Extracting Complex Nonlinear Response Surfaces from Deterministic Models

Davood Shahsavani and Anders GrimvallLinkping University

SAMO 2007, 21 Jun 2007

MotivationMany applications of computer code models require repeated model runs for different sets of inputs

Response surface methodologies can:

Help to learn about the modelFacilitate the development of computationally cheap decision support tools

SAMO 2007, 21 Jun 2007

The two steps in extracting response surfacesfrom computationally expensive computer-code modelsStep 1:Choose a suitable design of the computer experiment

Step 2:Choose an interpolation method that enables accurate prediction of the model output at previously untried inputs

SAMO 2007, 21 Jun 2007

Features of currently used techniques for extracting reponse surfaces The design criteria favour regular fractional factorial designs or an almost uniform coverage of the input domain

The interpolation is usually based on a global model for the entire input domain

SAMO 2007, 21 Jun 2007

Study objective

Extraction of response surfaces whose curvature varies strongly over the input domain

Our designs are space-filling and have a particularly good coverage of regions in which the response surface is rough or strongly nonlinear

We fit local models to the responses computed for subsets of design points

SAMO 2007, 21 Jun 2007

Splitting procedure

SAMO 2007, 21 Jun 2007

Chart1

00.4010.20

11.40.41.41.21

11.40.41.41.21.2

00.4010.20.2

00.4010.20

Sheet1

x-shifty-shift

0.50.4

FrontBackBottomTopCentre

000.50.400010.750.7

010.51.40.50.40.51.4

111.51.41.50.41.51.4

101.50.41011

000.50.40001

First splitSecond split

0.250.20.50

0.251.20.51

1.251.20.751.2

1.250.20.750.2

0.250.20.50

0.6250.60.8750.6

0.8750.80.3750.6

Sheet1

Sheet2

Sheet3

Design points after two splits

SAMO 2007, 21 Jun 2007

A sequential design algorithm for box-shaped input domains1. Initiate the design algorithm by selecting a (slightly extended) corner-centre design

2. Start a loop in which the input domain is split into sub-boxes, and new corners and centres are added to the design

A measure of roughness or nonlinearity is used to determine which box that shall be split into two halvesA direction criterion is used to determine in which direction the selected sub-box shall be split

SAMO 2007, 21 Jun 2007

Quantifying the roughness of a response surface

The integrated roughness Rf(D) of a function f (x1, .xp) in a subset D of the input domain is usually defined as

The integrated roughness of a second order polynomial

is

We estimate the integrated roughness of any function by computing

where are estimated parameters in a fitted polynomial

SAMO 2007, 21 Jun 2007

A roughness measure that focuseson integrated absolute errors in linear predictors

Consider a box in which the ith side has length hi

We estimate the roughness of the response surface by computing

SAMO 2007, 21 Jun 2007

Possible splits of a box

SAMO 2007, 21 Jun 2007

Splitting direction

Suppose that the sub-box D* shall be split, and let

be the polynomial fitted to data in or on the border of that box

Then we split the selected sub-box along the kth coordinate where

SAMO 2007, 21 Jun 2007

Design points for a simple response surface

f(x1, x2) = x15 + x25

SAMO 2007, 21 Jun 2007

Chart1

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0.96875

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x1

x2

Design points generated by our algorithm

test

Interpolation ErrorNo. NeighborhoodSSE

52.448853681360

63.762885615560

67.797097384160

48.699214794160

64.072458181560

42.037295092360

85.245654404860

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45.544533144960

60.894045273160

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60.487538926460

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60.525067139960

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44.080568554760

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75.69022586260

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81.249051699260

48.001282134660

57.348680723860

56.907460668160

60.53291578460

94.72231874760

66.476362145560

98.233466640860

53.222056213660

61.787011292860

59.186609948860

93.034832092360

44.079531323860

46.509436190960

62.378662474460

87.376797160760

59.629689436960

83.083261830460

71.793990885960

57.682896992660

42.85496555960

88.224275205360

67.962919276860

64.365121698860

40.849114757960

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56.853217739960

49.420415176160

56.057205188960

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41.1412797960

36.911135934960

78.689699909860

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75.172356756460

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110.927736473260

50.497926101460

54.373223096160

53.025302796860

77.882450177860

99.771612967460

Evaluationset

911

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0.001420.00191

0.003760.03602

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Efficiency

-0.0774972204

52.442386-0.0064672795

63.7632990.0004131404

67.794437-0.0026608136

48.690329-0.0088860037

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40.406919-0.0101834468

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71.090644-0.0036421762

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54.365110-0.0081133285

53.018979-0.0063235558

77.878552-0.0038980442

99.770683-0.0009303475

Efficiency

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Estimation set

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0.2500.00097656250.250.843750.4286073148

