Advanced SI
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., Fourierlog deconvolution, elemental mapping), image processing (e.g., jumpratio 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 xray spectroscopy (EDS) or EELS lowloss 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

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

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

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

Additional output

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

Reduced Chisquared – Shows the reduced chisquared (goodness of fit) parameter

Fit Uncertainties – Outputs the fit uncertainties



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

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

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


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

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


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

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

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

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

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


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

The computation then proceeds and the optimum fit is output in a new image display
Nonlinear least squares fitting
Nonlinear least squares (NLLS) fitting involves fitting models to spectral features to quantify the spectral peak properties. Nonlinear 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

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

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

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

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


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

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

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


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)

Ensure a single spectrum is front most, with an active NLLS fit region selected, when you choose this menu item

Click on the appropriate Constrain parameter value check box

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


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

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

Next, select Apply Model to Parent Spectrum Image

This applies the NLLS fitting on a pixelbypixel basis to the parent spectrum image you associate with the front most exploration spectrum

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


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

The associated parent spectrum image must also be open

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


This dialog enables you to specify the properties that are output by the routine

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

Fit Model Output – Outputs an individual computed model for each fitting region you specify


Once you specify the output preferences, then the computation will proceed on a pixelbypixel 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 multipleleastsquares fitting. Ultramicroscopy. 26:393 – 404.