Are the ocean and present atmosphere leftovers from the original atmosphere of earth?

The ocean and the present atmosphere themselves are not leftovers from the original atmosphere of Earth. However, they do contain components within them that give us and scientists ideas of what Earth's past atmosphere was like. 

Some example include:

1) Ice cores - scientists may go to Antartica or Alaska and stick this metal tube into the ice to remove what is known as an "ice core". This is then taken back to the lab for analysis. So what happens is that this ice as we know is constantly melting and freezing with different seasons and climate change. So, when the ice starts to crystallize, particles like carbon dioxide, sulfur etc may get trapped as bubbles in the ice. This is what scientists look for in ice coring and this is how they know that Carbon dioxide levels were relatively high back then (Not as high compared to now). 

2) Ocean sediment cores - this is the exact same process of the ice cores except they take cores of sediments (esspecially calcium carbonate that contains oxygen). As you know, CaCO3 is found in the exoskeletons of organisms that die and build up on the ocean floor. 

Those are some of the main ways that the present atmosphere and oceans have leftovers from the original atmosphere of Earth. 

Hope that helped!