References [5, 6] presented the first record for full glacial–interglacial period from an ice core drilled in the Russian Vostok station in Antarctica. However, it is important to establish criteria for controlling the quality of this new science that we claim could eventually lead to a Climate Prediction Engine (CPE), based on verified causes.

between gravity and inertial forces, with the angular momentum and action held constant as a function of mass and the gravitational constant [10]. By contrast, the molecular states in the global ecosystem are dominated by transient energy flows, oscillating in a complex set of time scales as short as hourly to as long as geological epochs. Such variations in energy flows invite frequency analysis, characterized

*Kalman Filter Harmonic Bank for Vostok Ice Core Data Analysis and Climate Predictions*

A "snowball" Earth [11], more or less covered by ice and snow as the ice cores were fashioned clearly has diminished total energy in terms of molecular quantum states affecting molecular vibration, rotation, translation as well as sustaining the more coherent circulatory motions in the atmosphere and the ocean, transferring thermal energy towards the poles. A warm Earth as at present has all these flows in higher quantum states, raising their entropy as we have quantified for atmospheric molecules elsewhere [12]. Variation in physical parameters such as temperature is obvious but complex in causes, with any longer-term trends overlying daily or

One state property that does not vary over time on Earth except geologically is atmospheric pressure, given that it is effectively based on the weight of the atmosphere, apart from variation for water, some 4% during the El Nino cycle [13]. The current IPCC consensus concludes that the most significant warming is from greenhouse gas content, such as that of CO2, methane and nitrous oxide, but this is not necessarily the major controlling cause of increasing temperature, given how negotiable thermal energy is. Globally, we are conducting a large uncontrolled experiment to see if CO2 is so important. Although originally thought that the CO2

An additional causative factor controlling the atmospheric pressure of CO2 that seems to have been overlooked is varying acidity of waters, which can also respond to temperature. In warm periods with high oxygen levels, acidification by oxidation of reduced sulfur, nitrogen and carbon compounds is favored [14]. Alkaline conditions are favored anaerobically. The reduction of dimethyl sulfate, a major oxidant in sea water, can also lead to the evolution of reduced sulfur compounds, which can be converted to sulfuric acid in the atmosphere by ultraviolet oxidation. The alkaline pH of the ocean about 8.2 is effectively the same as that obtained in aerated water equilibrated with limestone (CaCO3). Sources of alkaline carbonated salts in the ocean is basaltic volcanic rocks and soils, in contrast to more acidic granitic

We have shown using equilibrium theory that a lowering in surface ocean pH of 0.01 units can lead to an increased CO2 pressure in the atmosphere of 8–10 ppm by volume. So, a change of 100 ppm could occur locally if the ocean surface or water on land was acidified by 0.10 unit, say from pH 8.20 to 8.10 as observed recently. Much larger local changes could occur, from a catastrophic event such as major volcanic eruption. The annual oscillation in CO2 pressure at 3500 m altitude on Mona Loa, Hawaii could be a result of the change in the high surface temperature of the nearby ocean between winter and summer, rather than imbalance between photosynthetic assimilation and respiratory evolution of CO2, often claimed in general climate models. The CO2 pressure in Hawaii peaks after winter in May when the sea temperature is about 23°C and reaches its minimum in late October when the sea is about 27°C. A peculiarity of calcite is that its solubility declines with higher temperature [15], so its precipitation removing CO2 from the atmosphere could partly or fully explain the oscillation. By contrast, at the sampling site at Cape Grim in Tasmania, where the annual sea temperature has a mean temperature

data from ice cores might be considered as proof of its causal role in global warming, but given that changes in CO2 lag temperature changes its resultant release or absorption from solution in sea water is more likely the cause of

with Fourier transforms to provide underlying information.

seasonal trends as a function of latitude.

*DOI: http://dx.doi.org/10.5772/intechopen.94263*

correlations in the relationship.

deposits that tend to acidify water.

**39**

A 420,000 year record was constructed from the [5] study on a 3 km deep core of ice (**Figure 1**). Another source of similar ice core data is available from European EPICA drilling project [7] which lasted from 1998 until 2005. EPICA data are comparable with Vostok data. In this paper we focus on specific analysis related to only two of the Vostok data variables, namely relative temperature and CO2 content. We also touch upon possible effect of the dust on various points along Vostok timeline. We will include other data at a later date. Data sets used are from [8] and, with corrections, [9]. The variation of atmospheric CO2, temperature and dust are shown in **Figure 1** together with our definition of four cycles (C1, C2, C3, C4) formed from the maxima of the variables. These cycles could also be defined starting from the variable minima. Before presenting very detailed time and frequency analysis of specific subset of Vostok data, we make some general observations related to climate fluctuations of CO2, temperature and dust.

