Changes to the Fa notebook (#36)

* Changes to the frontal_ablation notebook

Hello,
I added Ben's suggestions and also did some editions to the text. 

Let me know if you agree

* Add files via upload

notebooks/oggm-tuto/inversion_with_frontal_ablation.ipynb

@@ -18,7 +18,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "In this notebook we are illustrating how OGGM searches for a frontal calving flux which is compatible with mass conservation, ice thickness estimation based on Glen's flow law, and the calving parameterization of [Ooerlemans and Nick (2005)](https://www.cambridge.org/core/journals/annals-of-glaciology/article/minimal-model-of-a-tidewater-glacier/C6B72F547D8C44CDAAAD337E1F2FC97F).\n",
+    "In this notebook we are illustrating how OGGM searches for a frontal calving flux which is compatible with mass conservation, ice thickness estimation based on Glen's flow law, and the calving parameterization of [Oerlemans and Nick (2005)](https://www.cambridge.org/core/journals/annals-of-glaciology/article/minimal-model-of-a-tidewater-glacier/C6B72F547D8C44CDAAAD337E1F2FC97F).\n",
     "\n",
     "For more details, see [Recinos et al. (in review)](https://www.the-cryosphere-discuss.net/tc-2018-254/)."
    ]
@@ -109,7 +109,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "**Results of the inversion without calving:**"
+    "**Let's first see the results of the inversion without calving:**"
    ]
   },
   {
@@ -132,13 +132,13 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We use a simple calving law borrowed from [Ooerlemans and Nick (2005)](https://www.cambridge.org/core/journals/annals-of-glaciology/article/minimal-model-of-a-tidewater-glacier/C6B72F547D8C44CDAAAD337E1F2FC97F), which relates the frontal calving flux $F$ to the frontal ice thickness $H$, the water depth $W_d$ and the front width $w$\n",
+    "We use a simple calving law borrowed from [Oerlemans and Nick (2005)](https://www.cambridge.org/core/journals/annals-of-glaciology/article/minimal-model-of-a-tidewater-glacier/C6B72F547D8C44CDAAAD337E1F2FC97F), which relates the frontal calving flux $F_{calving}$ to the frontal ice thickness $H_f$, the water depth $d$ and the terminus width $w$\n",
     "\n",
-    "$$F = k H W_d w$$\n",
+    "$$F_{calving} = k H_f d w$$\n",
     "\n",
-    "With $F$ in km$^3$ yr$^{-1}$ and $k$ a calibration parameter (default 2.4 yr$^{-1}$).\n",
+    "With $F_{calving}$ in km$^3$ yr$^{-1}$, $k$ a calibration parameter (default 2.4 yr$^{-1}$) and $d =$ $H_{f}$ - $E_{t}$ ($E_{t}$ being the free board).\n",
     "\n",
-    "As explained in |, ice conservation methods applied to tidewater glaciers *must* take into account this mass-flux at the terminus, otherwise the ice thickness is underestimated. In fact, the default OGGM ice thickness inversion procedure assumes an ice flux of zero at the terminus: "
+    "As explained in [Recinos et al. (in review)](https://www.the-cryosphere-discuss.net/tc-2018-254/), ice conservation methods applied to tidewater glaciers *must* take into account this mass-flux at the terminus, otherwise the ice thickness is underestimated. In fact, the default OGGM ice thickness inversion procedure assumes an ice flux of zero at the terminus: "
    ]
   },
   {
@@ -155,9 +155,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Here, `flux` is the calving flux in km$^3$ yr$^{-1}$, `free_board` is the height of ice above water (i.e. above 0 m a.s.l), `thick` the frontal ice thickness in m (equal to zero per construction in OGGM), `water_depth` the water depth in meter (negative here because there is no real calving front and to compensate the non-existent free board), `width` the front width in m.\n",
+    "Here, `flux` is the calving flux ($F_{calving}$) in km$^3$ yr$^{-1}$, `free_board` is the height of ice above water ($E_{t}$, i.e. above 0 m a.s.l), `thick` the frontal ice thickness ($H_{f}$) in m (equal to zero per construction in OGGM), `water_depth` the water depth in m (negative here because of the zero ice thickness and a terminus elevation equal to the freeboard imply a positive bedrock elevation), `width` the front width in m.\n",
     "\n",
-    "Now let's see how this calving flux would change if we increase the ice thickness (while keeping the free board fixed, because this is the only thing we know \"for sure\": the altitude of the glacier front):"
