# Single Scattering Albedo The single scattering albedo is $$a=1$$ , and the total optical depth $$\tau^{\star}=0.01$$ . The setting satisfies single-scattering approximation, $$a \times \tau^{\star}=0.01\ll1$$. The incident solar flux is an unpolarized light, $$F=\[0.5, 0.5, 0, 0]$$. In the following case example, the angles are set as follows, $$\theta^{\prime}=0^{o}, \mu^{\prime}=1, \phi^{\prime}=0^{o} ; \theta=\left\[0^{o}, 180^{\circ}\right], \mu=\[-1,1], \phi=60^{o}$$, where $$\theta^{\prime}, \phi^{\prime}$$ are the polar angle and azimuth angle of the incident light $$\Omega^{\prime}$$ , and the $$\theta , \phi$$ are for the corresponding observe angles $$\Omega$$ . ```python a = 1; tau = 0.01; F = [.5;.5;0;0]; thetap = 0.00001; mup = cos(thetap*pi/180); phip = 0; phi = 60; ``` Compute the Legendre-Gauss nodes ( $$ctheta$$ ) and weights ( $$Weight$$ ) on an interval \[-1,1] with truncation order nthetas ( $$ntheta=400$$ ) using an outside function `lgwt`, and retrieve the corresponding $$thetas$$. Function `lgwt` takes integral using Legendre-Gauss Quadrature. ```python nthetas = 400; [ctheta, Weight] = lgwt(nthetas,-1,1); theta = acos(ctheta)*180/pi; ``` The angle $$\Theta$$ between the directions of incidence $$\Omega$$ and observation $$\Omega^{\prime}$$ is given by $$ \cos \Theta=\cos \theta^{\prime} \cos \theta+\sin \theta^{\prime} \sin \theta \cos \left(\phi^{\prime}-\phi\right) $$ ![Illustration of the relationship between Cartesian and spherical coordinates.](/files/-Llhq082K7WWtT3-pEvM) There are two representations for the phase matrix, $$\mathbf{P}*{S}$$ and $$\mathbf{P}*{C}$$, where $$\mathbf{P}*{S}$$ is used for the Stocks vector representation $$\mathbf{I}*{S}=\[I, Q, U, V]^{T}$$, and $$\mathbf{P}*{C}$$ is for $$\mathbf{I}*{C}=\left\[I\_{ |}, I\_{\perp}, U, V\right]^{T}$$ representation (or Chandrasekhar's representation). The connection between these two representations is simply $$\mathbf{I}*{S} = \mathbf{DI}*{C}$$ , where the matrix $$\mathbf{D}$$ is given by: $$ \mathbf{D} \equiv\left(\begin{array}{cccc}{1} & {1} & {0} & {0} \ {1} & {-1} & {0} & {0} \ {0} & {0} & {1} & {0} \ {0} & {0} & {0} & {1}\end{array}\right) $$ The phase matrix $$\mathbf{P}*{C}$$ in $$\mathbf{I}*{C}=\left\[I\_{ |}, I\_{\perp}, U, V\right]^{T}$$ representation is related to the phase matrix $$\mathbf{P}*{S}$$ in $$\mathbf{I}*{S}=\[I, Q, U, V]^{T}$$ as $$\mathbf{P}*{C}=\mathbf{D}^{-1} \mathbf{P}*{S} \mathbf{D}$$, and $$\mathbf{D}^{-1}$$is given by: $$ \mathbf{D}^{-1}=\left(\begin{array}{cccc}{\frac{1}{2}} & {\frac{1}{2}} & {0} & {0} \ {\frac{1}{2}} & {-\frac{1}{2}} & {0} & {0} \ {0} & {0} & {1} & {0} \ {0} & {0} & {0} & {1}\end{array}\right) $$ ```python cTheta = mup*ctheta+sqrt((1-mup^2)*(1-ctheta.