Variations in global climate measured by temperature change automatically involve the thermal energy content and heat capacity of the atmosphere. These highly variable systems can be contrasted with the energy content of conservative physical systems like planetary orbits, where the principle of least action defines the trajectories favored; conservative systems can be reliably modeled as an interaction

**Figure 1.** *Vostok data variation in global relative temperature and CO2 content [8, 9].*

### *Kalman Filter Harmonic Bank for Vostok Ice Core Data Analysis and Climate Predictions DOI: http://dx.doi.org/10.5772/intechopen.94263*

between gravity and inertial forces, with the angular momentum and action held constant as a function of mass and the gravitational constant [10]. By contrast, the molecular states in the global ecosystem are dominated by transient energy flows, oscillating in a complex set of time scales as short as hourly to as long as geological epochs. Such variations in energy flows invite frequency analysis, characterized with Fourier transforms to provide underlying information.

A "snowball" Earth [11], more or less covered by ice and snow as the ice cores were fashioned clearly has diminished total energy in terms of molecular quantum states affecting molecular vibration, rotation, translation as well as sustaining the more coherent circulatory motions in the atmosphere and the ocean, transferring thermal energy towards the poles. A warm Earth as at present has all these flows in higher quantum states, raising their entropy as we have quantified for atmospheric molecules elsewhere [12]. Variation in physical parameters such as temperature is obvious but complex in causes, with any longer-term trends overlying daily or seasonal trends as a function of latitude.

One state property that does not vary over time on Earth except geologically is atmospheric pressure, given that it is effectively based on the weight of the atmosphere, apart from variation for water, some 4% during the El Nino cycle [13]. The current IPCC consensus concludes that the most significant warming is from greenhouse gas content, such as that of CO2, methane and nitrous oxide, but this is not necessarily the major controlling cause of increasing temperature, given how negotiable thermal energy is. Globally, we are conducting a large uncontrolled experiment to see if CO2 is so important. Although originally thought that the CO2 data from ice cores might be considered as proof of its causal role in global warming, but given that changes in CO2 lag temperature changes its resultant release or absorption from solution in sea water is more likely the cause of correlations in the relationship.

An additional causative factor controlling the atmospheric pressure of CO2 that seems to have been overlooked is varying acidity of waters, which can also respond to temperature. In warm periods with high oxygen levels, acidification by oxidation of reduced sulfur, nitrogen and carbon compounds is favored [14]. Alkaline conditions are favored anaerobically. The reduction of dimethyl sulfate, a major oxidant in sea water, can also lead to the evolution of reduced sulfur compounds, which can be converted to sulfuric acid in the atmosphere by ultraviolet oxidation. The alkaline pH of the ocean about 8.2 is effectively the same as that obtained in aerated water equilibrated with limestone (CaCO3). Sources of alkaline carbonated salts in the ocean is basaltic volcanic rocks and soils, in contrast to more acidic granitic deposits that tend to acidify water.

We have shown using equilibrium theory that a lowering in surface ocean pH of 0.01 units can lead to an increased CO2 pressure in the atmosphere of 8–10 ppm by volume. So, a change of 100 ppm could occur locally if the ocean surface or water on land was acidified by 0.10 unit, say from pH 8.20 to 8.10 as observed recently. Much larger local changes could occur, from a catastrophic event such as major volcanic eruption. The annual oscillation in CO2 pressure at 3500 m altitude on Mona Loa, Hawaii could be a result of the change in the high surface temperature of the nearby ocean between winter and summer, rather than imbalance between photosynthetic assimilation and respiratory evolution of CO2, often claimed in general climate models. The CO2 pressure in Hawaii peaks after winter in May when the sea temperature is about 23°C and reaches its minimum in late October when the sea is about 27°C. A peculiarity of calcite is that its solubility declines with higher temperature [15], so its precipitation removing CO2 from the atmosphere could partly or fully explain the oscillation. By contrast, at the sampling site at Cape Grim in Tasmania, where the annual sea temperature has a mean temperature

