Having selected the data vector to be analyzed (here our **'soi' **vector with the **SOI **time series) and the value of a sampling interval, a **Default **button is provided as a guide to set the rest of MTM parameters. By selecting the **Coherence** check box, user can choose another vector to compute MTM coherence with the **Data **vector. Both vectors should have the same length.

Results of **MTM **analysis, i.e. **harmonic F-test, Spectrum or Coherence, **are stored in a matrix with a name specified in **Spectrum** field. In addition, related **MTM confidence levels** (**F-test, Spectrum or Coherence***) *are stored in a matrix with the name obtained by prefixing "**conf**_" to a **Spectrum **name,** **and can be accessed in** Data I/O **tool. If results from several** MTM** calculations have been stored in different matrices, the parameters used in a particular **MTM** analysis will be restored in **GUI** by simply selecting correspondent matrix from a **Spectrum** pop-up list.

**All (default)****Narrowband****Hamonic**

Option **``All''** indicates that MTM spectrum will be tested both for the presence of **narrowband **signals, whose significance is measured by their amplitude relative to the estimated noise background, and for the presence of ``**harmonic**'' signals, which are significant as measured by the Thomson variance ratio test for periodic signals (**F-test**). Spectral peaks that pass both tests simultenously will form the **Reshaped** spectrum (see below **Advanced** settings), and together with the **Raw ** MTM spectral estimate are stored as results. Also, the frequencies of highly significant **narrowband** signals will be shown in MTM components table, that can be selected for subsequent Reconstruction.

Option**``Harmonic'' **will test the spectrum only for the presence of periodic signals, and the results of **F-test ** and its confidence levels are stored as MTM results. **MTM components** table will contain frequencies of highly significant **F-test **signals that can be selected for subsequent Reconstruction.

Here user can change various settings from their defaults, control **MTM** **Plot Options**, as well as reconstructing signal at selected significant frequencies.

Here the user can change threshold for significance** **of harmonic peak detection (**F-test** , **90%,95%,99%,99.5% and 99.9%**) used in the **``Reshaping''** procedure.

__The `Frequency Range' settings__

In addition **``Misfit Criterion''**
can be set to either a **log fit** or a **linear fit**:

This table will contain list of the frequencies of highly significant signals (greater than the 90% level ) either from MTM spectrum, MTM harmonic test or MTM coherence, which can be selected for subsequent MTM reconstruction.

If **Signal ** option is set to **All** or **Narrowband** , the **Components** table will contain a list of the central frequencies of narrowband signals (with their significance) in the MTM spectral estimate, identified as significant relative to the specified null hypothesis. For the **Harmonic** option of **Signal** it will contain
the ``harmonic'' signals which are significant as measured by
**MTM Harmonic test (F-test**) for periodic signals.

A maximum of the 200 most significant signals are stored.

** Plot Options**:

User can choose to plot different types of spectra, as well as confidence levels (**spectral, F-test or coherence**) by pressing on **Plot Options** button of **Advanced** **options **window:

To calculate the power spectrum of our SOI series, we set the parameter for the **``Reshape''** to 90%,
and click the **Compute** button.
Then, we click the **Plot** button to view the spectrum which should look like the following:

The above figure shows estimated continuous MTM spectrum or ``reshaped spectrum'' (the spectrum with estimated contributions of harmonic signals removed). Also, three additional smooth curves are shown, in increasing vertical progression, for the 90%, 95%, and 99% significance levels relative to the estimated noise background. Finally, the estimated harmonic or periodic component of the spectrum can be also displayed, i.e. the plot shows the estimated continuous spectrum, along with a curve that rises above it to indicate the portion of the spectrum above the continuous background associated with a period signal. The latter component is shown as a narrow spike of breadth equal to the spectral estimation bandwidth.

As we can see from the **MTM Components** table:

the MTM analysis identifies two highly significant peaks, one centered at * f=0.0146*, and another centered at

Results of **F-test **for harmonic signals can be stored and plotted by performing MTM analysis with **Harmonic **option chosen in **Signal **settings:

The list of **F-test** significant frequencies in the **MTM Components** table differs significantly from the narrowband signals:

which explains why no peaks have been reshaped in ENSO frequency band, and is another manifestation of the quasi-periodic nature of ENSO.

(Note: ** In a demo version**,

The **Components** table on the **MTM Advanced** options panel allows the reconstruction of the original time series from selected frequencies by clicking the **'Compute'** button in **Reconstruction**. The **``Plot''** button plots the sum of the reconstructed components against the original time series. The name of vector with reconstruction is specified by the user in** Result **field of

After selecting significant MTM spectral frequencies, clicking the **Compute** and **Plot **buttons in

By selecting the **Coherence** check box, user can choose a 2nd vector to compute MTM coherence with the **Data **vector, and store it in MTM results. Both vectors should have the same length, and the coherence will be computed with the specified number of tapers, resolution, and other relevant settings. Below are shown **Coherence **estimates of original SOI with MTM-filtered series,

and SSA-filtered series:

As also indicated by the MTM Components table,

SSA reconstruction provides for a very high coherence in a much wider frequency band then MTM, which is mainly restricted to the frequencies used in the reconstruction. This result is an agreement with the Blackman-Tukey cross-spectrum of SSA-filtered time series and original SOI, and also demonstrates SSA use as a very effective frequency band filter.