Attenuation Matrix and its Sensitivity in a Water Quality Credit Trading Program
Introduction
Water quality credit trading program is an efficient and low-cost option for
reducing water pollution for nutrients.
In most cases, point-sources and non-point-sources are not in the same area.
The dislocation results in different effect in reducing one unit of pollutant by
both users for downstream water quality. Usually, the upstream users will
have to reduce more than one unit of pollutant in order to get the same
downstream water quality as reduced directly by downstream user. This is
due to the fact that river itself can dilute, absorb and transform the excess
nutrients. The ratio of the amount of pollutant being removed by the river to
its original load in upstream is the attenuation coefficient. Due to the
complexity of river networks and the difficulties for instream attenuation
experiment, attenuation coefficients are often calculated through modeling.
While the calculation of attenuation coefficients is crucial for determining the
upstream-downstream trading ratio, studies seldom look into the sensitivity
and uncertainty of this value.
This study is one of the first to focus on the sensitivity and uncertainty
analysis of attenuation coefficients. Watershed Analysis Risk Management
Framework (WARMF) model is used for watershed modeling and analysis.
Probability Collocation Method with variance decomposition (PCM-VD)
(Zheng, et al, 2011) is used for sensitivity and uncertainty analysis.
Flow Chart for Attenuation Matrix
Sensitivity and Uncertainty Analysis
Xiaoli
Methods description
Attenuation Coefficients Calculation
Rebecca
1
Dorsey ,
Briana
1
Seapy ,
Arturo
1
Keller
xchen@bren.ucsb.edu • 1Bren School of Environmental Science and Management, University of California, Santa Barbara, CA, United States
The ratio of the amount of pollutant being removed by river to its original load in
upstream is the attenuation coefficient. In WARMF, the attenuation is calculated by
adding an extra point source to upstream and compares the loading difference in
both upstream and downstream to determine the attenuation coefficients.
Attenuation downstream
1
Chen ,
Figure A: Nitrate Concentration for Still River
Model Calibration
Figure B: TP Concentration for Upper Great Miami
Amount of extra load removed along thestream

Amount of extra load (Original load differencein the upstream )
Amount of extra load left in the downstream
1 
Amount of extra load (Original load differencein the upstream)
1 
1 
Flowratedstream, with  Concendstream, with  Flowratedstream, without  Concendstream, without
Flowrateustream, with  Concenustream, with  Flowrateustream, without  Concenustream, without
Q j , with ,k  C j , with ,k  Q j , without ,k  C j , without ,k
Qi , with ,k  Ci , with ,k  Qi , without ,k  Ci , without ,k
Figure D: Nitrate for Great Miami River at Bear Creek
Figure C: Flow for Great Miami River at Bear Creek
......(1.1)
The subscription ‘dstream’ means downstream, ‘upstream’ means upstream,
‘with’ means with the extra point source, ‘without’ means without the extra
point source, ‘k’ means it is the attenuation in Day k.
It should be pointed out that the direct uses of eq. (1.1) will lead to
misunderstanding and errors for attenuation calculation. It is because: 1. the
actual extra load left in downstream is a result of river attenuation during the
time for water traveling from upstream to downstream. However, the one
calculated in eq. (1.1) is just a value for a single day, no matter how long the real
traveling time is; 2. part of the load in water will be absorbed by sediment and
released later in the following days. Keeping these in mind, a better way to
calculate is to use the accumulated load difference in upstream and downstream
during simulation period of time (D days):
D
Accumulated Attenu i , j  1 
 (Q
k 1
D
j , with , k
 (Q
k 1
i , with , k
Figure E: TP for Great Miami River at Bear Creek
 C j , with,k  Q j , without ,k  C j , without ,k )
...(1.2)
 Ci , with,k  Qi , without , k  Ci , without , k )
Methods description
Probability Collocation Method
Attenuation Results
Table 1: River Networks for Attenuation Matrix, each element is a HUC-10
number representing sub-basin in Great Miami Watershed
PCM is a stochastic response surface method developed by Tatang al. By using Polynomial
Chaos Expansion (PCE) to approximate the model output, it captures the changes in
output by using different orders of single variable as well as their crossing terms. In PCM,
Hermite polynomial is used for Polynomial Chaos Expansion.