^2))*cos((phip-phi)/180*pi); Theta = acos(cTheta)*180/pi; D = [1 1 0 0;... 1 -1 0 0;... 0 0 1 0;... 0 0 0 1]; ``` In this test case, we only include one wavelength at 500 nm, number of moment 300+1, and load the six columns of greek constants from `Mie_tool output` file, without delta fit. ```python number of wavelength numwl = 1; % number of moment nmom = 300+1; fname1 = 'mie_output_small2.dat'; fid1 = fopen(fname1); rawdata1(1, :) = textscan(fid1, '%f %f %f %f %f %f', nmom, 'headerLines', 3); % rawdata1(2, :) = textscan(fid1, '%f %f %f %f %f %f', nmom, 'headerLines', 4); % rawdata1(3, :) = textscan(fid1, '%f %f %f %f %f %f', nmom, 'headerLines', 4); % rawdata1(4, :) = textscan(fid1, '%f %f %f %f %f %f', nmom, 'headerLines', 4); % rawdata1(5, :) = textscan(fid1, '%f %f %f %f %f %f', nmom, 'headerLines', 4); % rawdata1(6, :) = textscan(fid1, '%f %f %f %f %f %f', nmom, 'headerLines', 4); % rawdata1(7, :) = textscan(fid1, '%f %f %f %f %f %f', nmom, 'headerLines', 4); fclose(fid1); rawdata2 = cell2mat(rawdata1); µdata2 = zeros(nmom, 6, numwl);µ ``` The expression for the Fourier coefficients can be written as $$ \mathbf{A}^{m}\left(\tau, u^{\prime}, u\right)=\sum\_{\ell=m}^{2 N-1} \mathbf{P}*{\ell}^{m}(u) \boldsymbol{\Lambda}*{\ell}(\tau) \mathbf{P}\_{\ell}^{m}\left(u^{\prime}\right) $$ where $$ \boldsymbol{\Lambda}*{\ell}(\tau)=\left(\begin{array}{cccc}{\alpha*{1}^{\ell}} & {\beta\_{1}^{\ell}} & {0} & {0} \ {\beta\_{1}^{\ell}} & {\alpha\_{2}^{\ell}} & {0} & {0} \ {0} & {0} & {\alpha\_{3}^{\ell}} & {\beta\_{2}^{\ell}} \ {0} & {0} & {-\beta\_{2}^{\ell}} & {\alpha\_{4}^{\ell}}\end{array}\right) $$ The matrices $$P\_{\ell}^{m}( \pm \mu)$$ are defined through $$ \mathbf{P}*{\ell}^{m}(u)=\left(\begin{array}{cccc}{P*{\ell}^{m, 0}(u)} & {0} & {0} & {0} \ {0} & {P\_{\ell}^{m,+}(u)} & {P\_{\ell}^{m,-}(u)} & {0} \ {0} & {P\_{\ell}^{m,-}(u) P\_{\ell}^{m,+}(u)} & {0} \ {0} & {0} & {0} & {P\_{\ell}^{m, 0}(u)}\end{array}\right) $$ with $$P\_{\ell}^{m, \pm}(u)=\frac{1}{2}\left\[P\_{\ell}^{m,-2}(u) \pm P\_{\ell}^{m, 2}(u)\right]$$. The functions $$P\_{\ell}^{m, 0}(u)$$ and $$P\_{\ell}^{m, \pm 2}(u)$$ are the generalized spherical functions. We define $$P\_{\ell}^{m,n}(u)=0 \quad \text { for } \ell<\max {|m|,|n|}$$. For $$\ell>\max {|m|,|n|}$$ the functions are calculated as follow ($$i$$ is the imaginary unit). $$ \begin{aligned} P\_{m}^{m, 0}(u)=(2 i)^{-m}\left\[\frac{(2 m) !}{m ! m !}\right]^{1 / 2}\left(1-u^{2}\right)^{m / 2} \quad(m \geq 0) \end{aligned} \\ \begin{aligned} P\_{2}^{0, \pm 2}(u)=\frac{-1}{4} \sqrt{6}\left(1-u^{2}\right) \quad(m=0)\end{aligned} \\ \begin{aligned} P\_{2}^{1, \pm 2}(u)=\pm(2 i)^{-1}\left(1-u^{2}\right)^{1 / 2}(1 \pm u) \quad(m=1)\end{aligned} \\ \begin{aligned} P\_{m}^{m, \pm 2}(u)=&-(2 i)^{-m}\left\[\frac{(2 m) !}{(m+2) !(m-2) !