# 6.8.7.1. TSURFER Appendix A: Sensitivity/Uncertainty Notation

In the following expressions, the notation E[X] represents the expected value of random variable X, which is equal to the integral of X weighted by its probability density function over the range of allowable values.

## 6.8.7.1.1. Basic variables

I =

number of integral response (experiment and applications) used in GLLS analysis

M =

number of nuclear data parameters used in transport calculations (i.e., number of unique nuclide-reaction pairs multiplied by the number of energy groups)

$$\boldsymbol{\alpha}$$ =

M dimensional vector of prior nuclear data parameters, where component-i = $$\alpha$$i

A =

M by M diagonal matrix of prior nuclear data parameters, where diagonal element A(i,i) = $$\alpha$$i

m =

I dimensional vector of prior measured responses, where component-i = mi

M =

I by I diagonal matrix of prior measured responses, where diagonal element M(i,i) = mi

$$\mathbf{k}(\boldsymbol{\alpha})$$ =

I dimensional vector of prior calculated responses obtained with nuclear data $$\boldsymbol{\alpha}$$, where component I = ki

K =

I by I diagonal matrix of prior calculated responses, where diagonal element K(i,i) = ki

$$\mathbf{F}_{\mathbf{m} / \mathbf{k}}$$ =

I by I diagonal matrix of “E/C” values =

(6.8.55)$\begin{array}{l} \mathbf{M} \mathbf{K}^{-1}=\mathbf{K}^{-\mathbf{1}} \mathbf{M} \end{array}$

where diagonal element

(6.8.56)$\mathrm{F}_{\mathrm{m} / \mathrm{k}}(\mathrm{i}, \mathrm{i})=\frac{\mathrm{m}_{\mathrm{i}}}{\mathrm{k}_{\mathrm{i}}}$

$${{\mathbf{\hat{F}}}_{\mathbf{m/k}}}$$ =

I by I diagonal matrix, where diagonal element

(6.8.57)$\mathrm{F}_{\mathrm{m} / \mathrm{k}}\left(\mathrm{i}, \mathrm{i}\right)=\frac{\mathrm{m}_{\mathrm{i}}}{\mathrm{k}_{\mathrm{i}}}$

for a relative-formatted response and $$\mathrm{F}_{\mathrm{m} / \mathrm{k}}(\mathrm{i}, \mathrm{i})=1$$

$$\boldsymbol{\alpha}'$$ =

M dimensional vector of adjusted nuclear data parameters produced by GLLS procedure

m’ =

I dimensional vector of adjusted measured responses produced by GLLS procedure

$$\mathbf{k}'(\boldsymbol{\alpha}')$$ =

I dimensional vector of adjusted calculated responses obtained with modified nuclear data $$\boldsymbol{\alpha}'$$

Note

$$\mathbf{k}'(\boldsymbol{\alpha}') = \mathbf{m}'$$, due to GLLS adjustment procedure.

$$\mathbf{\tilde{d}}\,$$ =

original absolute discrepancy vector = $$\mathbf{k}-\mathbf{m}$$ , where component-i= $${{k}_{i}}-{{m}_{i}}$$

d =

original relative discrepancy vector = $$\mathbf{K}^{-1}(\mathbf{k}-\mathbf{m})$$ , where component-i = $$\left(\mathrm{k}_{\mathrm{i}}-\mathrm{m}_{\mathrm{i}}\right) / \mathrm{k}_{\mathrm{i}}$$

$$\mathbf{\hat{d}}$$

original mixed absolute-relative discrepancy vector, where component-i = $$\left(\mathrm{k}_{\mathrm{i}}-\mathrm{m}_{\mathrm{i}}\right) / \mathrm{k}_{\mathrm{i}}$$ for a relative-formatted response and $$\left(\mathrm{k}_{i}-\mathrm{m}_{\mathrm{i}}\right)$$ for an absolute-formatted response

$$[\boldsymbol{\Delta} \boldsymbol{\alpha}]$$ =

M dimensional vector of relative variations in nuclear data = $$\mathbf{A}^{-1}\left(\boldsymbol{\alpha}^{\prime}-\boldsymbol{\alpha}\right)$$ where component-i = $$\frac{\alpha_{i}^{\prime}-\alpha_{i}}{\alpha_{i}}$$