Usually, truncated PCE with only lower order terms can already provide a good
approximation. In this study, a two-order truncated PCE is used. For example, a twoorder truncated PCE with two variables could be expressed as:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
yˆ  a0  a11  a22  a3    1  a4    1  a512 ...(2.1)
2
1
2
2
Study Area
where yˆ is the truncated approximation; i is the first order Hermite polynomial term;
Great Miami Watershed is on the boarder of Ohio and Indiana, consisting
of 40 HUC-10 sub-catchments and one lake, covering an area of 13879.8
km2. Agriculture landuse consists large part in this area, with 23.24% of
total land for soybean production, 19.8% for corn and 1.19% for winter
wheat. Based on EPA database, this watershed has 166 major point
sources, including several major wastewater treatment plants.
i2  1is the second order Hermite polynomial term; and i j is the second order crossing
term in the form of multiplication of two first-order terms. The coefficients in front of each
Hermite polynomial terms (ai ) are to be determined by either determined linear equation
system or regression.
Zheng  2011 integrated PCM with Sobol’ variance-decomposition  VD  and derived the

Mi
i 1
j 1
Figure F: TKN for Great Miami River at Bear Creek
total variation as V ( y )   ai2  Pij !...(2.2)
River Networks With HUC10 Number
Upstream ------------------------------------------------------------ downstream
0
1
2
3
4
5
6
7
8
0508000105 0508000106 0508000107 0508000108 0508000120 0508000201 0508000204 0508000207 0508000209
0508000106 0508000107 0508000108 0508000120 0508000201 0508000204 0508000207 0508000209
0508000103 0508000104 0508000107 0508000108 0508000120 0508000201 0508000204 0508000207 0508000209
0508000104 0508000107 0508000108 0508000120 0508000201 0508000204 0508000207 0508000209
0508000107 0508000108 0508000120 0508000201 0508000204 0508000207 0508000209
0508000108 0508000120 0508000201 0508000204 0508000207 0508000209
0508000115 0508000116 0508000118 0508000119 0508000119 0508000201 0508000204 0508000207 0508000209
0508000117 0508000118 0508000119 0508000119 0508000201 0508000204 0508000207 0508000209
0508000116 0508000118 0508000119 0508000119 0508000201 0508000204 0508000207 0508000209
0508000118 0508000119 0508000119 0508000201 0508000204 0508000207 0508000209
0508000119 0508000119 0508000201 0508000204 0508000207 0508000209
0508000119 0508000201 0508000204 0508000207 0508000209
0508000109 0508000112 0508000113 0508000114 0508000201 0508000204 0508000207 0508000209
0508000110 0508000111 0508000113 0508000114 0508000201 0508000204 0508000207 0508000209
0508000110 0508000111 0508000113 0508000114 0508000201 0508000204 0508000207 0508000209
0508000112 0508000113 0508000114 0508000201 0508000204 0508000207 0508000209
0508000111 0508000113 0508000114 0508000201 0508000204 0508000207 0508000209
0508000113 0508000114 0508000201 0508000204 0508000207 0508000209
0508000114 0508000201 0508000204 0508000207 0508000209
0508000120 0508000201 0508000204 0508000207 0508000209
0508000201 0508000204 0508000207 0508000209
0508000202 0508000203 0508000207 0508000209
0508000203 0508000207 0508000209
0508000204 0508000207 0508000209
0508000207 0508000209
0508000205 0508000206 0508000209
0508000206 0508000209
0508000208 0508000209
0508000301 0508000302 0508000304 0508000306 0508000308 0508000209
0508000303 0508000304 0508000306 0508000308 0508000209
0508000302 0508000304 0508000306 0508000308 0508000209
0508000304 0508000306 0508000308 0508000209
0508000305 0508000306 0508000308 0508000209
0508000307 0508000308 0508000209
0508000306 0508000308 0508000209
0508000308 0508000209
0508000209
Table 2: Attenuation Matrix for TN,
with point source flow=0.1 cum/s,
Nitrate Load = 100 kg/d
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0.24
0.15
0.05
0.17
0.07
0.17
0.06
0.09