Greenland where intensive ice core drilling has occurred since 1956, with several countries supporting more than a score of different drilling projects. Currently, intensive ice core drilling is being conducted in other areas as well, so an even larger data set is anticipated. In previous papers [1–4] the history and limitations of ice core drilling are described in detail. Our purpose in this paper is to employ Vostok ice core data for time and frequency related analyses to set a basis for improving prediction of climate variations. The data sets include derivations of relative temperature, carbon dioxide (CO2), methane (CH4), oxygen, dust and solar variation (insolation), during the past 420,000 years. Because isotopic fractionation of oxygen-18 and deuterium in snowfall is temperature dependent and a strong spatial correlation exists between mean annual temperature and mean isotopic ratios it is possible to derive ice-core climate records. References [5, 6] presented the first record for full glacial–interglacial period from an ice core drilled in the Russian Vostok station in Antarctica. However, it is important to establish criteria for controlling the quality of this new science that we claim could eventually lead to a

A 420,000 year record was constructed from the [5] study on a 3 km deep core of ice (**Figure 1**). Another source of similar ice core data is available from European EPICA drilling project [7] which lasted from 1998 until 2005. EPICA data are comparable with Vostok data. In this paper we focus on specific analysis related to only two of the Vostok data variables, namely relative temperature and CO2 content. We also touch upon possible effect of the dust on various points along Vostok timeline. We will include other data at a later date. Data sets used are from [8] and, with corrections, [9]. The variation of atmospheric CO2, temperature and dust are shown in **Figure 1** together with our definition of four cycles (C1, C2, C3, C4) formed from the maxima of the variables. These cycles could also be defined starting from the variable minima. Before presenting very detailed time and frequency analysis of specific subset of Vostok data, we make some general observa-

Climate Prediction Engine (CPE), based on verified causes.

*Glaciers and the Polar Environment*

tions related to climate fluctuations of CO2, temperature and dust.

*Vostok data variation in global relative temperature and CO2 content [8, 9].*

**Figure 1.**

**38**

Variations in global climate measured by temperature change automatically involve the thermal energy content and heat capacity of the atmosphere. These highly variable systems can be contrasted with the energy content of conservative physical systems like planetary orbits, where the principle of least action defines the trajectories favored; conservative systems can be reliably modeled as an interaction of 15°C with variation between 11 and 19, a range which does not lead necessarily to precipitation of calcite, there is only slight evidence of an oscillation. While the burning of fossil fuels would be the main current cause of pH variation in the ocean, any other acidifying processes in the atmosphere or on land such as from agriculture, could also be controlling influences.

For all of the above reasons regarding the complexity of causes of climate change, we consider it would be beneficial if the ice core data could be subjected to careful Fourier frequency analysis, yielding detailed data regarding long term mechanisms of variation in climate, also pointing to an appropriate dynamic modeling of the underlying processes. In this chapter we focus specifically on CO2 Vostok data for C1 as defined in **Figure 1**, as an example of our CPE approach, extending this to CO2 for C2, C3 and C4, or their combinations. Other data such as temperature, methane, dust, and any other available ice core data such as EPICA, can be analyzed similarly.

## **2. Methodology**

The eminent statistician Fisher [16] was an early exponent of testing statistical significance by harmonic analysis. Some preliminary spectral analysis has been conducted on Vostok ice core data set as reported in [17–19]. In this paper, a detailed spectral analysis in R and Excel was applied to Barnola et al. data set [8, 9]. In such analysis, time series are decomposed into underlying sine and cosine functions to establish the most important frequencies. Various texts on Fast Fourier Transforms (FFT) were also sources for frequency determination. These approaches allow construction of periodograms quantifying the contributions of the individual frequencies to the time series regression. The methodology developed for bioinformatics [20] has general application for time series and was employed in this study. In this paper we perform our own time and frequency analysis using R and Excel. Trudinger et al. [21, 22] have also applied Kalman filter analysis to ice core data. This allowed a more rigorous analysis of CO2 variability for the Law Dome ice cores of Antarctica over the recent 1000 years than the usual deconvolution method. These authors pointed out that the Kalman filter allows better calculation of uncertainties in the deduced sources. The uncertainties correspond to the selected range of frequencies. They claimed [22] that it allows investigation of statistical properties that are directly related to physical properties.

data sets including current data may be very useful in estimating global near and far future effects of factors like temperature and CO2 content an ultimate goal in this