$$[\mathbf{\Delta m}]$$ =

I dimensional vector of relative variations in measured responses = $$\mathbf{M}^{-1}\left(\mathbf{m}^{\prime}-\mathbf{m}\right)$$ where component-i = $$\frac{\text{m}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{m}}_{\text{i}}}}{{{\text{m}}_{\text{i}}}}\to \frac{\text{k}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{m}}_{\text{i}}}}{{{\text{m}}_{\text{i}}}}$$

$$[\mathbf{\Delta m}]$$ =

I dimensional vector of absolute variations in measured responses = $$\mathbf{m}^{\prime}-\mathbf{m}$$ where component-i $$\text{m}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{m}}_{\text{i}}}\to \text{k}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{m}}_{\text{i}}}$$

$$[\mathbf{\Delta} \hat{\mathbf{m}}]$$ =

I dimensional vector of mixed absolute-relative variations in measured responses, where component-i = $$\frac{\text{m}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{m}}_{\text{i}}}}{{{\text{m}}_{\text{i}}}}$$ for a relative-formatted response and $$\text{m}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{m}}_{\text{i}}}$$ for an absolute-formatted response

$$[\boldsymbol{\Delta} \mathbf{k}]$$ =

I dimensional vector of relative variations in calculated responses = $$\mathbf{K}^{-1}\left(\mathbf{k}^{\prime}-\mathbf{k}\right)$$ where component-i = $$\frac{\text{k}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{k}}_{\text{i}}}}{{{\text{k}}_{\text{i}}}}$$

$$[\boldsymbol{\Delta} \mathbf{k}]$$ =

I dimensional vector of absolute variations in calculated responses = $$\mathbf{k'}-\mathbf{k}$$, where component-i = $$\text{k}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{k}}_{\text{i}}}$$

$$[\boldsymbol{\Delta} \hat{\mathbf{k}}]$$ =

I dimensional vector of mixed absolute-relative variations in calculated responses, where component-i = $$\text{k}{{\text{ }\!\!'\!\!\text{ }}_{\text{i}}}-{{\text{k}}_{\text{i}}}$$ for an absolute formatted response

## 6.8.7.1.2. Sensitivity Relations

$$\widetilde{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}$$ =

I by M absolute sensitivity matrix; where element $$\widetilde{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}(\mathrm{i}, \mathrm{n})=\alpha_{\mathrm{n}} \frac{\partial \mathrm{k}_{\mathrm{i}}}{\partial \alpha_{\mathrm{n}}}$$

$$\mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}$$ =

I by M relative sensitivity matrix = $$\mathbf{K}^{-1} \mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}$$, where element $$\mathbf{S}_{k \alpha}(i, n)=\frac{\alpha_{n}}{k_{i}} \frac{\partial k_{i}}{\partial \alpha_{n}}$$.

$$\hat{\mathbf{S}}_{\mathbf{k}\boldsymbol{\alpha}}$$ =

I by M mixed absolute-relative sensitivity matrix, where element $$\hat{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}(\mathrm{i}, \mathrm{n})=\frac{\alpha_{\mathrm{n}}}{\mathrm{k}_{\mathrm{i}}} \frac{\partial \mathrm{k}_{\mathrm{i}}}{\partial \alpha_{\mathrm{n}}}$$ if response-i is relative-formatted and $$\hat{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}(\mathrm{i}, \mathrm{n})=\alpha_{\mathrm{n}} \frac{\partial \mathbf{k}_{\mathrm{i}}}{\partial \alpha_{\mathrm{n}}}$$ if response-i is absolute-formatted

(6.8.58)$\begin{split}\begin{array}{l} {[\boldsymbol{\Delta} \mathbf{k}]=\quad \mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}[\mathbf{\Delta} \boldsymbol{\alpha}]} \\ {{[\boldsymbol{\Delta} \mathbf{k}}]=\mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}[\mathbf{\Delta} \boldsymbol{\alpha}]} \\ {[\boldsymbol{\Delta} \hat{\mathbf{k}}]=\hat{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}[\mathbf{\Delta} \boldsymbol{\alpha}]} \end{array}\end{split}$

