**8.1 First harmonic H1**

The harmonic analysis of Cycle C1 Vostok data is presented in Section 5. In this Section we proceed and build a specific first harmonic H1 KFHO, as an example of CPE set up, with the initial conditions:

$$
\hat{\mathbf{x}}\_1(\mathbf{0}/\mathbf{0}) = \mathbf{A}\_1 \cos \left(\theta\_1\right), \hat{\mathbf{x}}\_2(\mathbf{0}/\mathbf{0}) = -2\pi f\_1 A\_1 \sin \left(\theta\_1\right) \tag{31}
$$

were A1 and θ<sup>1</sup> are first harmonic amplitude and phase. In **Table 10** we show H1 KFHO gains from Eqs. (15), (16) for the first seven time samples. The boldfaced values indicate the moment (after 5 time samples) when the gains become constant. These constant gains can be used in the KFHO design for its simplicity. **Table 11** summarizes various filter parameters. The results for predicted and corrected state estimates *x*^<sup>1</sup> and *x*^<sup>2</sup> for 128 time samples are shown in **Figure 12**. The reference harmonic H1 data is calculated using standard cosine function and it is the

**x1(0) x2(0)** Initial Initial 231.0068 0 26.42065 1.20707 13.48622 0.81265 4.1098 1.643843 6.050971 0.45685 0.754028 1.87793 4.844518 0.7715 2.016297 1.070199

Q11 100 Q22 100 Q12 49.759 R 25 a 1.99759

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

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

"measurement" as in (9) (**Table 12**).

**Table 11.** *Parameters.*

**Figure 12.**

**Table 12.** *Initial values.*

**55**

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

*Complete data, H1 harmonic, predicted and corrected states.*

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


**Table 10.** *Filter gains.* *Kalman Filter Harmonic Bank for Vostok Ice Core Data Analysis and Climate Predictions DOI: http://dx.doi.org/10.5772/intechopen.94263*


**Table 11.**

**Figure 11** indicates this arrangement with KFHO where we have:

*x t* ^ð Þ¼ *<sup>=</sup><sup>t</sup>* � <sup>1</sup> <sup>X</sup>

precise than predicted with the real data available.

*Glaciers and the Polar Environment*

(following inverse Fourier Transform) as in (27) with:

*y t*ðÞ¼ <sup>X</sup> *N*

realistic) environment.

**8.1 First harmonic H1**

CPE set up, with the initial conditions:

*Boldfaced entries indicate constant Kalman Filter gains.*

**Table 10.** *Filter gains.*

**54**

1 *yi*

**8. Vostok cycle C1 CO2 Kalman filter harmonic oscillators**

*N*

1

*N*

1

General idea here as compared to a simple sum of harmonic cosine signals

ðÞ¼ *<sup>t</sup>* <sup>X</sup> *N*

1

is to accommodate stochasticity of the underlying Vostok measurement data as well as a simple linear structure of Kalman Filters Harmonic Oscillator, and its ability to make predictions for the signal future values in the probabilistic (more

The harmonic analysis of Cycle C1 Vostok data is presented in Section 5. In this Section we proceed and build a specific first harmonic H1 KFHO, as an example of

**Optimal Constant Gain** k1(+) k2(+) N/A N/A 0.96149 0.48075 0.93616 0.29119 0.93633 0.29926 0.93578 0.30137 **0.9357 0.30151 0.9357 0.30151 0.9357 0.30151**

*x*^1ð Þ¼ 0*=*0 *A*<sup>1</sup> *cos* ð Þ *θ*<sup>1</sup> , *x*^2ð Þ¼� 0*=*0 2*π f* <sup>1</sup>*A*<sup>1</sup> *sin* ð Þ *θ*<sup>1</sup> (31)

representing the total KFHB predicted state estimate. These can be used for short and long term prediction purposes for CO2, temperature or other variables of interest. We can similarly define a set of corrected state estimates which are more

*x t* ^ð Þ¼ *<sup>=</sup><sup>t</sup>* <sup>X</sup>

*x*^*i*ð Þ *t=t* � 1 (28)

*x*^*i*ð Þ *t=t* (29)

*Ai cos*ð Þ *ωit* þ *θ<sup>i</sup>* (30)

*Parameters.*

**Figure 12.** *Complete data, H1 harmonic, predicted and corrected states.*


**Table 12.** *Initial values.*

were A1 and θ<sup>1</sup> are first harmonic amplitude and phase. In **Table 10** we show H1 KFHO gains from Eqs. (15), (16) for the first seven time samples. The boldfaced values indicate the moment (after 5 time samples) when the gains become constant. These constant gains can be used in the KFHO design for its simplicity. **Table 11** summarizes various filter parameters. The results for predicted and corrected state estimates *x*^<sup>1</sup> and *x*^<sup>2</sup> for 128 time samples are shown in **Figure 12**. The reference harmonic H1 data is calculated using standard cosine function and it is the "measurement" as in (9) (**Table 12**).
