How does one perform a missed obligatory prayer (Qada) when intending it?

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If a person intends a missed prayer by specifying it in their heart, such as 'Dhuhr of today,' they do not need to specify it as Qada or Ada. If such a person intended it as Ada (on-time performance) but it turns out the time had already passed, the prayer counts as Qada without a specific intention for Qada. Conversely, if one believes the time has passed and intends it as Qada, but it turns out the time was still valid, it counts as Ada without an intention for Ada. This is analogous to a captive who fasts a month believing it to be Ramadan; if it coincides or is the month following, the obligation is fulfilled.