0.11
0.02
0.01
0.07
0.26
0.10
0.09
0.02
0.04
0.08
0.07
0.08
0.13
0.21
0.06
0.07
0.17
0.35
0.06
0.06
0.29
0.10
0.11
0.04
0.07
0.13
0.14
0.12
2
0.34
0.20
0.19
0.22
0.24
0.21
0.13
0.10
0.12
0.03
0.08
0.19
0.29
0.13
0.13
0.08
0.11
0.13
0.18
0.21
0.19
0.25
0.14
0.17
3
0.38
0.33
0.24
0.38
0.29
0.31
0.14
0.10
0.13
0.10
0.20
0.24
0.32
0.19
0.18
0.13
0.17
0.24
0.24
0.27
0.25
0.36
0.39
0.12
0.13
0.17
0.23
0.19
0.22
0.41
0.23
0.24
0.24
0.31
Sensitivity and Uncertainty Analysis Results
If fixed 1 , the variance would be V ( E ( yˆ1 ))  a  2a ; if fixed  2 , the variance
Table 6: Sensitivity Analysis Results using Probability
Collocation Method, with 30 parameters and 931 runs. Using
sub-catchment 0508000105 as an example
2
3
would be V ( E ( yˆ2 ))  a22  2a42 . The Sobol’s sensitivity indices could be estimated as:
a12  2a32
V ( E ( yˆ1 ))
S1 
 2
...(2.3)
2
2
2
2
V ( yˆ )
a1  a2  2a3  2a4  a5
V ( E ( yˆ 2 ))
a22  2a42
S1 
 2
...(2.4)
2
2
2
2
V ( yˆ )
a1  a2  2a3  2a4  a5
a
V ( E ( yˆ1   2 ))  V ( E ( yˆ1 ))  V ( E ( yˆ 2 ))
S12 
 2
...(2.5)
2
2
2
V ( yˆ )
a1  a2  2a  2a4  a5
2
5
2
3
The total sensitivity can be calculated as: ST1  S1  S12 , and ST2  S 2  S12 .
As a global sensitivity analysis method, PCM is significantly better in computational
efficiency, compared with traditional Monte Carlo based method. PCM has been used in
groundwater flow and soil conductivity uncertainty analysis (Li and Zhang [2007]; Shi
and Zhang [2010]) and then been introduced to water quality sensitivity analysis by
Zheng al. [2011].
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Parameters Included
Productivity, Corn
Litter Fall Rate, Corn
Productivity, Soybean
Litter Fall Rate, Soybean
Precipitation Weighting factor
Soil erosivity factor
Reaction Rate Nitrification,soil
Reaction Rate Nitrification,surface
Denitrification, Soil
Denitrification, Surface
Thickness, Layer 1
Initial Moisture, Layer 1
Field Capacity, Layer 1
Saturation Moisture, Layer 1
Horizontal Hydraulic Conductivity, Layer 1
Vertical Hydraulic Conductivity, Layer 1
Adsoption NO3, Layer 1
Thickness, Layer 2
Adsoption NO3, Layer 2
Initial Depth
Manning's n
Detachment Velocity Exponent
Initial sediment depth
Bed diffusion rate
Reaction Rate Nitrification, 1/d, water
Reaction Rate Deniitrification, 1/d, water
Reaction Rate Nitrification, 1/d, bed
Reaction Rate Deniitrification, 1/d, bed
Adsoption, Nitrate, Water
Adsoption, Nitrate, Bed
Sensitive Parameters (in descend order)
Reaction Rate Deniitrification, 1/d, water
Manning's n
Precipitation Weighting factor
Initial sediment depth
Initial Depth
Adsorption, Nitrate, Bed
Field Capacity, Layer 1
Bed diffusion rate
Saturation Moisture, Layer 1
Thickness, Layer 1
Adsoption, Nitrate, Water
Vertical Hydraulic Conductivity, Layer 1
Detachment Velocity Exponent
Thickness, Layer 2
Initial Moisture, Layer 1
Horizontal Hydraulic Conductivity, Layer 1
soil erosivity factor
Reaction Rate Deniitrification, 1/d, bed
Adsoption NO3, Layer 1
Reaction Rate Nitrification,surface
Reaction Rate Nitrification, 1/d, water
Productivity, Corn
Denitrification, Soil
Adsoption NO3, Layer 2
Reaction Rate Nitrification,soil
Reaction Rate Nitrification, 1/d, bed
Litter Fall Rate, Soybean
Litter Fall Rate, Corn
Productivity, Soybean
Denitrification, Surface
Total Sensitivity
0.8281
0.1927
0.0013
3.9993E-04
2.8086E-04
2.5209E-04
9.2398E-05
7.0409E-05
5.7040E-05
4.0520E-05
1.1070E-05
9.4692E-06
7.3987E-06
4.7223E-06
8.3028E-07
2.4681E-07
9.9648E-08
5.5416E-08
2.6677E-10
1.3219E-10
1.2273E-10
7.6240E-11
7.3517E-11
7.3122E-11
6.1945E-11
5.5224E-11
5.1039E-11
4.2955E-11
3.0682E-11
1.3824E-25
5
0.50
0.43
0.45
0.51
0.42
0.39
0.17
0.22
0.28
0.26
0.33
6
0.56
0.47
0.53
0.54
0.46
0.40
0.32
0.32
0.26
0.32
0.34
0.42 0.46
0.35 0.38
0.36 0.45
0.30
0.37
0.38
For example, the total variance of expression 1.2  is: V  yˆ   a12  a22  2a32  2a42  a52 .