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0.56250.68750.2099037170.3750.906250.6186956465

0.68750.68750.30718040470.906250.1250.6113103926

0.56250.56250.11262702940.906250.3750.6186956465

0.68750.56250.2099037170.750.93750.9615011215

0.812500.3540925980.81250.906250.965372473

0.81250.50.3853425980.81250.968751.2073077857

0.781250.250.29201486710.93750.750.9615011215

0.843750.250.42860731480.906250.81250.965372473

0.50.81250.3853425980.968750.81251.2073077857

00.81250.3540925980.50.968750.8844651878

0.250.781250.29201486710.3750.9531250.794007373

0.250.843750.42860731480.3750.9843750.9316943279

0.3750.1250.00744628910.968750.50.8844651878

0.3750.3750.0148315430.9531250.3750.794007373

10.93751.7241964340.9843750.3750.9316943279

0.8750.93751.237105369600.968750.8532151878

0.93750.906251.33547630910.1250.9531250.7866221191

0.93750.968751.57741162180.1250.9843750.9243090739

00.250.00097656250.9687500.8532151878

0.1250.1250.00006103520.9531250.1250.7866221191

0.1250.3750.00744628910.9843750.1250.9243090739

0.68750.1250.15362071990.68750.906250.7648700774

0.68750.3750.16100597380.68750.968751.0068053901

0.1250.68750.15362071990.906250.68750.7648700774

0.3750.68750.16100597380.968750.68751.0068053901

0.750.93750.96150112150.937511.724196434

0.81250.906250.9653724730.906250.968751.4644950628

0.81250.968751.20730778570.968750.968751.7064303756

0.93750.750.96150112150.1250.843750.4276612699

0.906250.81250.9653724730.3750.843750.4350465238

0.968750.81251.20730778570.843750.1250.4276612699

0.68750.906250.76487007740.843750.3750.4350465238

0.68750.968751.00680539010.906250.56250.6675933897

0.906250.68750.76487007740.968750.56250.9095287025

0.968750.68751.00680539010.56250.906250.6675933897

0.56250.906250.66759338970.56250.968750.9095287025

0.56250.968750.90952870250.56250.81250.4104061127

0.68750.81250.5076828003

0.750.6250.3326721191

0.81250.56250.4104061127

0.81250.68750.5076828003

0.906250.906251.2225597501

0.968750.906251.4644950628

0.81250.750.5913972855

0.781250.81250.6451309025

0.843750.81250.7817233503

0.812511.354092598

0.781250.968751.1442534924

0.843750.968751.2808459401

10.81251.354092598

0.968750.781251.1442534924

0.968750.843751.2808459401

0.3750.8750.520324707

0.3750.93750.7316122055

0.31250.906250.6142601073

0.43750.906250.6273082793

0.93750.3750.7316122055

0.8750.3750.520324707

0.906250.31250.6142601073

0.906250.43750.6273082793

00.906250.611279875

0.1250.8906250.5603999132

0.1250.9218750.6658556564

0.9062500.611279875

0.8906250.1250.5603999132

0.9218750.1250.6658556564

Estimation set

0

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x1

x2

Design points generated by our sequential algorithm

Results

No. cuboidsNo. function evaluationsMax. nonlinearityNo. cuboidsNo. function evaluationsMax. nonlinearity