**No of Data Points Cycle C1 Cycle C2 Cycle C3 Cycle C4 Total** Temperature 74 136 99 54 363 **CO2 80** 135 99 49 363

*Kalman Filter Harmonic Bank for Vostok Ice Core Data Analysis and Climate Predictions*

Note that one such additional current data point may not make a large difference on the current C1, as far as harmonic analysis, which has long term time behavior built in, but the short term time effects may be more prominent. Our approach covers both short and long term effects and it can be used to perform sensitivity analysis using current climate data together with original Vostok data. The uneven data sampling times within each cycle) may pose some numerical issues as well for the analysis in the context of Kalman filter (KF) prediction methodology. There are various methods missing data problem and one of the simpler methods, yet effective one, is to enter missing data by some (linear or nonlinear) approximation method. We inject additional data using linear interpolation, also achieving Nyquist

sampling requirements and in the process minimize the number of required

harmonics used by the Kalman filter harmonic bank (KFHB). For C1 which originally had 80 data points for CO2 we added additional data points for the total of 128 suitable for FFT but also corresponding to the length of 128,000 years so far in the cycle.

An aim is to gain more insight into Vostok data in time and frequency domains, and check the corresponding amplitude and energy content in order to reduce the number of significant harmonic components (**Figure 2**). Other authors used similar energy consideration but in our case, we propose the simple and effective mathematical model of a stochastic harmonic oscillator, based on which a KFHB can be built and used to further analyze Vostok data as well as predict near and far future data behavior. Note that besides C1 other cycles can be used to improve the precision by combining two or three cycles based on past data for C1, C2. C3. Amplitude

research.

**41**

**Table 1.**

**Figure 2.**

*Climate prediction engine (CPE) logic.*

*Boldfaced entries point to CO2 and Cycle 1 related values.*

*DOI: http://dx.doi.org/10.5772/intechopen.94263*

*Number of original Vostok data points for each cycle.*

Our aim is to apply Kalman filter based methodology to define the logic for a Climate Prediction Engine (**Figure 2**) based on 4 data cycles, **Figure 1**. Cycle C1 is the most recent period, the current interglacial warming period nearing its maximum but still incomplete. The overall number of published data points for both variables (relative temperature and CO2) is 363. The individual cycles are determined by locating maximum absolute values for relative temperature and CO2. Individual cycles differ in the number of data points slightly. Also, because of a lag observed between the maxima for temperature and CO2 data, the number of data points differs slightly in two data sets in each cycle. This is also confirmed in our time correlation analysis below. **Table 1** summarizes the number of data points in each cycle. Note that each cycle is defined as top-to-top data values for both CO2 and temperature, resulting in a different number of data points for CO2 and temperature. However, the beginning of each interglacial where temperatures commences climbing might prove superior for some analyses, given that the controlling causes for reversal may be more consistent for these periods. Obviously, the current cycle is still evolving and new data may be added as required for further analysis. We can achieve that by appending the original Vostok data with additional current points, that might skew the previous natural data progression. Comparison of two

*Kalman Filter Harmonic Bank for Vostok Ice Core Data Analysis and Climate Predictions DOI: http://dx.doi.org/10.5772/intechopen.94263*

#### **Figure 2.**

of 15°C with variation between 11 and 19, a range which does not lead necessarily to precipitation of calcite, there is only slight evidence of an oscillation. While the burning of fossil fuels would be the main current cause of pH variation in the ocean, any other acidifying processes in the atmosphere or on land such as from agricul-

For all of the above reasons regarding the complexity of causes of climate change, we consider it would be beneficial if the ice core data could be subjected to careful Fourier frequency analysis, yielding detailed data regarding long term mechanisms of variation in climate, also pointing to an appropriate dynamic modeling of the underlying processes. In this chapter we focus specifically on CO2 Vostok data for C1 as defined in **Figure 1**, as an example of our CPE approach, extending this to CO2 for C2, C3 and C4, or their combinations. Other data such as temperature, methane, dust, and any other available ice core data such as EPICA,