## 6.8.7.1.3. Absolute covariances

$${{\mathbf{\tilde{C}}}_{\mathbf{mm}}}$$ =

I by I covariance matrix for prior measured experiment responses where element $${{\mathbf{\tilde{C}}}_{\mathbf{mm}}}$$(i,j) = $$E\left( \delta {{m}_{i}}\,\delta {{m}_{j}} \right)$$

$${{\mathbf{\tilde{C}}}_{\mathbf{kk}}}$$ =

I by I covariance matrix for prior calculated responses, where element $${{\mathbf{\tilde{C}}}_{\mathbf{kk}}}$$(i,j) = $$E\left( \delta {{k}_{i}}\,\delta {{k}_{j}} \right)$$

$${{\mathbf{\tilde{C}}}_{\mathbf{dd}}}$$ =

I by I covariance matrix for the discrepancies (k-m), where element $${{\mathbf{\tilde{C}}}_{\mathbf{dd}}}$$(i,j) = $$E\left( \delta {{d}_{i}}\,\delta {{d}_{j}} \right)$$ = $$\mathrm{E}\left(\delta\left(\mathrm{k}_{\mathrm{i}}-\mathrm{m}_{\mathrm{i}}\right) \delta\left(\mathrm{k}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}}\right)\right)$$

$${{\mathbf{\tilde{C}}}_{\mathbf{k'k'}}}$$ =

I by I covariance matrix for adjusted responses, where element $${{\mathbf{\tilde{C}}}_{\mathbf{k'k'}}}$$(i,j) = $$E\left( \delta k{{'}_{i}}\,\delta k{{'}_{j}} \right)$$

$$\boldsymbol{\sigma}_{\mathbf{m}}$$ =

I by I diagonal matrix containing standard deviations in prior measured responses, where diagonal element $$\widetilde{\sigma}_{\mathrm{m}}\left(\mathrm{i} \mathrm{i}\right)=\sqrt{\widetilde{\mathrm{C}}_{\mathrm{mm}}(\mathrm{i}, \mathrm{i})}$$

$$\boldsymbol{\sigma}_{\mathbf{k}}$$ =

I by I diagonal matrix containing standard deviations in prior calculated responses, where diagonal element $$\widetilde{\sigma}_{\mathrm{k}}(\mathrm{i}, \mathrm{i})=\sqrt{\widetilde{\mathrm{C}}_{\mathrm{kk}}(\mathrm{i}, \mathrm{i})}$$

$$\boldsymbol{\sigma}_{\mathbf{k}^{\prime}}$$

## 6.8.7.1.4. Relative covariances

$${{C}_{\mathbf{mm}}}$$ =

I by I relative covariance matrix for prior measured responses, = $$\mathbf{M}^{-1}\left[\tilde{\mathbf{C}}_{\mathbf{m m}}\right] \mathbf{M}^{-1}$$ $$C_{m m}(i, j)=\frac{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\mathrm{m}_{\mathrm{i}} \mathrm{m}_{\mathrm{j}}}$$

$${{C}_{\boldsymbol{\alpha \alpha}}}$$ =

M by M relative covariance matrix for prior nuclear data, where element $$\widetilde{C}_{\alpha \alpha}(i, j)$$ = $$\frac{\mathrm{E}\left(\delta \alpha_{\mathrm{i}} \delta \alpha_{\mathrm{j}}\right)}{\alpha_{\mathrm{i}} \alpha_{\mathrm{j}}}$$

$${{C}_{\mathbf{kk}}}$$ =

I by I relative covariance matrix for prior calculated responses = $$\mathbf{K}^{-1}\left[\mathbf{C}_{\mathrm{kk}}\right] \mathbf{K}^{-1}$$ where element $$C_{k k}(i, j)=\frac{\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})}{\mathrm{k}_{\mathrm{i}} \mathrm{k}_{\mathrm{j}}}$$

