**3. A climate prediction engine**

The block diagram in **Figure 2** summarizes our CPE. It seeks the ability to predict various quantities using both past data (from one or more cycles) as well as the current data available This also allows to run various sensitivity analysis by using a variety of future scenarios as far as CO2 and temperature, for example. The CPE can be applied to any ice core data, Vostok, EPICA or any other and for any variable, CO2, temperature, methane, O2, or dust.

Detailed visual inspection of maximum values in **Figures 2** and **3** indicates that CO2 lags relative temperature in C4, C3, and C2. In C1, the data may be a little skewed due to the number of recent data points where maximum values are still not

The differences in average sampling times obviously come from the number of data points collected and the duration of each individual cycle. To facilitate our general approach all other combination of available cycles data are also considered. The idea is enrich with the varied of all available data for prediction purposes and also training purposes of machine learning approach. For example **Table 4** indicates respectively average sampling times for C1 and C2 combined (C12), C2 and C3 combined (C23), as well as C2, C3 and C4 combined (C234), plus the total data set C1234. Note from **Table 4** that more cycles we add the more uniform average data sampling times become, until it becomes equivalent for both CO2 and temperature. Further refinement of average sampling time is given in

**Average sampling time in years Cycle C1 Cycle C2 Cycle C3 Cycle C4** Temperature 1703 800 865 1792 **CO2 1605** 813 862 1911

**Average sampl. time, years Cycles C12 Cycles C23 Cycles C234 Cycles C1234** Temperature 1087 827 1008 1149 **CO2 1108** 834 1020 1149

clearly identified in Vostok data (left-most CO2 data, **Figure 1**), as C1 is still evolving. Better lag estimates can be obtained by cross correlation analysis as done in Section 4 below. Another feature is the varied time differences when the data is obtained from ice core readings. In some cases difference between two data samples are only 400–500 years, but in some as much as 5000 years. To start the analysis for each cycle we calculated average time sampling values as summarized in **Table 3**. In Section 5 this will be corrected to some extent by data filling and determining an

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

ideal Nyquist and data sampling rate.

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

*Average data sampling time in years for each cycle.*

*Boldfaced entries point to CO2 and combined Cycle C12.*

*Combined cycles C12, C23, C234 and C1234 mean data sampling times in years.*

*Short term temperature and CO2 cross correlation.*

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

Section 5.2 bellow.

**Table 3.**

**Table 4.**

**43**

**Figure 3.**

The CPE consists of three major blocks. First one is a data base of various ice core data, both original, conditioned and inserted as required to make the data more uniformly distributed across the time span of each cycle. The second block is data analysis, in time and frequency domains, including various correlation measures. The third block is the prediction engine which consists of a set of oscillators (KFHO) that produce a KFHB (Section 8). Various predictions as well as sensitivity analysis are performed in this block. Finally prediction parameters are fine-tuned by original ice core data vs. the prediction errors. This is primarily done to fine tune KFHO gains. Section 8 contains all the mathematical details of KFHB.

## **4. Average sampling time analysis**

First, we considered the entire set of Vostok data regarding time sampling. This will yield some initial indication of important issues in dealing with the data and how to perform further Vostok data filling to minimize the effects of non-uniform data distribution. As **Table 1** indicates the number of data points and the corresponding time periods for each cycle is quite different. This has to be taken into account when analysis is performed, as far as machine learning use of individual cycles for the benefit of estimating on going C1 data values. **Table 2** summarizes approximate duration of each cycle based on our definition of four cycles.


#### **Table 2.** *Climatic cycles approximate duration in years.*

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

**Figure 3.**

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

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).

The block diagram in **Figure 2** summarizes our CPE. It seeks the ability to predict various quantities using both past data (from one or more cycles) as well as the current data available This also allows to run various sensitivity analysis by using a variety of future scenarios as far as CO2 and temperature, for example. The CPE can be applied to any ice core data, Vostok, EPICA or any other and for any

The CPE consists of three major blocks. First one is a data base of various ice core data, both original, conditioned and inserted as required to make the data more uniformly distributed across the time span of each cycle. The second block is data analysis, in time and frequency domains, including various correlation measures. The third block is the prediction engine which consists of a set of oscillators

(KFHO) that produce a KFHB (Section 8). Various predictions as well as sensitivity analysis are performed in this block. Finally prediction parameters are fine-tuned by original ice core data vs. the prediction errors. This is primarily done to fine tune

First, we considered the entire set of Vostok data regarding time sampling. This will yield some initial indication of important issues in dealing with the data and how to perform further Vostok data filling to minimize the effects of non-uniform

corresponding time periods for each cycle is quite different. This has to be taken into account when analysis is performed, as far as machine learning use of individual cycles for the benefit of estimating on going C1 data values. **Table 2** summarizes approximate duration of each cycle based on our definition of four cycles.

**Duration in year**s **Cycle C1 Cycle C2 Cycle C3 Cycle C4** Temperature 127,726 115,156 86,462 96,782 **CO2 128,399** 109,800 86,148 95,587

KFHO gains. Section 8 contains all the mathematical details of KFHB.

data distribution. As **Table 1** indicates the number of data points and the

climate related societal and technological decisions.

variable, CO2, temperature, methane, O2, or dust.

**4. Average sampling time analysis**

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

*Climatic cycles approximate duration in years.*

**Table 2.**

**42**

**3. A climate prediction engine**

*Glaciers and the Polar Environment*

*Short term temperature and CO2 cross correlation.*

Detailed visual inspection of maximum values in **Figures 2** and **3** indicates that CO2 lags relative temperature in C4, C3, and C2. In C1, the data may be a little skewed due to the number of recent data points where maximum values are still not clearly identified in Vostok data (left-most CO2 data, **Figure 1**), as C1 is still evolving. Better lag estimates can be obtained by cross correlation analysis as done in Section 4 below. Another feature is the varied time differences when the data is obtained from ice core readings. In some cases difference between two data samples are only 400–500 years, but in some as much as 5000 years. To start the analysis for each cycle we calculated average time sampling values as summarized in **Table 3**. In Section 5 this will be corrected to some extent by data filling and determining an ideal Nyquist and data sampling rate.

The differences in average sampling times obviously come from the number of data points collected and the duration of each individual cycle. To facilitate our general approach all other combination of available cycles data are also considered. The idea is enrich with the varied of all available data for prediction purposes and also training purposes of machine learning approach. For example **Table 4** indicates respectively average sampling times for C1 and C2 combined (C12), C2 and C3 combined (C23), as well as C2, C3 and C4 combined (C234), plus the total data set C1234. Note from **Table 4** that more cycles we add the more uniform average data sampling times become, until it becomes equivalent for both CO2 and temperature. Further refinement of average sampling time is given in Section 5.2 bellow.


#### **Table 3.**

*Average data sampling time in years for each cycle.*


#### **Table 4.**

*Combined cycles C12, C23, C234 and C1234 mean data sampling times in years.*
