update SDR equation natcap/invest#2379

Author: emlys

source/en/sdr.rst

@@ -174,10 +174,16 @@ where :math:`d_i` is the length of the flow path along the *i*\ th cell accordin
 
 **Step 2.** The SDR ratio for a pixel :math:`i` is then derived from the conductivity index :math:`IC` following (Vigiak et al., 2012):
 
-.. math:: SDR_i = \frac{SDR_{max}}{1+\exp\left(\frac{IC_0-IC_i}{k}\right)}
+.. math::
+    SDR_i = \left\{\begin{array}{lr}
+        1 & \text{if } i \text{ is a stream pixel} \\
+        \frac{SDR_{max}}{1+\exp\left(\frac{IC_0-IC_i}{k}\right)} & \text{otherwise}\\
+        \end{array}\right\}
     :label: sdr
 
-where :math:`SDR_{max}` is the maximum theoretical SDR, set to an average value of 0.8 (Vigiak et al., 2012), and :math:`IC_0` and :math:`k` are calibration parameters that define the shape of the SDR-IC relationship (which is an increasing function). The effect of :math:`IC_0` and :math:`k` on the SDR is illustrated below:
+Note that because :math:`D_{dn}` is 0 on streams, :math:`IC` is undefined, and so :math:`SDR` would be undefined if not for the special case. It is necessary for SDR to be defined in streams for the subsequent calculations, which recurse upslope from the stream. The value of 1 is intuitively reasonable because it represents 100% sediment delivery - no sediment is trapped as it flows from one stream pixel to another.
+
+:math:`SDR_{max}` is the maximum theoretical SDR, set to an average value of 0.8 (Vigiak et al., 2012), and :math:`IC_0` and :math:`k` are calibration parameters that define the shape of the SDR-IC relationship (which is an increasing function). The effect of :math:`IC_0` and :math:`k` on the SDR is illustrated below:
 
 .. figure:: ./sdr/ic0_k_effect.png
 

source/es/sdr.rst

@@ -172,7 +172,11 @@ donde :math:`d_i` es la longitud de la trayectoria del flujo a lo largo de la *i
 
 **Paso 2.** La tasa SDR para un píxel :math:`i` se deriva entonces del índice de conductividad :math:`IC` siguiendo a Vigiak et al. (2012):
 
-.. math:: SDR_i = \frac{SDR_{max}}{1+\exp\left(\frac{IC_0-IC_i}{k}\right)}
+.. math::
+    SDR_i = \left\{\begin{array}{lr}
+        1 & \text{if } i \text{ is a stream pixel} \\
+        \frac{SDR_{max}}{1+\exp\left(\frac{IC_0-IC_i}{k}\right)} & \text{otherwise}\\
+        \end{array}\right\}
     :label: sdr
 
 donde :math:`SDR_{max}` es la máxima SDR teórica, fijada en un valor medio de 0,8 (Vigiak et al., 2012), y :math:`IC_0` y :math:`k` son parámetros de calibración que definen la forma de la relación SDR-IC (que es una función creciente). A continuación se ilustra el efecto de :math:`IC_0` y :math:`k` en el SDR:

source/zh/sdr.rst

@@ -177,7 +177,11 @@ Renard 等，1997）。
 
 **步骤2** 栅格:math:`i` 中的泥沙输移比（SDR）可以通过如下the conductivity index :math:`IC` 指数计算 (Vigiak et al., 2012):
 
-.. math:: SDR_i = \frac{SDR_{max}}{1+\exp\left(\frac{IC_0-IC_i}{k}\right)}
+.. math::
+    SDR_i = \left\{\begin{array}{lr}
+        1 & \text{if } i \text{ is a stream pixel} \\
+        \frac{SDR_{max}}{1+\exp\left(\frac{IC_0-IC_i}{k}\right)} & \text{otherwise}\\
+        \end{array}\right\}
     :label: sdr
 
 式中 :math:`SDR_{max}` 表示理论最大SDR值,为平均值0.8 (Vigiak et al., 2012), :math:`IC_0` 和 :math:`k` 为定义SDR-IC 关系的校准参数 (递增函数)。 :math:`IC_0` 和 :math:`k` 与SDR的关系如下图所示：