}\right]^{1 / 2} \times(1-u)^{(m \mp 2) / 2}(1+u)^{(m \pm 2) / 2} \quad(m \geq 2) \end{aligned} $$ and the recurrence relations $$ \sqrt{(\ell+1)^{2}-m^{2}} P\_{\ell+1}^{m, 0}(u)=(2 \ell+1) u P\_{\ell}^{m, 0}(u)-\sqrt{\ell^{2}-m^{2}} P\_{\ell-1}^{m, 0}(u) $$ The equations above are taken from de Hann's 1987 paper, `eq. 74 to 81`, in $$\textit{The adding method for multiple scattering calculations of polarized light}$$. The rotations of reference plane are implicitly accounted for in the expansion method. In the following code, `line 3~line6` uses functions defined outside. `plm0` calculates cases with m=0 and n=0, `plm2` calculates cases with m=2 and n=2, `plmn2` calculates cases with m=2, n=-2, and `plm2` calculates cases with m=0 and n=2. ```python nterms = length(data2); p00 = plm0(0,nterms-1,ctheta); % m=0, n=0 p22 = plm2(2,nterms-1,ctheta); % m=2, n=2 p2n2 = plmn2(2,nterms-1,ctheta); %P subscript m=2, n=-2 p02 = plm2(0,nterms-1,ctheta); %P subscript m=0, n=2 ``` Read in the phase moments from the `'mie_output_small2.dat'` file to alphas and betas (`line 2~8`). Suppressing the $$a = b$$ dependence, the following equations are for cases of non-spherical particles. $$ \begin{aligned} a\_{1}(\Theta)=\sum\_{\ell=0}^{2 N-1} \alpha\_{1}^{\ell} P\_{\ell}^{0,0}(\cos \Theta) \\ a\_{2}(\Theta)+a\_{3}(\Theta)=\sum\_{\ell=2}^{2 N-1}\left(\alpha\_{2}^{\ell}+\alpha\_{3}^{\ell}\right) P\_{\ell}^{2,2}(\cos \Theta) \\ a\_{2}(\Theta)-a\_{3}(\Theta)=\sum\_{\ell=2}^{2 N-1}\left(\alpha\_{2}^{\ell}-\alpha\_{3}^{\ell}\right) P\_{\ell}^{2,-2}(\cos \Theta) \\ a\_{4}(\Theta) &=\sum\_{\ell=0}^{2 N-1} \alpha\_{4}^{l} P\_{\ell}^{0,0}(\cos \Theta) \\ b\_{1}(\Theta) &=\sum\_{\ell=2}^{2 N-1} \beta\_{1}^{\ell} P\_{\ell}^{0,2}(\cos \Theta) \\ b\_{2}(\Theta) &=\sum\_{\ell=2}^{2 N-1} \beta\_{2}^{\ell} P\_{\ell}^{0,2}(\cos \Theta) \end{aligned} $$ The general form of Stokes scattering matrix Fs (which is the ensemble-averaged Mueller matrix averaging over a small volume containing an ensemble of particles) is of the form: $$ \mathbf{F}*{S}(\Theta)=\left(\begin{array}{cccc}{a*{1}(\Theta)} & {b\_{1}(\Theta)} & {0} & {0} \ {b\_{1}(\Theta)} & {a\_{2}(\Theta)} & {0} & {0} \ {0} & {0} & {a\_{3}(\Theta)} & {b\_{2}(\Theta)} \ {0} & {0} & {-b\_{2}(\Theta)} & {a\_{4}(\Theta)}\end{array}\right) $$ Note that there are six independent elements for non-spherical particles. ```python for i = 1:1 %1:numwl data2(:, :, i) = rawdata2((1+nmom*(i-1):nmom*i), 1:end); alpha1(:, i, 2) = data2(:, 1, i); alpha2(:, i, 2) = data2(:, 2, i); alpha3(:, i, 2) = data2(:, 3, i); alpha4(:, i, 2) = data2(:, 4, i); beta1(:, i, 2) = data2(:, 5, i); beta2(:, i, 2) = data2(:, 6, i); a1(:, i, 1) = alpha1(:,i, 2)' * p00; % Eq.68 a4(:, i, 1) = alpha4(:,i, 2)' * p00; % Eq.71 b1(:, i, 1) = beta1(:,i, 