How can I combine multiple Overlay layers into a single one?

Question

Here I have a simple image of a cat:

I add a new layer on top of it and near its mouth area I add a small circle and apply gaussian blur to it, and set the mode of the layer to overlay to create this(I used this merely for demonstration):

Now, suppose that I wanted the same effect more intense and that I duplicated the same circle layer two times, to create this: (all of them in overlay mode)

Now, How do I keep the effect of the three overlay layers but make them only one single layer? If I merge them down, the effect goes away:

How can I do this in GIMP?

EDIT: Just for the clarification for other folks whom I directed to this post, another reason to want to do this is when we create a layer mask for the above layer with overlay we may have to copy and paste the layer mask to every overlay layer and making edits to layer masks even troublesome.

Answer 1

The equation of the Overlay mode is:

Where I is the composition of all layers under the layer in Overlay mode and M is the Overlay layer.

With 3 layers, the result of the top layer is E, the next layer applies the same formula (replacing I by E) and the third layer does this again. I'll let you figure out the resulting formula, and see if there is one of the blend modes that matches it. Answer: likely not.

Combining the layers in a group won't help, since this gives the same result as merging the layers and then putting the resulting layer in the blend mode of the group.

There may be ways to achieve equivalent effects, but they would be different depending on the image.

Answer 2

GIMP documentation has the equation how the result of overlay mode is calculated. It's a second order polynomial of original color intensity value. The result is a fourth order polynomial of the original image color intensity if the same overlay is inserted twice.

There's no fourth order functions in layer mode equations => no single layer with well selected layer mode can give the same result as inserting 2 or more identical overlays.

You have a theoretical possiblity to write a pixel level script which calculates the same function as inserting 2 or more identical overlays. Writing such script and giving any instructions how to write it and how to keep the result compatible with the non-extended GIMP is unfortunately well beyond my capablities.

ADD (inspired by the comments)

If somehow resembling visual effect is enough, you can try inserting a 50% grey layer behind your cat photo. Give to the cat photo mode=Hard Light. Paint yellow onto the grey layer.