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 a11 a22 a3 1 a4 1 a512 ...(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 ( yˆ1 )) 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 ( yˆ2 )) a22 2a42 . The Sobol’s sensitivity indices could be estimated as: a12 2a32 V ( E ( yˆ1 )) 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 ( yˆ1 2 )) V ( E ( yˆ1 )) 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.