The eminent statistician Fisher [16] was an early exponent of testing statistical significance by harmonic analysis. Some preliminary spectral analysis has been conducted on Vostok ice core data set as reported in [17–19]. In this paper, a detailed spectral analysis in R and Excel was applied to Barnola et al. data set [8, 9]. In such analysis, time series are decomposed into underlying sine and cosine functions to establish the most important frequencies. Various texts on Fast Fourier Transforms (FFT) were also sources for frequency determination. These

approaches allow construction of periodograms quantifying the contributions of the individual frequencies to the time series regression. The methodology developed for bioinformatics [20] has general application for time series and was employed in this study. In this paper we perform our own time and frequency analysis using R and Excel. Trudinger et al. [21, 22] have also applied Kalman filter analysis to ice core data. This allowed a more rigorous analysis of CO2 variability for the Law Dome ice cores of Antarctica over the recent 1000 years than the usual deconvolution method. These authors pointed out that the Kalman filter allows better calculation of uncertainties in the deduced sources. The uncertainties correspond to the selected range of frequencies. They claimed [22] that it allows investigation of

Our aim is to apply Kalman filter based methodology to define the logic for a Climate Prediction Engine (**Figure 2**) based on 4 data cycles, **Figure 1**. Cycle C1 is the most recent period, the current interglacial warming period nearing its maximum but still incomplete. The overall number of published data points for both variables (relative temperature and CO2) is 363. The individual cycles are determined by locating maximum absolute values for relative temperature and CO2. Individual cycles differ in the number of data points slightly. Also, because of a lag observed between the maxima for temperature and CO2 data, the number of data points differs slightly in two data sets in each cycle. This is also confirmed in our time correlation analysis below. **Table 1** summarizes the number of data points in each cycle. Note that each cycle is defined as top-to-top data values for both CO2 and temperature, resulting in a different number of data points for CO2 and temperature. However, the beginning of each interglacial where temperatures commences climbing might prove superior for some analyses, given that the controlling causes for reversal may be more consistent for these periods. Obviously, the current cycle is still evolving and new data may be added as required for further analysis. We can achieve that by appending the original Vostok data with additional current points, that might skew the previous natural data progression. Comparison of two

statistical properties that are directly related to physical properties.

ture, could also be controlling influences.

*Glaciers and the Polar Environment*

can be analyzed similarly.

**2. Methodology**

**40**

*Climate prediction engine (CPE) logic.*


#### **Table 1.**

*Number of original Vostok data points for each cycle.*

data sets including current data may be very useful in estimating global near and far future effects of factors like temperature and CO2 content an ultimate goal in this research.

Note that one such additional current data point may not make a large difference on the current C1, as far as harmonic analysis, which has long term time behavior built in, but the short term time effects may be more prominent. Our approach covers both short and long term effects and it can be used to perform sensitivity analysis using current climate data together with original Vostok data. The uneven data sampling times within each cycle) may pose some numerical issues as well for the analysis in the context of Kalman filter (KF) prediction methodology. There are various methods missing data problem and one of the simpler methods, yet effective one, is to enter missing data by some (linear or nonlinear) approximation method. We inject additional data using linear interpolation, also achieving Nyquist sampling requirements and in the process minimize the number of required harmonics used by the Kalman filter harmonic bank (KFHB). For C1 which originally had 80 data points for CO2 we added additional data points for the total of 128 suitable for FFT but also corresponding to the length of 128,000 years so far in the cycle.

An aim is to gain more insight into Vostok data in time and frequency domains, and check the corresponding amplitude and energy content in order to reduce the number of significant harmonic components (**Figure 2**). Other authors used similar energy consideration but in our case, we propose the simple and effective mathematical model of a stochastic harmonic oscillator, based on which a KFHB can be built and used to further analyze Vostok data as well as predict near and far future data behavior. Note that besides C1 other cycles can be used to improve the precision by combining two or three cycles based on past data for C1, C2. C3. Amplitude and energy analysis performed here reduces a number of required individual Kalman Filter Harmonic Oscillators (KFHO) as individual blocks for the KFHB. It is important to note that using this approach rather than just combining amplitudes and harmonic components directly as described for example in [23], is to accommodate stochasticity of the data and the overall probabilistic nature of the prediction problem using KFs. This approach adds to the robustness of the method and provides better probabilistic accuracy both for short and long term periods, as well as allowing for prediction sensitivity analysis using various values, such as CO2 levels as measured now and estimated for some future periods. Our methodology can be applied for both research as well as for policy making tools for the future climate related societal and technological decisions.

In this paper we perform time analysis of all cycles and C1 CO2 frequency analysis (boldfaced in **Table 1**) to illustrate the KFHB approach (Section 7).