$${{C}_{\mathbf{dd}}}$$

I by I relative covariance matrix for response discrepancies; = $$\mathbf{K}^{-1}\left[\mathbf{C}_{\mathbf{dd}}\right] \mathbf{K}^{-1}$$, where element $$C_{d d}(i, j)=\frac{\mathrm{C}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j})}{\mathrm{k}_{\mathrm{i}} \mathrm{k}_{\mathrm{j}}}$$

$$\boldsymbol{\sigma}_{\mathbf{m}}$$ =

I by I diagonal matrix containing relative standard deviations in measured responses, where diagonal element $$\sigma_{\mathrm{m}}\left(\mathrm{i}, \mathrm{i}\right)=\sqrt{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{i})}$$

$$\boldsymbol{\sigma}_{\mathbf{k}}$$ =

I by I diagonal matrix containing relative standard deviations in calculated responses, where diagonal element $$\sigma_{k^{\prime}}\left(\mathrm{i}, \mathrm{i}\right)=\sqrt{\mathrm{C}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i})}$$

$$\boldsymbol{\sigma}_{\boldsymbol{\alpha}}$$ =

M by M diagonal matrix containing standard deviations in nuclear data, where diagonal element $$\boldsymbol{\sigma}_{\boldsymbol{\alpha}}(\mathrm{i}, \mathfrak{i})=\sqrt{C_{\alpha \alpha}(i, i)}$$

## 6.8.7.1.5. Mixed absolute-relative covariances

If response-i and response-j are both absolute formatted, then

(6.8.59)\begin{split}\begin{aligned} \hat{\mathrm{C}}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j}) &=\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j}) \\ \hat{\mathrm{C}}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j}) &=\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j}) \\ \hat{\mathrm{C}}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j}) &=\mathrm{C}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j}) \\ \hat{\mathrm{C}}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j}) &=\mathrm{C}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j}) \end{aligned}\end{split}

Likewise, if both response-i and response-j are relative-formatted, then

(6.8.60)$\begin{split}\begin{array}{l} \hat{\mathrm{C}}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})=\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})}{\mathrm{k}_{\mathrm{i}} \mathrm{k}_{\mathrm{j}}} \\ \hat{\mathrm{C}}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})=\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\mathrm{m}_{\mathrm{i}} \mathrm{m}_{\mathrm{j}}} \\ \hat{\mathrm{C}}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j})=\mathrm{C}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j})}{\mathrm{d}_{\mathrm{i}} \mathrm{d}_{\mathrm{j}}} \\ \hat{\mathrm{C}}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j})=\mathrm{C}_{\mathrm{kk}^{\prime}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{kk}^{\prime}}(\mathrm{i}, \mathrm{j})}{\mathrm{k}_{\mathrm{i}}^{\prime} \mathrm{k}_{\mathrm{j}}^{\prime}} \end{array}\end{split}$

If response-i is absolute-formatted and response-j is relative-formatted, then

(6.8.61)$\begin{split}\begin{array}{l} \hat{\mathrm{C}}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})}{\mathrm{k}_{\mathrm{j}}} \\ \hat{\mathrm{C}}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\mathrm{m}_{\mathrm{j}}} \\ \hat{\mathrm{C}}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{dd}}(\mathrm{i}, \mathrm{j})}{\mathrm{d}_{\mathrm{j}}} \hat{\mathrm{C}}_{\mathrm{k}^{\prime}\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j})}{\mathrm{k}_{\mathrm{j}}^{\prime}} \end{array}\end{split}$

Similar expressions can be derived if response-i is relative-formatted, and response-j is absolute-formatted. The I by I diagonal matrices of standard deviation values are the following:

(6.8.62)$\begin{split}\hat{\sigma}_{\mathrm{k}}(\mathrm{i}, \mathrm{i})=\left\{\begin{array}{ll} \sigma_{\mathrm{k}}(\mathrm{i}, \mathrm{i}) & \text { absolute-formatted } \\ \sigma_{\mathrm{k}}(\mathrm{i}, \mathrm{i}) & \text { relative-formatted } \end{array}\right.\end{split}$
(6.8.63)$\begin{split}\hat{\sigma}_{\mathrm{m}}(\mathrm{i}, \mathrm{i})=\left\{\begin{array}{ll} \sigma_{\mathrm{m}}(\mathrm{i}, \mathrm{i}) & \text { absolute-formatted } \\ \sigma_{\mathrm{m}}(\mathrm{i}, \mathrm{i}) & \text { relative-formatted } \end{array}\right.\end{split}$
(6.8.64)$\begin{split}\hat{\sigma}_{\mathrm{d}}(\mathrm{i}, \mathrm{i})=\left\{\begin{array}{ll} \sigma_{\mathrm{d}}(\mathrm{i}, \mathrm{i}) & \text { absolute-formatted } \\ \sigma_{\mathrm{d}}(\mathrm{i}, \mathrm{i}) & \text { relative-formatted } \end{array}\right.\end{split}$
(6.8.65)$\begin{split}\hat{\sigma}_{\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i})=\left\{\begin{array}{ll} \sigma_{\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i}) & \text { absolute-formatted } \\ \sigma_{\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i}) & \text { relative-formatted } \end{array}\right.\end{split}$

## 6.8.7.1.6. Correlation matrices

$$\mathbf{R}_{\mathbf{kk}}$$ =

I by I correlation matrix for prior calculated responses, where element $$\mathrm{R}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})$$ = $$\frac{\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{k}}(\mathrm{i}, \mathrm{i}) \sigma_{\mathrm{k}}(\mathrm{j}, \mathrm{j})}=\frac{\mathrm{C}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{k}}(\mathrm{i}, \mathrm{i}) \sigma_{\mathrm{k}}(\mathrm{j}, \mathrm{j})}=\frac{\hat{\mathrm{C}}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})}{\hat{\sigma}_{\mathrm{k}}(\mathrm{i}, \mathrm{i}) \hat{\sigma}_{\mathrm{k}}(\mathrm{j}, \mathrm{j})}$$

$$\mathbf{R}_{\mathbf{mm}}$$ =

I by I correlation matrix for prior measured responses, where element $$\mathrm{R}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})$$ $$\frac{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{m}}(\mathrm{i}, \mathrm{i}) \sigma_{\mathrm{m}}(\mathrm{j}, \mathrm{j})}=\frac{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{m}}(\mathrm{i}, \mathrm{i}) \sigma_{\mathrm{m}}(\mathrm{j}, \mathrm{j})}=\frac{\hat{\mathrm{C}}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\hat{\sigma}_{\mathrm{m}}(\mathrm{i}, \mathrm{i}) \hat{\sigma}_{\mathrm{m}}(\mathrm{j}, \mathrm{j})}$$

$$\mathbf{R}_{\boldsymbol{\alpha} \boldsymbol{\alpha}}$$ =

M by M correlation matrix for prior nuclear data, where element $$\mathrm{R}_{\alpha \alpha}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\alpha \alpha}(\mathrm{i}, \mathrm{j})}{\sigma_{\alpha}(\mathrm{i}, \mathrm{i}) \sigma_{\alpha}(\mathrm{j}, \mathrm{j})}$$

$$\mathbf{R}_{\mathbf{k}^{\prime} \mathbf{k}^{\prime}}$$ =

I by I correlation matrix for adjusted responses, where element $$\mathrm{R}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}} \quad(\mathrm{i}, \quad \mathrm{j})$$ = $$\frac{C_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i}) \sigma_{\mathrm{k}^{\prime}}(\mathrm{j}, \mathrm{j})}=\frac{\mathrm{C}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i}) \sigma_{\mathrm{k}^{\prime}}(\mathrm{j}, \mathrm{j})}=\frac{\hat{\mathrm{C}}_{\mathrm{k}^{\prime} \mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{j})}{\hat{\sigma}_{\mathrm{k}^{\prime}}(\mathrm{i}, \mathrm{i}) \hat{\sigma}_{\mathrm{k}^{\prime}}(\mathrm{j}, \mathrm{j})}$$