Once acquired, you can treat a 3D electron energy loss spectroscopy (EELS) dataset $$I(E,x,y)$$ as a collection of spectra or sequence of images irrespective of acquisition mode. You can apply conventional electron energy loss spectral processing techniques (e.g., Fourier-log deconvolution, elemental mapping), image processing (e.g., jump-ratio imaging, MSA) or progress to use more advanced analysis techniques.

## Multiple linear least squares

You can use the multiple linear least squares (MLLS) method to fit a number of reference spectra and/or models to a spectrum. The reference spectra can be fitted as a linear combination.

$$F(E)=AE^{-r}+B_{a}S_{a}(E)+B_{b}S_{b}(E)+...$$

where

• $$S_{a}(E),S_{b}(E)...$$ = Reference models

• $$B_{a},B_{b}...$$ = Scaling coefficients

### Common uses for MLLS Include

• Separate overlapping EELS Edges – Extracts edge signal when background removal fails

• Spectral phase mapping – Map out the spatial distribution of a certain spectral shape (e.g., energy dispersive x-ray spectroscopy (EDS) or EELS low-loss distribution)

• Energy loss near edge structure (ELNES) fingerprinting – Use references to determine the spatial distribution of chemical states for an edge using references

• Anisotropic studies – Orientation and coordination mapping

### Workflow detail

1. The MLLS Fitting Preferences dialog in the Gatan Microscopy Suite® (GMS) 3 software contains commands for setting up and performing multiple linear least-squares fitting.

1. Use Fit Weights – Specifies the type of weighting to use when determining the least-squares fit parameters

2. Output fit as – Determines whether the fit is output as a coefficient or scaled to give the signal integral

1. Residual (Misfit) Signal – Displays all the fit parameters and their uncertainties, by which you may judge the quality of the fit

2. Reduced Chi-squared – Shows the reduced chi-squared (goodness of fit) parameter

3. Fit Uncertainties – Outputs the fit uncertainties

2. When you are ready to perform a MLLS fit of any spectra and/or models to a specific portion of the spectrum (e.g., analysis of overlapping edges and superimposed fine structure), select the Perform Fitting menu item.

1. Initiates the program to form a model function that consists of a linear combination of the specified spectra and/or models

2. The program then fits that model to the foreground spectrum when it adjusts the coefficient of each linear term to minimize the square deviation between the model and the selected spectrum

3. If the fit spectrum has one or more image slices, specify the spectra (or models) to use in the fit.

1. Specify at least two valid and appropriate spectra for the procedure to commence

4. Then confirm the range over which you wish to perform the fit.

1. Note that if you place a region of interest on the fit spectrum before you execute this command to specify the fitting region, then the values corresponding to the region of interest range will be in the appropriate dialog fields

2. If the reference spectra do not fully cover the range you specify, a suitable alert will appear

3. In the event that the reference spectra have dispersions different from the spectrum you want to fit to, they will be interpolated to the same dispersion

4. If the interpolation factor is deemed to be too extreme, then a warning will appear to inform the you that the reference spectra you provide might be inadequate for an accurate analysis

5. If you select the Compute from Data fit-weights option in the MLLS Fitting Preferences dialog; then specify the location of the original source data that the fit spectrum originates.

6. The computation then proceeds and the optimum fit is output in a new image display.

## Non-linear least-squares fitting

Non-linear least squares (NLLS) fitting involves fitting models to spectral features to quantify the spectral peak properties. Non-linear refers to the models being functions, rather than static references (c.f. MLLS fitting). The NLLS fitting tools within DigitalMicrograph® software allow you to fit one or more Gaussian peaks to a spectrum. Once fitted, the fitting parameters can be output (amplitude, center, height). You can apply the peak fitting to an entire spectrum image, hence fitting parameters can be shown as 2D maps. This provides a powerful tool for mapping peak shifts in a spectrum image.

### Workflow detail

1. The NLLS Fitting Preferences dialog in the GMS 2 software contains commands for setting up and performing multiple linear least-squares fitting.

2. Within this dialog, select the Fit multiple NLLS Models mode appropriate for your experiment.

1. Simultaneously – Fitting algorithm will attempt to find the optimal linear combination through least-squares fitting for all the specified fit models simultaneously

2. Sequentially – Causes the Gaussian models to be fitted individually in an ordered, sequential manner

3. Next, select Fit Gaussian to ROI button to assign the region of interest (ROI) you select as a Gaussian NLLS fit.

1. When you perform this function, ensure a single spectrum is frontmost, with a range ROI selected and positioned over the desired fitting range

2. This will designate the NLLS fitting region; it will have a label and solid outline, plus a Gaussian model will be fit and shown

4. Select the Constrain Model Parameters menu item to constrain one or more of the selected NLLS model fit parameters to its current or specified value(s).

1. Ensure a single spectrum is frontmost, with an active NLLS fit region selected, when you choose this menu item

2. Click on the appropriate Constrain parameter value checkbox

3. Once complete, select OK to close the dialog and update the fit model

5. Open the Output Fit Values to Results window to output the NLLS fitting parameters for the front most spectrum.

6. To initiate this routine, select this submenu item with the NLLS fitted spectrum of interest frontmost.

7. Next, select Apply Model to Parent Spectrum Image.

1. This applies the NLLS fitting on a pixel-by-pixel basis to the parent spectrum image you associate with the frontmost exploration spectrum.

2. The output will include the model fit properties as a line profile or map, respectively.

8. To perform the above operation, the system uses the frontmost spectrum imaging with one or more active NLLS fitting regions to make an exploration spectrum.

1. The associated parent spectrum image must also be open.

2. On initiation, the routine will first present you with the SI NLLS Fitting Output Options dialog.

9. This dialog enables you to specify the properties that are output by the routine.
1. Fit Parameter Output – Outputs and labels the individual NLLS model fit parameters (e.g., amplitude, center, and width for a Gaussian model) to a new image display.
2. Fit Model Output – Outputs an individual computed model for each fitting region you specify.

10. Once you specify the output preferences, then the computation will proceed on a pixel-by-pixel basis to perform the NLLS fitting, while it uses the fit regions and parameters you specify for the exploration spectrum.

## References

Leapman, R. D.; Swyt, C. R. Separation of overlapping core edges in electron energy loss spectra by multiple-least-squares fitting. Ultramicroscopy. 26:393 – 404.