1628.1252167.031252

2109.5100402832221011.67297363282

3127.038116455113129.91516113281

4152.478790283214168.20380449291

5191.242828369115195.72501420972

6221.242828369116224.97131347661

7250.709150917827255.73188066482

8280.631809420928294.26565194512

9300.091169960829333.56236291122

10340.0911699608110363.25394868852

11380.089453347211383.29851162592

12410.0507825986112413.25008630752

13430.0507825986113442.58738077851

14460.0244444982214472.33564455991

15490.0232465311215492.32489896012

16510.0138284639216522.32489896011

17550.0138284639217552.13110935882

18590.0121118501118591.81221524252

19620.0068664551219631.78527832032

20660.005109124120665.23452419081

21690.005109124121695.73188066481

22720.0050206299222724.26565194511

23760.0050206299123762.58738077851

24800.0039111569224791.81221524252

25820.0039111569225831.73493939281

26840.0033528763126851.73493939281

27860.0031258282127871.51775018312

28880.0026246956128901.51775018312

29920.0026246956129931.297673471

30960.0023217872230961.297673472

31980.0020811615231991.29284549371

321020.00171829022321021.29284549372

331050.00150813442331051.23191052121

341070.00150813442341071.23191052122

351090.00148217791351091.17919338081

361120.00148217792361121.13080586371

371150.00120303761371141.13080586371

381170.00120303762381161.11566190612

391190.00108951361391181.11566190611

401210.00108951362401201.09818060882

411230.00105403821411221.09818060881

421250.00105403821421250.96608303442

431270.00088319432431270.90439100751

441300.00060517341441300.8725137931

451340.0006040542451330.8725137931

461370.0006040542461360.87145277692

471390.00058696991471400.87145277691

481420.00058696991481440.86662480072

491450.00055974452491470.86662480072

501480.00054673731501500.78202486041

206627515450926658500001

Results

28.125

9.5100402832

7.0381164551

2.4787902832

1.2428283691

1.2428283691

0.7091509178

0.6318094209

0.0911699608

0.0911699608

0.089453347

0.0507825986

0.0507825986

0.0244444982

0.0232465311

0.0138284639

0.0138284639

0.0121118501

0.0068664551

0.005109124

0.005109124

0.0050206299

0.0050206299

0.0039111569

0.0039111569

0.0033528763

0.0031258282

0.0026246956

0.0026246956

0.0023217872

0.0020811615

0.0017182902

0.0015081344

0.0015081344

0.0014821779

0.0014821779

0.0012030376

0.0012030376

0.0010895136

0.0010895136

0.0010540382

0.0010540382

0.0008831943

0.0006051734

0.000604054

0.000604054

0.0005869699

0.0005869699

0.0005597445

0.0005467373

Max. nonlinearity

elementary values

Enter Factors number:911

Minimum values for factors:00

Maximum values for factors:11

Enter the path and a name for bach file:E:\Article2\batchbatch.bat

Enter path and a name for inca outoutE:\Article2\out1.dsd

9101112151618

00080000

0.010.010.0514010.10.05

INCA output file name

E:\Article2\out1.dsd

General

HeadingRiver Tweed 1994 - 2000

Landuse acronymsForestSVegUGSVegGNFSVegFArableUrban

Landuse categoriesForestSveg (Ungrazed)Urban

Start date01/01/1994

No. Observations2557

No. Sub-basins1

Land phase

Land phase parameters

Minimum valueMaximum valueCentral value

ParameterUnitForestSVegUgSVegGNFSVegFArableUrbanForestSVegUgSVegGNFSVegFArableUrbanForestSVegUgSVegGNFSVegFArableUrban

Land phase initial conditionSurface flowm3/day0.0010.0010.0010.0010.0000.0010.0010.0010.0010.0010.01000.0010.0010.0010.0010.0010.005000.001

Land phase initial conditionSubsurface flowm3/day0.0010.0010.0010.0010.0000.0010.0010.0010.0010.0010.01000.0010.0010.0010.0010.0010.005000.001

Land phase initial conditionSurface Nitratemg N / l1.0001.0001.0003.0000.0001.0001.0001.0001.0003.00010.00001.0001.0001.0001.0003.0005.000001.000

Land phase initial conditionSubsurface Nitratemg N / l1.5001.5001.5004.0000.0001.5001.5001.5001.5004.00010.00001.5001.5001.5001.5004.0005.000001.500