+    "Now let's see how this calving flux would change if we increase the ice thickness (while keeping the free board fixed, because this is the only thing we know \"for sure\": the surface elevation at the terminus.):"
    ]
   },
   {
@@ -187,15 +187,15 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The flux is zero as long as the ice isn't thick enough to reach water, after which the water depth is positive and calving occurs. The calving flux varies with $H$ as a polynome of degree 2.\n",
+    "The flux is zero as long as the ice isn't thick enough to reach water, after which the water depth is positive and calving occurs. The calving flux varies with $H$ as a polynomial of degree 2.\n",
     "\n",
     "We don't know which is the **real** value of the calving flux at this glacier. From here, let's make some very coarse assumptions:\n",
-    "- the Ooerlemans and Nick calving law is perfectly exact\n",
+    "- the Oerlemans and Nick calving law is perfectly exact\n",
     "- the tuning parameter $k$ is known\n",
     "- our glacier is in equilibrium (a fundamental assumption for mass-conservation inversion)\n",
-    "- ice flow at the front follows Glen's flaw law\n",
+    "- ice flow at the front follows Glen's flow law\n",
     "\n",
-    "Under these two assumptions, we are now going to show that there is one and only one value for the frontal thickness which complies with *both* the calving law and the ice thickness inversion model of OGGM.\n",
+    "Under these assumptions, we are now going to show, that there is **one and only one** value for the frontal thickness which complies with *both* the calving law and the ice thickness inversion model of OGGM.\n",
     "\n",
     "To test this hypothesis, we are going to compute a calving flux (from the calving law) for a range of frontal ice thicknesses, then give this flux back to the OGGM inversion model, which will use this flux to compute the frontal ice thickness according to the physics of ice flow (see the [documentation](https://docs.oggm.org/en/latest/inversion.html) or [Recinos et al. (in review)](https://www.the-cryosphere-discuss.net/tc-2018-254/) for more info). Let's see how this works:"
    ]
@@ -256,7 +256,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We already know that the calving law relates the thickness to the flux with a root of degree two (blue curve). Now, what explains the shape of the blue curve? It is Glen's flow law, which relates ice thickness to the flux with a 5th degree root (assuming n=3). This is the reason why there is one (and only one) non-zero solution to the problem of finding a calving flux which is compatible with both the calving law and the physics of ice deformation (under our simplified framework of course).\n",
+    "We already know that the calving law relates the thickness to the flux with a root of degree two (blue curve). Now, what explains the shape of **the orange curve** (OR BLUE??)? It is Glen's flow law, which relates ice thickness to the flux with a 5th degree root (assuming n=3). This is the reason why there is one (and only one) non-zero solution to the problem of finding a calving flux which is compatible with both the calving law and the physics of ice deformation (under our simplified framework of course).\n",
     "\n",
     "Note that adding sliding doesn't change the problem (we still solve a polynome of degree 5 in OGGM, [with a new term in degree 3](https://docs.oggm.org/en/latest/ice-dynamics.html)):"
    ]
@@ -283,7 +283,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "There are several ways to find this \"optimal\" thickness, where mass-conservation inversion and calving law are compatible. In OGGM, we implement an iterative procedure converging to this value in a few iterations:"
+    "There are several ways to find this \"optimal\" thickness, where mass-conservation inversion and the calving law are compatible. In OGGM, we implement an iterative procedure converging to this value in a few iterations:"
    ]
   },
   {
@@ -472,6 +472,7 @@
   }
  ],
  "metadata": {
+  "celltoolbar": "Raw Cell Format",
   "hide_input": false,
   "kernelspec": {
    "display_name": "Python 3",
@@ -488,7 +489,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.5.2"
+   "version": "3.6.5"
   },
   "toc": {
    "base_numbering": 1,