# Continuity of a quantity in a conical system to determine the velocity

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• tworitdash
tworitdash
My research is on radar images and the images are collected in several conical surfaces. These conical surfaces have the same origin, the same maximum length (max flare or max range), but different elevations angles. The images are collected on the surface of the cones only.

I want to determine the velocity field in 3D for this image. I have several measurements of the images in time. Usually, in the literature I have seen people using a continuity of the brightness itself (image) in time and space assuming that the reflection is conserved.

However, it is usually done in polar co-ordinates. It is due to the fact that the radial velocities (one component of the flow field in the polar coordinates) are also measured through a radar with Doppler effect.

I was wondering if it is possible to formulate it in a very generic way where I consider a conical coordinate system instead such that I can make use of all the cones? Or, is it a bad exercise ?

Furthermore, it is usually assumed that the energy is conserved so the following cost function is always imposed in optimizing for the flow fields:

$$J = \left(\frac{\partial \eta}{\partial t} + V_r \frac{\partial \eta}{\partial r} + V_{\theta} \frac{\partial \eta}{r \partial \theta} + V_{\phi} \frac{\partial }{r \sin(\theta) } \frac{\partial\eta}{\partial \phi} \right)^2$$.

However, if the total reflection is not conserved, how can I optimize for the flow field? Do I have to estimate a constant $A_0$ such that,

$$\frac{\partial \eta}{\partial t} + V_r \frac{\partial \eta}{\partial r} + V_{\theta} \frac{\partial \eta}{r \partial \theta} + V_{\phi} \frac{\partial }{r \sin(\theta) } \frac{\partial\eta}{\partial \phi} = A_0$$ ?, instead of 0 ?

The reflection or image is $\eta$, and the flow field is $\vec{V}$.

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