APPROACH
Before presenting the approach used in obtaining the point coefficients, a brief discussion of the SWAT outputs offers a better understanding on the logic behind the estimation of point coefficients. In SWAT, a watershed is delineated into several subbasins, with each subbasin being delineated further into several hydrological units which can be interpreted as fields. SWAT computes the ambient emissions at the each subbasin outlet as a function of the component fields’ emissions. Next, the ambient emissions from each subbasin are routed into a nonlinear and nonseparable way to determine the ambient water quality at the main outlet. Hence, SWAT provides emissions outputs at the field, the subbasin, and the main watershed level.
Step 1: Estimate a set of point coefficients that vary by subbasin using the watershed level output.
In this step, I estimate a set of point coefficients that is subbasin specific, implying that: (a) all the edge-of field reductions from a subbasin can be weighted by the same delivery coefficient, (b) a given abatement actions has the same edge-of-field impact in that subbasin (i.e., there is not nonseparability in at the subbasin level). Assuming that in a watershed there are S
subbasins and J abatement actions, ( ) can be rewritten as:
≅ ∑ ∑ ∑∈ ∑ , (1)
where is a vector 1 of abatement levels measured at the watershed’s outlet, is an index for the subbasins, d is the delivery coefficient of subbasin , the total number of fields in the
subbasin, and ∑∈ sums up the total abatement realized at the level of subbasin . The term sum ∑ represents the total emission reductions corresponding to
subbasin measured at the watershed exit. Next, the subbasin reductions are approximated as the sum of the total area allocated to the abatement action , ,weighted by a factor ,
∑ ∑∈ ∑ . (2)
It should be emphasized that measures the efficiency of practice at the subbasin and has the same value for all fields in that subbasin. By combining equations (2) and (3), and by
defining the product , the abatement function can be written as:
≅ ∑ ∑∈ ∑ ∑ ∑ ∑ ∑ ∗ ∗ , (3)
where is matrix, with each element , representing the area allocated to
abatement action , in the subbasin . Finally, the vector of point coefficients, , is obtained via regression:
∑ ∑ ∗ ~ 0,1 . (4)
The point coefficients estimates, , obtained above includes information on the delivery coefficients by definition. They can be interpreted as the marginal impact of abatement action on the total level of abatement given that it is implemented by a field located in the subbasin .
The above estimation considers that, within a given subbasin, an abatement action has the same impact regardless the location in a subbasin. For an accurate representation of the true abatement function, field-specific point coefficients are indicated. Next, using SWAT data obtained at field level, I estimate a set of point coefficients for each field.
Step 2: Estimate a set of point-coefficients for each field.
Using the output available for each field, the reduced emissions for each field, can be written as
∑ ~ 0,1 ∀ 1, … . (5) This step implies running a regression and obtaining a set of coefficients for each field. By running a regression for each field, the field characteristic are taken into account.
Let, be the point coefficient estimate for abatement action , given field . By aggregating at the subbasin level and assuming that the individual edge-of-field reduced emission within the same subbasin has the same impact on the abatement level at subbasin level, I retrieve the total reduced emissions at the subbasin level:
∑ ∑ ∑ . (6)
Next, for a given subbasin and each abatement action, I compute a weighted average of the estimated point coefficients, , where the weight is given by the field’s area, ,
∑ ∀ ∈ , (7)
where are of field in subbasin ; and total area for subbasin . Given the above notation, equation (7) is becomes
∑ ∑ ∑ , (8)
where ∑ .
Equations (8) and (3) are approximations of the same total reduced emissions in a given subbasin. Thus, for a given abatement action and a given subbasin , the following should hold on average : ≅ (i.e., the subbasin-level point coefficients should be an average measure of the field level point coefficients).
Step 3: Obtain the delivery coefficients.
∑ ∑ ∑ ∑ ∑ ∑ ∑ . (9) Equation (4) is equivalent to equation (3). Thus, by comparing the results with the results obtained in step 1, the following relation should hold: ≅ (i.e., the impact of the abatement action on the overall abatement should be equal to its field level impact weighted by the delivery coefficient). All the elements but the ’s are known in equation (4) . In this case, delivery coefficients can be obtained as ≅ (i.e., the ratio of the subbasin point-specific coefficients to the weighted average of field-specific point coefficients).
Finally, obtaining the delivery coefficients requires one more level of aggregation because there are abatement actions and only one delivery coefficient for each subbasin. Furthermore, since both and being obtained via ordinary least square have normal distributions, averaging over the ratios / has a Cauchy distribution. As the Cauchy distribution does not have finite moments of any order, the average of J’s ratio has no meaning. Instead, I instead rely on the median measure
̅ . (10)
Step 4: Using the delivery coefficients to obtain the final set of point coefficients.
For a given field and a given abatement action, a more refined set of point coefficients can be obtained by multiplying the point coefficients obtained in the second step, , with the delivery coefficients estimated above:
̅ , (11)
Alternative Approach to Computing the Delivery Coefficients
In the previous section, I show how to obtain the delivery coefficients and the field-specific point coefficients by using HRU and watershed-level SWAT outputs. This approach requires using data at a very fine scale. Alternatively, both the delivery coefficients and the point
coefficient can be obtained using data at a less refined scale such as watershed and subbasin, or watershed only.
Obtaining the delivery coefficients using SWAT watershed and subbasin level outputs
This is also a multistep approach, with the first step being identical with the one described in section 3.2.1, where the estimated point coefficients are subbasin specific and include the delivery coefficients.
Step 1: ≅ ∑ ∑ ∑ ∑ ∑ (12)
where .
The first step implies the estimation of a set of point coefficients for each subbasin using a single regression. Let be the set of point coefficients, and be the subset of point
coefficients for subbasin .
Step 2: In the second step, using the subbasin SWAT outputs, I estimate a set of point coefficients for each subbasin. The point coefficients can be interpreted as the marginal impact of an abatement level on the total abatement measured at subbasin level. The abatement function for subbasin , , can be written as a linear combination of subbasin specific weights , and the area allocated to a particular abatement action:
≅ ∑ ∀ 1, … , . (13)
The second step implies the estimation of a number of regressions equal to the number of subbasins in a watershed, . Let , be the set of point coefficients obtained for the subbasin.
Step 3: By combining equations (7) and (8), for a given subbasin s, the delivery coefficients can be obtained as46: / }
Step 4: Next, the delivery coefficients are applied to let obtain the final subbasin specific point coefficients: ∀ , ∀ .
Obtaining the delivery coefficients using SWAT watershed only outputs
This approach is also a multistep procedure, with the first step being described before. Step 1: Obtain via regression subbasin-specific point coefficients. Let the be the set of point coefficients, and be the subset of point coefficients for subbasin :
A x ≅ ∑ ∑ a X Xa . (14)
Step 2: A unique set of point coefficients is obtained for the entire watershed. The abatement function is approximated as a linear combination of weights, , assigned to each abatement action, with the weights being the same for each field, and the area allocated to that abatement action:
≅ ∑ . (15)
46 The median is used instead of the average because / is the ratio of two normal