What kind of scientific experiment can a satellite conduct to determine the internal stucture of a gas giant? I know that geophones and seismic waves can be used for a planetary lander on a rocky planet (e.g Insight lander). What other kind of wave/instrument can be used from a satellite to determine a gas giant’s interior composition? Would such an experiment require multiple satellites (e.g. to monitor refraction effects)?
@SteveLinton gave the primary investigations for determining a giant planet's internal structure: gravity field structure and magnetic field structure. One minor correction: while the round-trip propagation times ("ranging" data) are useful, the most accurate data are Doppler data of the spacecraft's velocity vs time, as described in Arv Kliore's paper on the Cassini radio science investigations (skip to the section on Celestial Mechanics investigations). Arv's paper is a good general reference for the rest of this post. Doppler measurements of velocity translate directly via differentiation into accelerations; more about that in a bit.
If all planets were perfectly spherically symmetric the celestial mechanics experiments would give information only about the total mass of a planet, since a spherically symmetric object produces a spherically symmetric gravity field. But all planets have asymmetries, especially fast rotators that become significantly oblate. All the giant planets are fast rotators so they are great subjects for study. For a given size and rotation rate, the higher the object's mass density, the less oblate it will be. This is why Saturn's oblateness is so much larger than Jupiter's despite its slower rotation rate. Oblate objects produce a gravity field that is cylindrically symmetric but not spherically symmetric, described by mathematical functions called spherical harmonics. With Radio Science Celestial Mechanics (RSCM) investigations (using Doppler and ranging data) we can measure the spherical harmonics to great accuracy.
As you descend deeper into a giant planet the local mass density naturally increases, responding to the competing influences of increasing pressure and temperature with depth. At times, when a compositional change or sudden decrease in the thermal lapse rate occurs, a more rapid increase in density can occur, leading to an approximation of a density contrast surface ("surface" in the sense of the surface of an abstract shape, not a liquid or solid surface!): above that surface the density is distinctly lower than below it. The density contrast surface will be oblate as well, and the degree of oblateness will be a function of the densities above and below. This oblate density structure also contributes to the spherical harmonics.
Good measurements of the spherical harmonics of a planet's gravity field allow inferring the vertical profile of mass density, the primary measure of interior structure, from the troposphere to the core. Integrating the appropriate equations relating such things as gravitational acceleration, pressure, and temperature to the equations of state of planetary materials (hydrogen, helium, methane, water, silicates, metals, etc.) downward from the troposphere allows roughly inferring composition as a function of depth, constrained by the measured density profile. If a proposed compositional profile, convolved with those equations, produces a density profile that disagrees with the measurements, then whoops!, back to the drawing board!
Magnetic fields add more information to the problem. Dynamo fields are generated at depths where you get significant electrical conductivity of the constituents there. Similar to equations of state, for most constituents of giant planets we have good experimental measurements of their conductivities (including mixtures) as a function of temperature, pressure, etc. Measuring the structure of a planet's dynamo magnetic field tells you the depth where it is generated, and that helps to constrain the vertical profiles of pressure, temperature, and composition.
How do we get the Doppler measurements for RSCM investigations? A Deep Space Network (DSN) station on Earth transmits to the spacecraft a radio signal whose frequency is very accurately determined by a hydrogen maser at the station. When that signal arrives at the spacecraft it has been Doppler shifted by the relative velocity along the line of sight between the DSN station and the spacecraft. The spacecraft then turns that signal around and retransmits it (transponds it) back to the DSN, doubling the Doppler shift on the original signal. At the DSN station the received signal is compared to the transmitted signal and the Doppler shift extracted, giving the relative velocity of the spacecraft with respect to the DSN station's antenna, along the line of sight. Huge programs take all the Doppler and ranging data (and any other relible data the investigators can scrounge!) as input, and then do best fits to the planet's total mass, its location in space, its spherical harmonics, the spacecraft's location with respect to the planet—a huge list of variables that are best-fit to the data. JPL's program is called "ODP" (not very exciting, eh?) and several other institutions have their own software.
I mentioned differentiating Doppler velocity measurements to get accelerations. (For those who haven't had calculus, this is essentially looking at how much the velocity changes from one measurement to the next, which is acceleration) There is a bit of a twist to this: since the velocities are only the components of velocity along the line of sight, so are the accelerations! The gravitational accelerations derived from the data are the components of the planet's gravity field along the line of sight. If the gravitational acceleration vector is perpendicular to that line of sight, you can't measure it! The component along the line of sight is zero! This is why RSCM data are best when the spacecraft's orbit plane aligns edge-on with the direction from the planet to Earth.
And that is why the RSCM data Cassini got in its final orbits are so exquisitely good: the orbit's periapsis, just next to grazing the atmosphere, was almost directly between Saturn and Earth. The spacecraft was immersed in the strongest regions of the gravity field, and the acceleration vectors were well aligned with the DSN-Cassini line of sight. Having flagship-class radio equipment didn't hurt, either! Juno won't do quite as well because their orbit plane is closer to face-on than edge-on, so there's a large angle between the gravitational acceleration vectors and the line of sight, even at periapsis.
Stay tuned! There's another potential measurement technique in the offing that might revolutionize studies of giant planet interiors! Planetary normal mode seismology is closely related to helioseismology, the discipline that has provided the great majority of our knowledge about the interior structure of the sun. It is based on things like turbulence in a planet's interior exciting normal mode oscillations, compared in the popular literature to "ringing like a bell". Doppler measurements of these oscillations tell you a lot about the interior structure of the "bell". The character of these oscillations depends on mass density, to be sure, but they also depend critically on the intermolecular and interatomic forces that give rise to acoustic velocity (the speed of sound), so these measurements are complementary to the gravity field measurements. The two are synergistic! Several groups in the US and Europe are working on instruments small enough to fly on spacecraft.
One uncertainty in this approach is the strength of the normal mode oscillations in a planet instead of in the sun. They depend somewhat on the amount of turbulence and other driving forces, which are very uncertain. One French group claims to have detected normal mode oscillations in Jupiter from Earth-based observations but there is some skepticism in the community. Measurements made in the near vicinity of a planet will be much more sensitive than ones made from Earth. If this method pans out, look for "Doppler imager" instruments on future spacecraft headed for any giant planet.
Just about any data concerning a planet can contribute to our understanding of its internal structure one way or another.
That said, probably the most powerful techniques at the moment are observation of the planets gravitational and magnetic fields and how they vary in space and time. Observing the gravitational field amounts to accurately tracking the position of the satellite which is done from Earth, by measuring (very accurately) the time for a radio signal to make the round trip to and from the probe. Magnetic fields can be measured directly.
I imagine multiple satellites would be nice -- ranging between a pair of satellites has been used to map the gravity of Earth and the Moon to unprecedented accuracy, while a fleet of 4 satellites has provided very high qulity mapping of Earth's magnetic field in 3D and over time.
There is a very clear description of both these instruments in the case of the Juno probe on this page. Juno is a particularly good example, both because it is the most recent outer planet mission, and because its main target is Jupiter itself, rather than the moons.
Another technique that has been used on moons of some gas giants is exact measurement of the rotation of the surface of the moon. The way this changes over time in response to tides from other moons can reveal whether there are liquid layers beneath the surface.