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I have a series of buffered points. I want to apply a gradient to the buffer edge so it appears the buffer just fades away - no discrete line. How can I create this effect?

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In the symbol selector you can choose 'Circular Gradient' then select edit symbol, here you can select the color ramp style and outline type. enter image description here

When selecting the outline type select 'Edit Symbol' again and choose the 'Null' option from the drop-down menu:

enter image description here

The result is a gradient buffer with no outline like this:

enter image description here

There may be other ways to accomplish this but this is one option.

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For a raster-based approach, you can use the Euclidean Distance tool. For this example, I changed the raster display to histogram equalized stretched (Layer Properties > Symbology > Stretch > Type : Histogram Equalize). The stretch type will give you control on how the buffers appear--you may want to use a custom histogram. Also, you will want to experiment with the proper cell size to find a balance between resolution and file size.

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