This question came up in the Online 10-day Retreat Nov 3, 2020, guided by Delson Armstrong. Day 1 was part of a daily 30-minute discussion on the suttas, the Dhamma talk and reflections.
Slightly edited to improve readability
Observation is just another synonym that I used for Mindfulness. This is knowing that your attention was swerving from one place to another, and then bringing it back to your object of meditation.
Investigation is more in relation to bringing up and understanding how this phenomenon was caused. It can be used in conjunction with attention rooted in reality, yoniso manasikara. Investigating into the phenomenon of Sloth & Torpor is essentially utilizing observation.
First and foremost, you have seen and recognized that your mind is tending towards Sloth & Torpor. You then investigate into what Factor needs to be brought up. In other words, whether you need to bring up Joy or a little more Effort and put more attention towards the object.
The synonym for Investigation, that I use, is understanding. The end result of investigation is understanding. Once you have investigated what is required in that process of the meditation, where you’re leaning towards Sloth & Torpor, you then understand that this is the Factor you need to bring up a little more, in order to balance it. With that understanding, you apply the effort to bring in Joy, Energy or Effort.
Watch it here