2)' * p02; % Eq.72 b2(:, i, 1) = beta2(:,i, 2)' * p02; % Eq.73 a2(:, i, 1) = .5*((alpha2(:,i, 2) + alpha3(:,i, 2))' * p22 + (alpha2(:,i, 2) - alpha3(:,i, 2))' * p2n2); %Eq.69+70 a3(:, i, 1) = .5*((alpha2(:,i, 2) + alpha3(:,i, 2))' * p22 - (alpha2(:,i, 2) - alpha3(:,i, 2))' * p2n2); %Eq.69+70 end ``` If we want to study spherical particles, there are four independent components, $$a\_{1}(\Theta), a\_{3}(\Theta), b\_{1}(\Theta), b\_{2}(\Theta)$$ , given $$a\_{2}(\Theta)=a\_{1}(\Theta), a\_{4}(\Theta)=a\_{1}(\Theta).$$ The code should be adjusted as follows $$ \mathbf{F}*{S}(\Theta)=\left(\begin{array}{cccc}{a*{1}(\Theta)} & {b\_{1}(\Theta)} & {0} & {0} \ {b\_{1}(\Theta)} & {a\_{1}(\Theta)} & {0} & {0} \ {0} & {0} & {a\_{3}(\Theta)} & {b\_{2}(\Theta)} \ {0} & {0} & {-b\_{2}(\Theta)} & {a\_{3}(\Theta)}\end{array}\right) $$ ```python for i = 1:1 %1:numwl data2(:, :, i) = rawdata2((1+nmom*(i-1):nmom*i), 1:end); alpha1(:, i, 2) = data2(:, 1, i); alpha2(:, i, 2) = data2(:, 2, i); alpha3(:, i, 2) = data2(:, 3, i); alpha4(:, i, 2) = data2(:, 4, i); beta1(:, i, 2) = data2(:, 5, i); beta2(:, i, 2) = data2(:, 6, i); a1(:, i, 1) = alpha1(:,i, 2)' * p00; % Eq.68 a4(:, i, 1) = alpha4(:,i, 2)' * p00; % Eq.71 b1(:, i, 1) = beta1(:,i, 2)' * p02; % Eq.72 b2(:, i, 1) = beta2(:,i, 2)' * p02; % Eq.73 a2(:, i, 1) = a1(:,i, 1) a3(:, i, 1) = a4(:,i, 1) end ``` {% hint style="info" %} Note: In here, a more precise way to do it is to output the a1 to a4 and b1, b2 directly from the mie code. The method above, to compute them from alpha1\~alpha4 and beta1 beta2 is actually taking a detour. {% endhint %} So far, we have calculated the scattering matrix $$\mathbf{F}*{S}$$. The phase matrix $$\mathbf{P}*{S}$$ is related to the scattering matrix by $$\mathbf{P}*{S}\left(u^{\prime}, \phi^{\prime}, u, \phi\right)=\mathbf{R}*{S}\left(\pi-\sigma\_{2}\right) \mathbf{F}*{S}(\Theta) \mathbf{R}*{S}\left(-\sigma\_{1}\right)$$. $$\mathbf{R}\_{S}$$ is a rotation matrix used to rotate reference planes, and it transforms phase matrix from the scattering plane to local meridian plane of reference. $$ \mathbf{R}\_{S}(\eta)=\left\[\begin{array}{cccc}{1} & {0} & {0} & {0} \ {0} & {\cos (2 \eta)} & {-\sin (2 \eta)} & {0} \ {0} & {\sin (2 \eta)} & {\cos (2 \eta)} & {0} \ {0} & {0} & {0} & {1}\end{array}\right] $$ Then use $$\mathbf{P}*{C}=\mathbf{D}^{-1} \mathbf{P}*{S} \mathbf{D}$$ (`line 10` in the following code snippet). $$\mathbf{F}*{C}(\Theta)=\mathbf{D}^{-1} \mathbf{F}*{S}(\Theta) \mathbf{D}$$, where $$\mathbf{F}\_{C}(\Theta)$$ is the Chandrasekhar's Stocks vector representation. Another way to find $$\mathbf{P}*{C}$$ is by first finding the transformed rotation matrix elements $$\tilde{\mathbf{R}}*{S}\left(\pi-\sigma\_{2}\right)$$ and $$\tilde{\mathbf{R}}*{S}\left(-\sigma*{1}\right)$$, and $$\mathbf{P}*{C}=\tilde{\mathbf{R}}*{S} (\pi-\sigma\_{2})\mathbf{P}*{S} \tilde{\mathbf{R}}*{S}(-\sigma\_{1})$$**.