Land phase initial conditionSurface Ammoniummg N / l0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0002.00000.0000.0000.0000.0000.0001.000000.000

Land phase initial conditionSubsurface Ammoniummg N / l0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0002.00000.0000.0000.0000.0000.0001.000000.000

Land phase initial conditionSurface drainage volumem3/km2450000.000450000.000450000.000450000.000100000.00000450000.000450000.000450000.000450000.000450000.000100000.00000450000.000450000.000450000.000450000.000450000.00010050000.00000450000.000

Land phase initial conditionSubsurface drainage volumem3/km25000000.0005000000.0005000000.0005000000.000100000.000005000000.0005000000.0005000000.0005000000.0005000000.00020000000.00005000000.0005000000.0005000000.0005000000.0005000000.00010050000.000005000000.000

Land phase parameterDenitrification ratem/day0.0010.0010.0020.0010.0000.0000.0010.0010.0020.0010.01000.0000.0010.0010.0020.0019.500000.000

Land phase parameterNitrogen fixationkg N/ha/day0.0000.0000.0000.0000.00000000.0000.0000.0000.0000.0000.01000.0000.0000.0000.0000.0000.000050.000

Land phase parameterPlant nitrate uptake ratem/day0.1500.1200.1000.0100.0000.0000.1500.1200.1000.0100.0500.0000.1500.1200.1000.01081.0000.000

Land phase parameterMaximum nitrate uptake ratekg N/ha/yr75.00042.000100.000105.00080.0000.00075.00042.000100.000105.000140.00000.00075.00042.000100.000105.000110.000000.000

Land phase parameterNitrate addition ratekg N/ha/day0.0000.0000.0000.3300.4300.0000.0000.0000.0000.330.43000.0000.0000.0000.0000.33050.000000.000

Land phase parameterNitrificationm/day0.0700.1000.0500.0200.0000.0000.0700.1000.0500.0200.10000.0000.0700.1000.0500.02027.000000.000

Land phase parameterMineralisationkg N/ha/day0.2500.2500.3200.5000.0000.0000.2500.2500.3200.5001.00000.0000.2500.2500.3200.500146.000000.000

Land phase parameterImmobilisationm/day0.0700.0700.0100.0050.0000.0000.0700.0700.0100.0050.10000.0000.0700.0700.0100.0050.050000.000

Land phase parameterAmmonium addition ratekg N/ha/day0.0000.0000.0000.3300.4300.0000.0000.0000.0000.3300.43000.0000.0000.0000.0000.33050.000000.000

Land phase parameterplant ammonium uptake ratem/day0.4500.3000.5000.0100.0000.0000.4500.3000.5000.0100.05000.0000.4500.3000.5000.01081.000000.000

Land phase parameterPlant growth start dayjulian day60.00060.00060.00060.00060.0000.00060.00060.00060.00060.00060.0000.00060.00060.00060.00060.000183.0000.000

Land phase parameterPlant growth perioddays245.000245.000245.000245.000190.0000.000245.000245.000245.000245.000190.0000.000245.000245.000245.000245.000183.0000.000

Land phase parameterFertiliser addition start dayjulian day0.0000.0000.00060.00060.0000.0000.0000.0000.00060.00060.0000.0000.0000.0000.00060.000183.0000.000

Land phase parameterFertiliser addition perioddays0.0000.0000.000184.000184.0000.0000.0000.0000.000184.000184.0000.0000.0000.0000.000184.000183.0000.000

Land phase parameterSoil moisture deficitmm100.000100.000100.000100.000100.000100.000100.000100.000100.000100.000170.0000100.000100.000100.000100.000100.000135.00000100.000

Land phase parameterMaximum temp. Difference4.5004.5004.5004.5004.5004.5004.5004.5004.5004.5004.5004.5004.5004.5004.5004.5004.5004.500

Land phase parameter9999.0009999.0009999.0009999.0009999.0009999.0009999.0009999.0009999.0009999.0009999.0009999.0009999.0009999.0009999.0009999.0009999.0009999.000











Land phase parameter-0.025-0.025-0.025-0.025-0.025-0.025-0.025-0.025-0.025-0.025-0.025-0.025-0.025-0.025-0.025-0.025-0.025-0.025