2
1
4
0.47
0.35
0.41
0.43
0.37
0.35
0.14
0.14
0.18
0.21
0.26
0.36
0.35
0.23
0.23
0.22
0.27
0.28
0.30
0.38
0.49
0.33
0.28
0.55
7
8
0.59 0.60
0.50
0.56 0.63
0.57
0.25 0.28 0.34
0.28 0.40
0.33 0.38
0.33
Table 3: Attenuation Matrix for TN,
with point source flow=0.1 cum/s,
Ammonia Load = 100 kg/d
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0.13
0.09
0.03
0.09
0.05
0.14
0.03
0.06
0.08
0.02
0.00
0.02
0.17
0.14
0.18
0.10
0.08
0.14
0.04
0.04
0.01
0.11
0.03
0.02
0.06
0.21
0.06
0.08
0.21
0.10
0.13
0.04
0.05
0.07
0.06
0.05
2
0.19
0.14
0.12
0.14
0.19
0.17
0.10
0.08
0.10
0.03
0.02
0.04
0.24
0.20
0.25
0.20
0.19
0.17
0.07
0.08
0.04
0.13
0.13
0.09
3
0.23
0.26
0.17
0.26
0.23
0.20
0.12
0.08
0.11
0.06
0.04
0.07
0.30
0.29
0.33
0.24
0.23
0.20
0.10
0.11
0.10
0.19
5
0.35
0.33
0.35
0.34
0.29
0.27
0.15
0.17
0.17
0.13
0.13
6
0.37
0.35
0.40
0.36
0.33
7
0.39
0.38
0.42
0.42
8
0.42
0.19
0.20
0.20
0.19
0.22
0.25
0.25
0.26
0.35
0.36
0.38
0.29
0.29
0.28
0.37
0.39
0.40
0.33
0.37
0.40
0.42
0.45
0.26
0.36
0.13
0.16
0.12
0.15
0.16
0.12
0.39
0.21
0.23
0.18
0.24
Figure G: Calculated vs Approximated Attenuation coefficients from
PCM, better fitness indicates the PCM method may better catch the
model responses. Using sub catchment 0508000105 and its
attenuation in downstream 0508000106 as an example.
R2=0.9997
4
0.32
0.30
0.31
0.30
0.26
0.23
0.12
0.12
0.14
0.10
0.07
0.13
0.33
0.32
0.36
0.27
0.27
0.22
0.18
0.16
0.47
0.27
0.30
0.53
Table 5: Attenuation for TN, point source
flow=0.1 cum/s, Phosphate Load = 100
kg/d, Ammonia Load = 100 kg/d
Table 4: Attenuation Matrix for TP,
with point source flow=0.1 cum/s,
Phosphate Load = 100 kg/d
0.45
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0.06
0.05
0.02
0.05
0.06
0.13
0.01
0.03
0.07
0.03
0.00
0.02
0.16
0.18
0.24
0.11
0.11
0.16
0.04
0.04
0.00
0.00
0.01
0.02
0.05
0.00
0.03
0.02
0.01
0.11
0.16
0.07
0.06
0.02
0.04
0.03
2
0.10
0.10
0.06
0.10
0.18
0.17
0.08
0.06
0.10
0.03
0.03
0.02
0.25
0.27
0.32
0.25
0.25
0.20
0.04
0.04
0.02
0.02
0.06
0.07
3
0.15
0.21
0.12
0.22
0.22
0.17
0.10
0.07
0.10
0.06
0.03
0.05
0.35
0.38
0.42
0.28
0.28
0.20
0.06
0.06
0.07
0.06
4
0.25
0.25
0.23
0.25
0.22
0.19
0.10
0.09
0.12
0.06
0.05
0.09
0.38
0.40
0.44
0.28
0.28
0.22
0.10
0.11
5
0.28
0.25
0.26
0.25
0.24
0.23
0.12
0.09
0.12
0.08
0.09
6
0.28
0.26
0.26
0.27
0.27
0.00
0.12
0.11
0.14
0.12
7
0.30
0.29
0.28
0.30
8
0.32
0.14
0.14
0.18
0.18
0.38
0.40
0.44
0.30
0.29
0.25
0.39
0.42
0.46
0.33
0.32
0.42
0.44
0.48
0.03
0.16
0.17
0.22
0.11
0.09
0.05
0.07
0.22
0.20
0.25
0.14
0.12
0.24
0.22
0.27
0.27
0.31
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0.14
0.09
0.02
0.10
0.06
0.16
0.02
0.08
0.11
0.03
0.00
0.01
0.19
0.12
0.17
0.07
0.09
0.14
0.07
0.03
0.02
0.12
0.02
0.03
0.05
0.23
0.04
0.02
0.24
0.08
0.12
0.04
0.06
0.07
0.06
0.04
2
0.21
0.14
0.13
0.15
0.20
0.19
0.12
0.10
0.14
0.03
0.02
0.04
0.23
0.19
0.24
0.18
0.20
0.19
0.10
0.08
0.05
0.14
0.07
0.08
3
0.25
0.27
0.18
0.28
0.23
0.23
0.14
0.11
0.14
0.06
0.05
0.07
0.29
0.27
0.32
0.23
0.25
0.23
0.14
0.11
0.10
0.20
4
0.34
0.29
0.30
0.30
