For the sake of simplicity, we further introduced the following notation to describe the soil isoline equation Display Formula
$[\rho r\rho n]=[a\u2032b\u2032]\rho n\u2032,$(24)
where $\rho n\u2032$ is a vector composed of the series $\rho n\u2032$, and $a\u2032$ and $b\u2032$ are vectors in a series of coefficients used to describe the polynomials of $\rho n\u2032$, defined as Display Formula$\rho n\u2032\u2254[1\rho n\u2032\rho n\u20322\cdots ]t,$(25)
Display Formula$a\u2032\u2254[a0\u2032(Rs,\omega )a1\u2032(Rs,\omega )a2\u2032(Rs,\omega )\cdots ],$(26)
Display Formula$b\u2032\u2254[b0\u2032(Rs,\omega )b1\u2032(Rs,\omega )b2\u2032(Rs,\omega )\cdots ],$(27)
where the coefficients $ai\u2032(Rs,\omega )$ and $bi\u2032(Rs,\omega )$ are defined using the Kronecker delta $\delta ij$Display Formula$ai\u2032(Rs,\omega )=\delta i0[cos(\theta )p0(Rs)]+(1\u2212\delta i0)[\u2212sin(\theta )\delta i1+cos(\theta )\omega 1\u2212ipi(Rs)],$(28)
Display Formula$bi\u2032(Rs,\omega )=\delta i0[sin(\theta )p0(Rs)+s0]+(1\u2212\delta i0)[cos(\theta )\delta i1+sin(\theta )\omega 1\u2212ipi(Rs)].$(29)
Note that the zeroth-order terms of the polynomials describing each reflectance are equal to the spectra of the soil underneath the vegetation canopy $\rho s$Display Formula$\rho s=[cos(\theta )p0(Rs)sin(\theta )p0(Rs)+s0].$(30)
Thus, all isolines approximated by any order will contain the true soil spectra and will agree exactly with the soil spectra, regardless of the approximation order, for the zero vegetation case.