**8.2 Other harmonics (H2, H4, H3, H7, H10, H5)**

The other harmonic calculations are done likewise, with specific amplitude and phase used and we will not give the obvious details here. The initial conditions are

scale, as in original Vostok data (in red). This can be improved by addition of more harmonics. The advantage which our CPE offers is its flexibility to treat both natural induced values of CO2 and other variables as well as modern times human

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

Other cycle contributions can be examined as a base for KFHB based machine learning training and testing. This is important due to different duration of various cycles and the uncertainty about the duration of the ongoing Cycle 1. Then older cycles may carry less useful data for the predictions about the current cycle, especially in a view of current global warming effects due to human activities not present in the climate past. On the other hand the old data carries useful short and long term information which can be used judiciously in the KFHB fine tuning, in particular due to its inherent periodicity. Hence multicycle analysis can be very useful for machine learning implementation of our CPE based on KFHB idea. The minimal choice of harmonics will allow us to devise a reasonably simple machine learning algorithm for training, testing and prediction purposes based on KFHB. For example the longer data sets in C234 can be used as a training data set, as can shorter C23, in order to predict the completion of C1, calculating prediction error E1,234 (Error in predicting C1 given C2, C3 and C4 cycles) or E1,23 (predicting C1 given only C2 and C3 data). This applies to both temperature and CO2. This can be repeated for other components in Vostok data set, such as methane, oxygen and insolation. Similarly for the European EPICA data set as well as set of cycles indicated in Milankovich theory [31]. Hence, once we predict C1 using C123 or C23, we obtain errors E1,234 and E1,23. Intuitively we can expect that E1,23 > E1,234, i.e. training based on larger data set ideally would produce smaller test and prediction errors. This has to be confirmed by the further analysis, in particular due to differ-

Besides our aim at producing an effective CPE methodology, the harmonic analysis spurred a variety of related thinking and ideas which we also summarize in this paper. Some further ones follow. Climate change on the time scales of the ice cores has been considered as consistent with the IPCC's hypotheses [11], focusing on permanent greenhouse gases, particularly CO2, methane and nitrous oxide. This has included the role of increasing water vapor but viewed only as a secondary amplifying factor. The effect of temperature on water vapor pressure is shown by the Clausius-Clapeyron equation, dictating an exponential increase in vapor pressure as temperature rises. But the role of water vapor in modern global warming is only considered in GCMs as a derivative of primary warming by permanent greenhouse gases. This may be in error, as modern irrigation is now adding an extra 4% of water to land surfaces from 1960. It is possible to estimate water vapor content of atmospheres of different eras from temperature data. We have also hypothesized a positive forcing from irrigation water [32, 33] in addition to other primary sources of warming such as the Milankovich astronomical cycles. This may prove a more reliable means of correlation using the link already established between water vapor

One feature of some of the ice core analyses is the irregular but rising increasing levels of dust as the planet became colder (**Figure 1**), possibly absorbing rather than scattering solar radiation. From frequency analysis, a marked impulse effect can be recognized, given that the peak of the dust samples in ice cores clearly coincides with the commencement of the interglacials, suggesting a role in initiating the warming process. Dust-climate feedbacks have recently been highlighted as having

produced effects attributed to global warming.

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

**9. Further considerations**

ent lengths of the various cycles.

responsible for more than 80% of the air heating.

**57**

#### **Figure 13.**

*Comparison of Vostok data vs. approximation with 7 harmonics.*

calculated in the equivalent way as given in (31) using different values for amplitude, frequency and the phase angle, using **Table 7** harmonic values. Once calculated, initial conditions drive the KFHO for each harmonic, with the various harmonics KFHO parameters similar to the **Tables 11**–**12** above for H1 and the other harmonic signals like in **Figure 12** but with different frequencies and phases. Next all of the predicted and corrected x1 and x2 estimates are combined per KFHB output in Eqs. (28) and (29). Resulting values are compared to the known original and inserted Vostok data and appropriate MAPE are generated to check the performance of the method. **Table 9** summarizes resulting MAPE for all 7 harmonics. Note that the results are based on constant filter gains in KFHB similar to the values in **Table 10** which are equivalent to the optimal time varying gains to at least 2nd decimal point. Parameters in **Table 11** were not optimized in any way and they can be tuned further based on errors obtained. This may be done so Q values would be adjusted per specific harmonic amplitude. We made a simple assumption that the standard model deviations and variances are of order of 100 for both states and all harmonics, and cross variance of order of *a*0*Q*11*=*2, as noted earlier. Table also indicates what we expected, namely that the corrected errors are smaller than the predicted ones.

Note that the values for y are just specific harmonic amplitudes calculated based on maximum amplitude and phase angle, per Eq. (20). The prediction errors correspond to one sampling interval, in our case 1011 years. This is large enough for long terms calculations and prediction. If we chose to have 2 or more sampling intervals the prediction errors will obviously increase.

As far as corrected errors, they correspond to the situation where we have a specific CO2 value to use for KFHB correction. For example if we chose to consider a current CO2 value we can append it to the beginning of the Vostok data, and we actually did that by adding 350 PPM value for CO2 'now' in front of the 'newest' Vostok data, more than 2000 years old. The CPE as we envisioned it allows for all sorts of scenarios, sensitivity analysis, 'what if'scenarios for all the variables, CO2, temperature, methane, and so on, EPICA and other ice core measurements, both for short term (100+ years) as well as long term (1000 years+). To give a visual impression of Vostok data approximated by first seven harmonics, see **Figure 13** above. Approximation by 7 strongest amplitude harmonics produces very smoothed Vostok data (in blue) without capturing very abrupt changes on a smaller time

scale, as in original Vostok data (in red). This can be improved by addition of more harmonics. The advantage which our CPE offers is its flexibility to treat both natural induced values of CO2 and other variables as well as modern times human produced effects attributed to global warming.
