τdelay <2 times 1, 149 years 2, 300 years

The calculation is approximate, primarily because the non-uniform distribution of the Vostok data. Some data is separated by only hundred years rather than thousands. This points to a need to make the data more uniform by inserting additional data. We address how to harmonize these data in Section 5. The total delay appears to be of order of 2000 or more years and not 100–200 years. That may be an important finding which can influence our thinking about the role of CO2 increase caused by human actions. **Figure 4** indicates entire data set auto and cross correlation. It is clear that the data exhibits some periodicity. To determine average time delays between relative temperature and CO2 for individual cycles we examine **Figures 5**–**8** which also have numerical values around zero lag. The first two diagrams in **Figure 5** are autocorrelations and they also indicate certain periodicity within the each cycle but obviously not as well as the entire data set. The third diagram shows cross correlation between two variables. The time delay can be read from cross correlation and for C1 it is less than one lag period but maybe more than zero lag, due to the non uniformity of data. We can estimate it as less than half of one period lag. From **Table 3** for both

#### **Figure 4.**

*Total long term temperature and CO2 autocorrelations (left to right, above) and cross correlation (bellow) indicating inherent data periodicity.*

**5. Time correlation analysis**

*Glaciers and the Polar Environment*

up and down parts is chosen.

**Table 5.**

**Table 6.**

**44**

than the warming part. See also **Figure 9** for C1 CO2.

*Cross correlation coefficient for individual and entire cycle.*

*Cross correlation coefficients for individual sub cycles.*

As a starting point in time cross correlation analysis we examine correlation coefficients single numbers that can be used as simple measure of cross correlation intensity between two variables. There are several coefficients named after their inventors such as Spearman, Pearson and Kendal [24] and they all indicate certain statistical properties that can relate two data series. **Table 5** summarizes standard cross coefficient between relative temperature and CO2 for individual cycles as well as for the entire Vostok data set. The intensity of the cross correlation is quite high, on average more than 0.8 for the entire set. If we split the cycles into up and down sub cycles we obtain **Table 6** which indicates cross correlation coefficients for up and down cycle parts. Overall these coefficients indicate bigger spread between up and down sub cycles, and are very sensitive to where the break between

In general, one of the coefficients (up or down) is considerably larger than the overall single cycle coefficient. This might indicate that the usefulness of the individual up and down cross correlation analysis may be limited of the current C1 cycle. In the context of machine learning methodology this points to putting less emphasis on the cycles from longer in the past compared to the ongoing C1 cycle. To complete this analysis, the down-up period (boldfaced) ratios in **Table 6** indicate that the descending period is on average 6 to 8.6 times longer than the ascending period during interglacials, given that the cooling period is that much longer

**Cross correlation coefficient Cycle C1 Cycle C2 Cycle C3 Cycle C4 Entire cycle C1234** Temperature vs. CO2 0.8389 0.8094 0.8069 0.8191 0.821

Cycle 1 Down Correlation 0.869 Down Years 108,328

Cycle 2 Down Correlation 0.8008 Down Years 97,087

Cycle 3 Down Correlation 0.8558 Down Years 72,931

Cycle 4 Down Correlation 0.8519 Down Years 83,533

Up Correlation 0.8262 Up Years 15,353 Total Correlation 0.8389 Total Years 123,681

Up Correlation 0.9014 Up Years 11,219 Total Correlation 0.8094 Total Years 108,306

Up Correlation 0.6503 Up Years 12,234 Total Correlation 0.8069 Total Years 85,165

