2.2.2. 分布式目标的距离方程
2.2.2. Distributed TargetForms of the Range Equation
并不是所有的散射现象都能被模拟为来自单点散射源的反射。
Not all scattering phenomena can be modeledas a reflection from a single point scatterer.
例如，地杂波最好建模为来自表面的分布式散射，而天气现象（如雨或冰雹）则建模为来自三维体的分布式散射。
Ground clutter, for example, is bestmodeled as distributed scattering from a surface, while meteorologicalphenomena such as rain or hail are modeled as distributed scattering from athree-dimensional volume.
雷达距离方程可以用一种通用的方法重新推导，这种方法适用于所有三种情况。
The radar range equation can be rederivedin a generalized way that accommodates all three cases.
式（2.3）仍然可用作推导的起点。
Equation (2.3) is still applicable as astarting point.
考虑分布式散射问题，由于天线增益随方位角和俯仰角变化，式（2.4）必须替换为另一个方程，从而考虑天线功率方向图P(θ, ϕ)在特定方向(θ, ϕ)上辐射的功率密度，即：
To consider distributed scatterers, andbecause the gain of the antenna varies with azimuth and elevation angle, Eq.(2.4) must be replaced with an equation that accounts for the effect of theantenna power pattern P(θ, ϕ) on the power density radiated in a particulardirection (θ, ϕ):
假设天线视距方向对应于θ =ϕ = 0。
Assume that the antenna boresightcorresponds to θ = ϕ = 0.
天线视距通常是最大增益的轴方向，因此P(0, 0) = G。
The antenna boresight is normally the axisof maximum gain so that P(0, 0) = G.
现在考虑距离和角度坐标(R,θ, ϕ)上增量体积为dV的散射情况。
Now consider the scattering from anincremental volume dV located at range and angle coordinates (R, θ, ϕ).
假设体积单元的增量RCS为dσ平方米，dσ一般随空间位置变化。
Suppose the incremental RCS of the volumeelement is dσ square meters, and that dσ in general varies with position inspace.
dV的后向散射功率增量为
The incremental backscattered power from dVis
如前所述，dσ的定义是假设该散射功率是各向同性的，然后由天线有效孔径接收，并根据到达角进行调整变化。
As before, dσ is defined such that it isassumed this power is reradiated isotropically, and then collected by theantenna effective aperture, adjusted for the angle of arrival.
在替换有效孔径并考虑损耗后，这会导致接收到的增量功率
After substituting for effective apertureand accounting for losses, this results in an incremental received power of
同样，该功率是在电磁波传输2R/c秒后接收到的。
Again, this power is received 2R/c secondsafter transmission.
通过对空间所有的电磁波进行积分得到总接收功率，从而获得一个通用的雷达距离方程。
The total received power is obtained byintegrating over all space to obtain a generalized radar range equation.
式（2.16）中，被积分的体积V包括整个三维空间。
In Eq. (2.16), the volume of integration Vis all of three-dimensional space.
然而，来自所有距离的后向散射能量并不能同时到达雷达。
However, the backscattered energy from allranges does not arrive simultaneously at the radar.
如第1.4.2节所述，只有在距离分辨率单元ΔR内的散射体对任何给定时刻的雷达接收机输出有显著贡献。
As discussed in Sec. 1.4.2,only scatterers within a single range resolution cell of extent ΔR contributesignificantly to the radar receiver output at any given instant.
因此，更合适的通用雷达距离方程形式给出了接收回波功率随时间变化的函数为：
Thus, a more appropriate form of thegeneralized radar range equation gives the received power as a function of time
其中，ΔR是以距离R0为中心的分辨率单元，Ω表示角坐标上的积分。
where ΔR is the range interval of the resolutioncell centered at range R0 and Ω represents integration over theangular coordinates.
——本文译自Mark A. Richards所著的《Fundamentals of Radar Signal Processing（Second edition）》
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