How to choose a good start point for a least square fitting problem, if the result is highly depended on a good initial guess

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A two-dimensional pattern have five parameters. I hope to perform a least square fitting of this pattern to solve for the unknown parameters.

The problem is, whether we could find a good result is highly depend on the initial guess. The only thing I can think of is to just randomly choose initial guess, then I pick the one with the least square error as the result? Any better strategies for choosing the initial guess?