Up Correlation 0.827 Up Years 12,241 Total Correlation 0.8191 Total Years 95,774

**Ratio 7.05582**

**Ratio 8.653802**

**Ratio 5.961337**

**Ratio 6.824034**

relative temperature and CO2 the average data sampling times are 1703 and 1605 years putting the absolute delay at around 800–850 years or less. Similar approximate estimates can be done for other cycles. Reading from C2 and C3 cross correlations, **Figures 7** and **8** and **Table 3**, by the same consideration the delay is less than half the cycle, which translates into 400–435 years on average, or less. For C4 (**Figure 8** and **Table 3**) the delay appears to be of order of one cycle sampling time, a delay around 1800 years. To get a more precise approximations we would need more uniform data and finer resolution around the zero lag where the cross correlation is at its maximum. Note that on the individual cycle up and down parts this delay may differ from the average cycle level, also indicated by larger correlation coefficients spread in **Table 6**. We can identify no specific pattern in these coefficients regarding up and down sub cycles having larger or smaller coefficient. Overall, there is a significant CO2 delay across total Vostok data C1234 compared to the relative temperature. For individual cycles a more detailed cross correlation analysis should be done, especially following data insertion. We made very rough estimates above.

#### **Figure 5.**

*Cycle C1 temperature and CO2 autocorrelations (left to right, above) and cross correlation (bellow) indicating some periodicity within the cycle C1 itself.*

For practical reasons, the most important figure to keep in mind is the current cycle

*Cycle C3 temperature and CO2 cross correlation indicating some periodicity within the cycle C3 itself.*

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

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

*Cycle C4 temperature and CO2 cross correlation indicating some periodicity within the cycle C4 itself.*

Note at the end of this Section that the way various cycles are defined (**Figure 1**) also affects the analysis. In a follow up work we aim at repeating this analysis by defining cycles from minim to minimum CO2 values. Choice of maximum or minimum values may assist in determining what triggers cycle reversals. We have early indications that the dust might have a significant role in this reversal.

C1 delay which points to 800–850 years or less.

**Figure 7.**

**Figure 8.**

**Figure 9.**

**47**

*Cycle C1 CO2 content.*

#### **Figure 6.**

*Cycle C2 temperature and CO2 cross correlation indicating some periodicity within the cycle C2 itself.*

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

**Figure 7.**

relative temperature and CO2 the average data sampling times are 1703 and 1605 years putting the absolute delay at around 800–850 years or less. Similar approximate estimates can be done for other cycles. Reading from C2 and C3 cross correlations, **Figures 7** and **8** and **Table 3**, by the same consideration the delay is less than half the cycle, which translates into 400–435 years on average, or less. For C4 (**Figure 8** and **Table 3**) the delay appears to be of order of one cycle sampling time, a delay around 1800 years. To get a more precise approximations we would need more uniform data and finer resolution around the zero lag where the cross correlation is at its maximum. Note that on the individual cycle up and down parts this delay may differ from the average cycle level, also indicated by larger correlation coefficients spread in **Table 6**. We can identify no specific pattern in these coefficients regarding up and down sub cycles having larger or smaller coefficient. Overall, there is a significant CO2 delay across total Vostok data C1234 compared to the relative temperature. For individual cycles a more detailed cross correlation analysis should be done, especially following data insertion. We made very rough estimates above.

*Glaciers and the Polar Environment*

*Cycle C1 temperature and CO2 autocorrelations (left to right, above) and cross correlation (bellow) indicating*

*Cycle C2 temperature and CO2 cross correlation indicating some periodicity within the cycle C2 itself.*

**Figure 5.**

**Figure 6.**

**46**

*some periodicity within the cycle C1 itself.*

*Cycle C3 temperature and CO2 cross correlation indicating some periodicity within the cycle C3 itself.*

#### **Figure 8.**

*Cycle C4 temperature and CO2 cross correlation indicating some periodicity within the cycle C4 itself.*

**Figure 9.** *Cycle C1 CO2 content.*

For practical reasons, the most important figure to keep in mind is the current cycle C1 delay which points to 800–850 years or less.

Note at the end of this Section that the way various cycles are defined (**Figure 1**) also affects the analysis. In a follow up work we aim at repeating this analysis by defining cycles from minim to minimum CO2 values. Choice of maximum or minimum values may assist in determining what triggers cycle reversals. We have early indications that the dust might have a significant role in this reversal.