** The radiative transfer equitation for diffuse polarized radiation, described in terms of the Stokes equations is given by $$ \mu \frac{d \vec{I}(\tau, \mu, \phi)}{d \tau}=\vec{I}(\tau, \mu, \phi)-\frac{a(\tau)}{4 \pi} \int\_{0}^{2 \pi} d \phi^{\prime} \int\_{-1}^{1} d \mu^{\prime} \vec{P}\left(\tau, \mu, \phi ; \mu^{\prime}, \phi^{\prime}\right) \vec{I}\left(\tau, \mu^{\prime}, \phi^{\prime}\right)+\vec{S}(\tau, \mu, \phi) $$ Ignore the multiple-scattering term, we have $$\mu \frac{d \vec{I}(\tau, \mu, \phi)}{d \tau}=\vec{I}(\tau, \mu, \phi)+\vec{S}(\tau, \mu, \phi)$$ The source term is defined as $$\vec{S}(\tau, \mu, \phi)=\frac{a(\tau) F^{s}}{4 \pi} \vec{P}\left(\mu^{\prime}, \phi^{\prime} ; \mu, \phi\right) e^{-\tau / \mu\_{0}}$$ The diffused intensity in upward direction can be derived as $$\vec{I}*{d}^{+}(\tau, \mu, \phi)=\frac{a(\tau) \mu*{0} F^{s}}{4 \pi\left(\mu+\mu\_{0}\right)} \vec{P}\left(\mu^{\prime}, \phi^{\prime} ; \mu, \phi\right)\left\[e^{-\tau / \mu\_{0}}-e^{-\left\[\left(\tau^{*}-\tau\right) / \mu+\tau^{*} / \mu\_{0}\right]}\right]$$ For the intensity at the top of the layer ( $$\tau=0$$ ), the above equation becomes $$ \vec{I}*{d}^{+}(0, \mu, \phi)=\frac{a(\tau) \mu*{0} F^{s}}{4 \pi\left(\mu+\mu\_{0}\right)} \vec{P}\left(\mu^{\prime}, \phi^{\prime} ; \mu, \phi\right)\left\[1-e^{-\left(\tau^{*} / \mu+\tau^{*} / \mu\_{0}\right)}\right] $$ Isolate the Azimuthal dependence, we get the Chandraskhar's representation of $$\vec{I}\_{d}^{+}(\tau, \mu, \phi)$$ $$ \begin{aligned} \mathbf{I}(\tau, u, \phi)=& \sum\_{m=0}^{2 N-1}\left{\mathbf{I}*{c}^{m}(\tau, u) \cos m\left(\phi*{0}-\phi\right)+\mathbf{I}*{s}^{m}(\tau, u) \sin m\left(\phi*{0}-\phi\right)\right} \ \mathbf{Q}*{\delta}(\tau, u, \phi)=& \sum*{m=0}^{2 N-1}\left{\mathbf{Q}*{c \delta}^{m}(\tau, u) \cos m\left(\phi*{0}-\phi\right)+\mathbf{Q}*{s \delta}^{m}(\tau, u) \sin m\left(\phi*{0}-\phi\right)\right} \end{aligned} $$ As for the Stocks representation, we just need to read out the $$\vec{I}\_{d}^{+}(0, \mu, \phi)$$ calculated using the equation from the above. $$ \mathbf{I}*{C}=\left\[I*{\ell}, I\_{r}, U, V\right]^{T}=\mathbf{I}*{S}=\left\[I*{ |}, I\_{\perp}, U, V\right]^{T} $$ ````python for j = 1:length(phi) for i = 1:length(theta) Fs(:,:,(j-1)*181+i) = [a1(i,1), b1(i,1), 0, 0; b1(i,1), a2(i,1), 0, 0; 0, 0, a3(i,1), b2(i,1); 0, 0, -b2(i,1), a4(i,1)]; [Rs1(:,:,(j-1)*181+i), Rs2(:,:,(j-1)*181+i)] = Rs(mup, ctheta(i), cTheta(i)); Fc(:,:,(j-1)*181+i) = D^-1 * Fs(:,:,(j-1)*181+i) * D; Ps(:,:,(j-1)*181+i) = Rs2(:,:,(j-1)*181+i) * Fs(:,:,(j-1)*181+i) * Rs1(:,:,(j-1)*181+i); Pc(:,:,(j-1)*181+i) = D^-1 * Ps(:,:,(j-1)*181+i) * D; Rs1_1(:,:,(j-1)*181+i) = D^-1 * Rs1(:,:,(j-1)*181+i) * D; Rs2_1(:,:,(j-1)*181+i) = D^-1 * Rs2(:,:,(j-1)*181+i) * D; Pc_1(:,:,(j-1)*181+i) = Rs2_1(:,:,(j-1)*181+i) * Fc(:,:,(j-1)*181+i) * Rs1_1(:,:,(j-1)*181+i); Id(:,i,j) = a*mup/4/pi/(abs(ctheta(i))+mup)*(1-exp(-(tau/abs(ctheta(i))+tau/mup)))*Pc(:,:,(j-1)*181+i)*F; Im(j,i) = 1*(real(Id(1,i,j))+real(Id(2,i,j))); Qm(j,i) = 1*(real(Id(1,i,j))-real(Id(2,i,j))); Il(j,i) = real(Id(1,i,j)); Ir(j,i) = real(Id(2,i,j)); U(j,i) = real(Id(3,i,j)); V(j,i) = real(Id(4,i,j)); end ``` some codes to plot the graphs ``` end ```` Now it's time to scale it to two-slab and multi-slab cases. ### Single Multi-layer Slab For a multilayered slab the corresponding solution is $$ \begin{aligned} \mathbf{I}*{p}^{+}(\tau, \mu, \phi)=& \frac{\mu*{0}}{\left(\mu\_{0}+\mu\right)} \sum\_{n=p}^{L}\left{\mathbf{X}*{n}^{+}(\mu, \phi)\right.\ &\left. \times\left\[e^{-\left\[\tau*{n-1} / \mu\_{0}+\left(\tau\_{n-1}-\tau\right) / \mu\right]}-e^{-\left\[\tau\_{n} / \mu\_{0}+\left(\tau\_{n}-\tau\right) / \mu\right]}\right]\right} \end{aligned} $$ with $$\tau\_{n-1}$$ replaced by $$\tau$$ for n=p. p is the layer at which level that we are trying to evaluate, $$\tau\_{p}$$is the optical depth from the top of the slab to the p-th layer, and n is the total number of layer. ### Single, Two-layer slab $$\tau$$ is the depth measured from the top slab to any layer p. The subscript numbers get larger as more layers are accumulated downwards. As an example, for a 2 layer slab, with $$\tau\_{1}=0.0025, \tau\_{2}=0.005, \text{and we are evaluating an optical depth at }\tau=0.0024, \text{which means }p=1, \text{the equations are}$$ $$ \begin{array}{l}{I\_{1}^{+}=\frac{\mu\_{0}}{\left(\mu\_{0}+\mu\right)} X\_{1}\left(e^{-\frac{\tau}{\mu{0}}}-e^{-\left(\frac{\tau\_{1}}{\mu\_{0}}+\frac{\tau\_{1}-\tau}{\mu}\right)}\right)} \ \frac{\mu\_{0}}{\left(\mu\_{0}+\mu\right)} {+X\_{2}\left(e^{-\left(\frac{\tau\_{1}}{\mu\_{0}}+\frac{\tau\_{1}-\tau}{\mu}\right)}-e^{-\left(\frac{\tau\_{2}}{\mu\_{0}}+\frac{\tau\_{2}-\tau}{\mu}\right)}\right)} \end{array} $$ If evaluate on the bottom layer, say $$\tau=0.0026, \text{which means }p=2$$ $$ I\_{2}^{+}=\frac{\mu\_{0}}{\left(\mu\_{0}+\mu\right)} X\_{2}\left(e^{-\frac{\tau}{\mu\_{0}}}-e^{-\left(\frac{\tau\_{2}}{\mu\_{0}}+\frac{\tau\_{2}-\tau}{\mu}\right)}\right) $$ Therefore, to achieve this functionality, just need to (1) Evaluate the range to know what p is\ (2) Loop from p to n ![2 Layer, tau=0.0026](/files/-Lpnrh0h6Q01TniN_JJP) ![1 Layer, tau=0.0026](/files/-LpnrqCAezFbDomGET3o) The results are the same. Similar results can be achieved --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://louisazhou.gitbook.io/notes/single-scattering-albedo.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.