0.26
0.26
0.14
0.13
0.16
0.10
0.08
0.13
0.32
0.32
0.37
0.26
0.29
0.25
0.18
0.15
5
0.35
0.33
0.32
0.34
0.28
0.29
0.16
0.17
0.21
0.13
0.14
6
0.38
0.35
0.36
0.36
0.31
7
0.39
0.37
0.38
0.41
8
0.40
0.21
0.20
0.24
0.17
0.24
0.23
0.27
0.28
0.35
0.36
0.40
0.28
0.31
0.29
0.37
0.39
0.42
0.32
0.34
0.38
0.40
0.44
0.42
0.18
0.23
0.15
0.22
0.50
0.23
0.26
0.54
0.40
0.27
0.39
0.10
0.15
0.11
0.16
0.10
0.10
Discussion
In-stream reaction rates and channel geometry are the most sensitive parameter sets for TN attenuation, followed
by soil properties and the precipitation weighting factor. This is in contrast to the sensitivity of TN concentration,
which was mostly influenced by soil properties and precipitation weighting.
Parameters like in-stream denitrification, Manning’s number, stream initial depth, initial sediment depth, river bed
adsorption for nitrate and bed diffusion rate consist 6 out of 14 sensitive parameters (parameters with total
sensitivity over 1E-06 were considered sensitive parameters). Instream denitrification rate will determine the
transformation of nitrate to N2. Manning’s number directly determines the velocity of flow in the channel, which
has a great influence on the residence time of pollutant. River bed adsorption for nitrate is a key parameter for
transient storage in river. Other parameters like bank stability factor and bed diffusion rate affect the sediment
transport.
Soil properties are still sensitive to some extent in this study. Parameters like field capacity, thickness, saturation
moisture and hydraulic conductivity show some sensitivity in the test. But they are not dominant ones. Note that
in this study, only soil properties of layer 1 were included in the sensitivity analysis, but not for other layer. This is
due to previous experiments which showed that soil layer 1 was more sensitive than other layers.
Catchment biological parameter were not sensitive in attenuation study. Parameters like litter fall rate and
productivity which shows great sensitivity in TN concentration study had very little impact in attenuation. This is
because in the attenuation sensitivity study, the response reflects the ability to absorb, transform and transport
total nitrogen to downstream, not just the concentration of solute in the water.
Reference:
Zheng Yi, Wang Weiming, Han Feng, and Ping Jing, 2011: Uncertainty assessment for watershed water quality modeling: A Probability Collocation Method based approach. Advances in
Water Resources, 34:887-898.
Tatang MA, Pan WW, Prinn RG, McRae GJ. An efficient method for parametric uncertainty analysis of numerical geophysical model. J Geophys Res Atmos 1997; 102(D18):21925-32.
Li, H., and D. Zhang (2007), Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods, Water Resour. Res., 43, W09409, DOI:
10.1029/2006WR005673.
Shi, L., D. Zhang, L. Lin, and J. Yang (2010), A multiscale probabilistic collocation method for subsurface flow in heterogeneous media, Water Resour. Res., 46, W11562, doi:
10.1029/2010WR009066.