If you attach your post to your 8" concrete patio, you have four key things to consider:
- Local building code concerns
- Lateral load
- Uplift load
- Soil concerns
Local building code
I was surprised to find that my local building & planning department had requirements for a simple pergola. "Gazebo/pergola type structures are usually not of typical “light frame construction” and are treated as pole construction projects that require “structural engineering” done by an Idaho licensed engineer. Height restrictions may apply in some areas."
Lateral load
Setting your posts down into the ground gives your pergola a lot of ridigity when it gets pushed sideways from wind. I haven't seen any plans for pergolas where no pier or footing is recommended.
Your rough sketch shows three parallel posts. If you don't set your posts down into the ground to secure them, these posts could tilt sideways with wind load. Consider adding "knee braces" where your posts connect to the horizontal beams. For knee braces, see this answer.
Recommendations from a sun shade company
To get an idea for securing posts for a simple pergola, a sun shade company suggested setting 6"x6" posts into the ground at least 36", with concrete surrounding the post (30" diameter):
Footing depth should be a minimum of 36 inches deep and 30 inches
square. For example, if your post height is higher than 8 feet, we
recommend a hole depth 40% of the height above ground.
Link to PDF
This sun shade company noted they are happy to take calls, so you may consider asking your future sun shade supplier what they recommend.
Beware of freezing
In a great post by Piffin over at FineHomeBuilding, the results of freezing and tilting posts are discussed:
...look at it this way, if the ground heaves and the slab tilts a quarter
inch, you may not notice it much underfoot, but if that tilt plays into
the column at ten feet high, that could result in a 3 or 4 inch lean,
depending on how everything is tied together.
If you have minimal or no ground freezing, this is less of a concern.