for your geometry. This is not the cross-polarised response that the Ez will give you as a good estimate (there is a small offset in the components and it is not exactly cross-pol) of GPR cross-polarised response.
The vertical component above a metal target approaches a minimum directly over it. It looks asymmetric because it is the field received above the target irrespectively of the position of the transmitter. In your Ex scan the apex of the hyperbola is at the mid-point of your Tx and Rx as it should. However, the vertical component is at a minimum directly over the target. That means that it should appear before the apex and when the Rx itself is above the target and not when the mid-point of the Tx and Rx is over it.
I have done a similar analysis in my PhD thesis very very long time ago looking in the response of the magnetic fields but the concepts are the same.
If you are interested in the cross-polarised response that some GPRs provide then you are looking at the wrong component. In this case you need the Ez one. Your model is otherwise OK.
Hope this helps
Antonis