Are these ghost light spots and vertical grain indication that my scratched polarizer is ruined?

Question

Earlier, I unfortunately dropped my polarizer. There doesn't seem to be any damage save for a light scratch which I've uncovered only after a very thorough inspection. Nonetheless, I'm worried that there is damage worse than that.

The following feature all the same subject, a Christmas Tree in front of my department's building. This is my first time shooting suchlike subjects so I'm unsure if my polarizer is damaged or if the effects are something I should expect given my subject.

Set-up: Sony SLT-A35 with 55mm lens. Only thing on my lens is a Kenko polarizer.

First look at those "ghost colors" at the top left part of the tree. No matter how I turn the polarizer, they just won't go. But farther away from the subject and it's fine; they don't appear. And yes, I tried removing the polarizer while in this frame and distance and the ghost colors did disappear.

For example: http://chadestioco.deviantart.com/art/Christmas-Tree-Scrap-270316470

Should be pretty obvious what I'm talking about.

They respond to the polarizer the way glare does; that is they decrease/increase in intensity but never go away, at least not at that particular frame.

I have a close-up shot where the ghost colors didn't appear. See "Christmas Lights Scrap 2" in my DA scrapbook.

And then there's another thing that bothers me: some grain in my other shots. I'm not sure what/how they look like to you but it doesn't look like anything ISO-related to me.

Full picture at http://chadestioco.deviantart.com/gallery/#/d4gxt9o . Enlarge/view full image, around lower left of the tree, running from the grass lawn to the sidewalk ledge. Hope you can get what I'm talking about.

This is the most distinct I can find. As far as I've inspected my other shots, they all run vertical.

Also, I have some non-dark shots with my dropped polarizer and I don't see anything troubling with them. See "Front Facing Facade" in my DA gallery.

So, is my polarizer dead? Suggestions as to what I do with it?

Answer

1) extra images are just reflections from the polarizer. How can you tell? When you rotate it, they don't move. Therefore the angle doesn't matter, nor does the scratch matter. When you remove it, they go away. Therefore it must be causing it, but it is not related to the scratch or the angle being polarized.

2) The noise is just electronic noise. The higher you go in ISO the worse the noise gets. You can use noise reduction in post processing to remove it, but this has a price, you'll also loose fine detail and the image can start looking plasticky. You shot at 1600 ISO. Don't do that unless there isn't any other way to get the shot. Try again at ISO 100. Yeah, the shutter speed won't be 1/13, it will be much longer, so use a tripod.

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