The power returned to the radar is much less than the power transmitted by the radar. Imagine the sun as a transmitting radar and the planets are the hydrometeors. Some of the sun's energy goes from the sun and is reflected off the planets back toward the sun. This energy that makes it back to the sun is only the slightest of a tiny fraction of the original energy the sun emitted.

The return that makes it back to the radar is a function of several variables. The radar equation will show us these variables.

The radar equation (power received back to radar) is =
(Pi^3*Pt*g^2*O*o*h*K^2*l*z) / 1024*Ln(2)*wavelength^2*r^2

Where Pt is power transmitted, g is gain, 0 and o are beam widths, h is pulse length, K is refraction term, l is attenuation term, z is radar reflectivity factor, wavelength is the wavelength used by the radar and r is the radius from the radar to the precipitation echoes.

Most of the variables in the radar equation are constants for any given radar, thus the equation simplifies to:

Pr (power returned to radar) = (c2*z)/r^2

c2 is the single value of all the constants put together. This constant will be different for different radars and we will go over how power received can be increased by using a different radar later in this essay.

First we will look at the power received as a function of the radar reflectivity factor and the radius from the radar to the precipitation echoes. Notice in the equation Pr = (c2*z)/r^2, that z is in the numerator while r is in the denominator. Since z is in numerator, if z increases while r remains constant then the power returned to the radar must increase. This makes sense because z increases by increasing the size or number of hydrometeors. The return to the radar will be stronger for larger and more numerous drops for any given radius to the hydrometeors. Since r is in the denominator, when r increases and z is constant then the power returned must decrease. This makes sense because an object further from the radar will receive less radar radiation to scatter off of it than an object closer to the radar. Think of the planet examples again. Mercury and Venus get much more solar energy scattering off of them than Pluto does since Mercury and Venus are closer to the sun. Thus, both the size/number of hydrometeors and the distance (radius) to those hydrometeors determines how much radiation is scattered back to the radar.

We mentioned earlier that some of the constants in the radar equation are different for different radars. These terms include the power transmitted, gain, beam widths, pulse length and wavelength. If a term is in the numerator of the radar equation and that term is increased, then the power returned to the radar should increased. An example exception to this is when increasing one term in the numerator causes another term in the numerator to decrease more than the original term was increased. Let's go through intuitively how a different radar will cause the returned energy to the radar to increase.

Power transmitted: If the radar emits more energy than it is intuitive there will be more energy to scatter back toward the radar if hydrometeors are present. Thus, more powerful radars are going to receive more backscattered radiation. For example, if our sun in the solar system increased it's power transmission then the earth would receive more solar radiation and more solar radiation would be scattered off of the earth.

Gain and beam widths: These terms are intimately links because changing one can change the others. The more confined a beam is the more energy that will be within that beam. If there is more energy within a beam then there will be more scattering of radiation off of the hydrometeors that beam intersects. Increasing the gain will increase the power returned. Increasing the beam widths however will decrease the power returned because the increase in beam width is more than offset by the decrease in gain caused by the beam being more spread out.

Pulse length: Pulse length is a function of how long the radar emits radiation within a beam. For example, a flashlight that is turned on for 30 seconds will emit more total radiation than a flashlight turned on for 15 seconds. Increasing the pulse length will increase the returned energy to the radar.

Wavelength: Wavelength is in the denominator of the radar equation. Thus, when wavelength increases then the power returned decreases. Thus, radars that emit shorter wavelength radiation will get a more powerful return. Shorter wavelength radiation has more energy than longer wavelength radiation.

The Pi term and 1024*Ln(2) term are simply numbers thus they are always constant.

The last two terms we need to discuss are the attenuation (l) term and the complex index of refraction term (K^2). Attenuation is power loss due to radar radiation absorbing into the atmosphere or less radiation being able to scatter back toward the radar do to the presence of hydrometeors. For any given radar, this term varies depending on the weather conditions thus this term is often ignored and set to a constant of 1 since multiplying the radar equation by 1 yields the same result. Attenuation does have a function of the wavelength of radar used. Shorter wavelength radars will attenuate more than longer wavelength radars.

The complex index of refraction is a function of the material state of the hydrometeor. Generally less energy will be scattered off of ice than liquid water. With the same mass, there will be less returned radiation from dry snow than from rain.