Every ground plane I’ve ever seen has been connected to.. a ground!

The “active” part of the 1/4 wave is usually connected to the center conductor of the coax, the ground plane is connected to the shield and all ground planes are connected together.
That picture is a very wierd combination of a half wave horizontal V dipole and a ground plane. It “may” work but I have my doubts about whether it’s worth the effort as a simple V dipole works quite well.

It doesn’t take much to grab the vhf NOAA sats. I’ve done it with everything from a rubber ducky to a 250 foot long wire.

let’s think of the antenna requirements for low earth orbit. The following assume “normal” reception, not using  specialized (and far more expensive) equipment to squeeze the last second out of each pass.
1) It must be omnidirectional, hear equally well from all directions, because we don’t want to have to aim the antenna at the satellite and track it during each pass.
2) it should have best gain toward horizons and can have less gain directly overhand. The satellite is furthest away and there’s more atmosphere to absorb the signal when the satellite is lower in the sky. Overhead, it’s closer.
3) It should be able to receive both horizontal and vertical polarization. As the satellite passes, the angle of its antenna in relation to the angle of the receive antenna will change.
4) Gain isnt as important as a good “view” to the horizon in all directions. Gain won’t make up for being in the shadow of an object.

in order of complexity, here’s the three common antennas that meet the criteria.

a straight “V” antenna with one element connected to the center conductor and the other to the shield of the coax works quite well as a novice antenna.
pros: really easy to make. Decent performance, moderately omnidirectional in both vertical and horizontal planes, 
cons: changing polarization will cause brief sharp nulls during some phases.

a common vertical ground plane.
pros: can be made out of a panel mount “UHF” connector and a metal coat hanger, completely omnidirectional in the horizontal plane, null directly overhead in vertical plane, pretty good toward the horizon.
Cons: polarization is vertical, expect brief deep nulls as polarization shifts.

a “turnstyle” antenna which is basically two horizontal dipoles as right angles to each other.
pros: omnidirectional, slightly better performance than V dipole
cons: can be difficult to tune and match, changing polarization will cause brief sharp nulls during some phases.

a QFH antenna.
pros: omnidirectional on horizontal and vertical directions, polarization issues are minimal.
cons: mechanically complex, larger, ugly (according to my wife).

The antenna that’s being proposed violates requirement number 2. The ground plane, if configured correctly) will tend to boost signals overhead at the expense of weaker signals toward the horizon. At best, playing with spacing and lengths it could be forced to have a ‘lobe’ toward the horizon, but it would have dead spots as the vertical angle changes. Not a desirable situation for this type of reception. It’s a very strange he hybrid of a V and a ground plane. As a V is normally a dipole with both elements driven, it’s hard to see what the connection to the ground plane should be.

If your starting out, keep it simple, get the antenna outdoors with as large a view of the sky as possible. You will be surprised at how good the pictures are.
if you decide to get serious about this mode, I would jump straight to a QFH and not mess about with small signal improvements for lots of work.

Don’t get bent out of shape about “proper” coax. Normal, inexpensive, cable/satellite TV RG6 is just fine in most situations. Same with “LNA’s”. Unless you have a hearing impaired receiver, you should see pictures without and if you do have a deaf receiver, your better off saving your money and put it toward something decent.