update tf equations

simpeg · Sep 4, 2024 · b4f4140 · b4f4140
1 parent a23e4d2
commit b4f4140
Showing 1 changed file with 18 additions and 24 deletions.
diff --git a/docs/tutorials/tf_calculation.ipynb b/docs/tutorials/tf_calculation.ipynb
@@ -26,14 +26,6 @@
     "\n"
    ]
   },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "id": "3150836d-8d1d-4c6f-950a-5ffee05a0930",
-   "metadata": {},
-   "outputs": [],
-   "source": []
-  },
   {
    "cell_type": "markdown",
    "id": "9ad2c8f4-6dcc-44ee-915f-304b7aaf11c9",
@@ -72,12 +64,13 @@
     "This is three equations in six unknowns, but each of the three equations can be solved separately for pairs of tensor elements (rows of Equation 1).  \n",
     "\n",
     "Extracting the first equation from the system we write:\n",
+    "\n",
     "\\begin{equation*}\n",
     "\\begin{bmatrix} E_x \\end{bmatrix} \n",
     "= \n",
     "\\begin{bmatrix} Z_{xx} & Z_{xy} \\end{bmatrix} \n",
     "\\begin{bmatrix} H_x \\\\ H_y \\end{bmatrix}\n",
-    "\\label{eq:MTTF} \\tag{2a}\n",
+    "\\label{eq:2a} \\tag{2a}\n",
     "\\end{equation*}\n",
     "\n",
     "\n",
@@ -96,7 +89,7 @@
     "H_{x,1} & H_{x,2} & ... & H_{x,N} \\\\\n",
     "H_{y,1} & H_{y,2} & ... & H_{y,N} \\\\\n",
     "\\end{bmatrix}\n",
-    "\\label{eq:EXTF} \\tag{2b}\n",
+    "\\label{eq:2b} \\tag{2b}\n",
     "\\end{equation*}\n",
     "$\n",
     "\n",
@@ -197,7 +190,7 @@
     "\\left<H_{x} H_{x}^* \\right> & \\left<H_{x} H_{y}^* \\right> \\\\\n",
     "\\left<H_{y} H_{x}^* \\right> &  \\left<H_{y} H_{y}^* \\right> \n",
     "\\end{bmatrix}\n",
-    "\\label{eq:EXTFHDaggerSolutionStep} \\tag{3b}\n",
+    "\\label{eq:3b} \\tag{3b}\n",
     "\\end{equation*}\n",
     "\n",
     "\n",
@@ -349,15 +342,15 @@
     "=\n",
     "\\frac{\\left< E_{x} H_{x}^{*} \\right> \\left<H_{y} H_{y}^* \\right> - \\left<E_{x} H_{y}^{*} \\right> \\left<H_{y} H_{x}^* \\right> }\n",
     "{Det(HH^{\\dagger})}\n",
-    "\\label{eq:EXTFHDaggerSolutionStepq} \\tag{6a}\n",
+    "\\label{eq:6a} \\tag{6a}\n",
     "\\end{equation*}\n",
     "\n",
     "\\begin{equation*}\n",
     "Z_{xy}\n",
     "=\n",
     "\\frac{\\left<E_{x} H_{y}^{*} \\right> \\left<H_{x} H_{x}^* \\right> - \\left< E_{x} H_{x}^{*} \\right> \\left<H_{x} H_{y}^* \\right>}\n",
     "{Det(HH^{\\dagger})}\n",
-    "\\label{eq:EXTFHDaggerSolutionStep} \\tag{6b}\n",
+    "\\label{eq:6b} \\tag{6b}\n",
     "\\end{equation*}"
    ]
   },
@@ -427,7 +420,7 @@
     "C\n",
     "=\n",
     "\\begin{bmatrix} A^{*}_{1} & B^{*}_{1}  \\\\ A^{*}_{2} & B^{*}_{2} \\\\ \\vdots & \\vdots \\\\ A^{*}_{n} & B^{*}_{n}  \\end{bmatrix} \n",
-    "\\label{eq:EXTFHDaggerSolutionStep} \\tag{8}\n",
+    "\\label{eq:8} \\tag{8}\n",
     "\\end{equation*}\n",
     "\n",
     "Go back to Equation 3a, but instead of using $H^{\\dagger}$, use $C$, made of column vectors A, B"
@@ -704,7 +697,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c961d3f1-2476-4485-bacd-6825533f77b7",
+   "id": "8dcfb9c8-0de7-4029-b170-ae206c670da5",
    "metadata": {},
    "source": [
     "# Noise and Bias\n",
@@ -718,18 +711,18 @@
     "\n",
     "So the first set of equations reduces to $Z_{xy}$ only, .. repeating Equation 13b here\n",
     "\n",
+    "\n",
     "\\begin{equation*}\n",
     "Z_{xy} =\n",
     "\\frac{\n",
-    "\\left< E_{x} A^{*} \\right> \\left<H_{x} B^* \\right>  \n",
-    "-\\left<E_{x} B^{*} \\right> \\left<H_{x} A^* \\right> \n",
+    "\\left<E_{x} A^{*} \\right> \\left<H_{x} B^* \\right>\n",
+    "\\text{-} \\left<E_{x} B^{*} \\right> \\left<H_{x} A^* \\right> \n",
     "}\n",
     "{\n",
     "\\left<H_{y} A^* \\right> \\left<H_{x} B^* \\right> \n",
-    "-\n",
-    "\\left<H_{y} B^* \\right> \\left<H_{x} A^* \\right>\n",
+    "\\text{-} \\left<H_{y} B^* \\right> \\left<H_{x} A^* \\right>\n",
     "}\n",
-    "\\label{eq:13b_again} \\tag{13b_again}\n",
+    "\\label{eq:A13b} \\tag{A13b}\n",
     "\\end{equation*}\n",
     "\n",
     "\n",
@@ -746,7 +739,7 @@
     "-\n",
     "\\left<H_{y} H_y^* \\right> \\left<H_{x} H_x^* \\right>\n",
     "}\n",
-    "\\label{eq:7b_again} \\tag{7b_again}\n",
+    "\\label{eq:A7b} \\tag{A7b}\n",
     "\\end{equation*}\n",
     "\n",
     "\n",
@@ -767,6 +760,7 @@
     "\\label{eq:tmp0} \\tag{tmp0}\n",
     "\\end{equation*}\n",
     "\n",
+    "\n",
     "by **Assumption 0** first two terms in the denominator are also zero, so:\n",
     "\n",
     "\n",
@@ -1229,9 +1223,9 @@
  ],
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-   "display_name": "aurora",
+   "display_name": "mt_env",
    "language": "python",
-   "name": "aurora"
+   "name": "mt_env"
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   "language_info": {
    "codemirror_mode": {
@@ -1243,7 +1237,7 @@
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    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.10"
+   "version": "3.10.13"
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