Land phase parameterSoil waater residence timeday2.3002.3002.3002.3000.5002.3002.3002.3002.3002.3005.0002.3002.3002.3002.3002.3002.5002.300

Land phase parameterGround residence timeday23.00023.00023.00023.00010.00023.00023.00023.00023.00023.000200.00023.00023.00023.00023.00023.000100.00023.000

Land phase parameterMaximum soil retention volumemm0.4500.4500.4500.4500.1000.4500.4500.4500.4500.4501.0000.4500.4500.4500.4500.4500.0500.450

Instream initial

Instream initial conditions

flowm3/s0.010000

Nitrate concentrationmg N/l1.500000

Ammonium concentrationmg N/l0.050000

Reach descriptors

Reach descriptors and inputs

Minimum valueMaximum valueCentral valueInput

Subbasin nameLength (m)QNO3NH4QaQbNitDenitQaQbNitDenitQaQbNitDenit

Norham00.0000.0000.0000.0200.6700.0000.0000

Subcatchment descriptors

Subcatchment descriptors and inputs

LanduseMinimum valueMaximum valueCentral value

Subbasin nameArea (km2)ForestUrbanBaseflow indexNO3dryNO3wetNH4dryNH4wetBaseflow indexNO3dryNO3wetNH4dryNH4wetNO3dryNO3wetNH4dryNH4wet

Norham658000010000.0003.0003.0003.0003.0000.0003.0003.0003.0003.000

Norham

mymax

561.5675871934

345.6244042075

144.3079462037

74.3663034351

65.0367007403

36.2179662988

26.2107054971

23.6133106357

23.0541445743

21.1581761437

mymax

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p

00.2500.500.0125801400100.0500.05

00.50.50.6250.0250.05801400100.0500.05

0.50.6250.50.62500.02580140010.050.100.05

0.50.62500.50.0250.0580140010.050.100.05

0.750.8750.50.6250.01250.025801400100.0500.05

0.750.812500.500.05801400100.100.05

00.50.750.812500.05801400100.100.05

0.750.8750.750.87500.05801400100.100.05

0.50.6250.750.87500.05801400100.100.05

0.8750.937500.500.05801400100.100.05

00.50.8750.937580140

0.87510.750.87580140

0.87510.50.62580140

0.50.6250.875180140

0.87510.875180140

0.750.8750.8751

00.50.6250.75

0.6250.7500.5

0.50.6250.6250.75

0.250.500.5

0.6250.750.8751

0.87510.6250.75

0.9375100.5

00.50.93751

0.750.8750.6250.75

0.6250.750.750.875

0.6250.750.6250.75

0.6250.750.50.625

0.81250.87500.5

00.50.81250.875

0.0050.010.02656250.028125

00.0050.02656250.028125

0.0050.010.0218750.0234375

0.0050.010.0406250.04140625

00.0050.006250.0125

0.0050.010.046093750.046875

00.0050.046093750.046875

0.0050.010.03906250.03984375

0.0050.010.044531250.0453125

00.0050.044531250.0453125

0.0050.010.03750.03828125

0.0050.010.042968750.04375

00.0050.042968750.04375

0.0050.010.02343750.025

00.0050.02343750.025

0.0050.010.041406250.0421875

00.0050.041406250.0421875

0.0050.010.03593750.03671875

0.0050.010.018750.0203125

0.0050.010.039843750.040625

00.0050.039843750.040625

0.0050.00750.0343750.0359375

0.0050.010.038281250.0390625

00.0050.038281250.0390625

0.0050.00750.03281250.034375

0.0050.010.036718750.0375

00.0050.036718750.0375

0.0050.010.02031250.021875

0.0050.00750.031250.0328125

00.0050.02031250.021875

0.0050.010.006250.0125

0.00750.010.0343750.0359375

0.0050.010.0156250.01875

00.0050.035156250.0359375

0.0050.010.02968750.03125

0.00750.010.03281250.034375

00.0050.033593750.034375

0.0050.010.0281250.0296875

0.00750.010.031250.0328125

00.0050.032031250.0328125

Efficiency5

true value0.8987867613

83.28074773.237862783760.948955100.8595282715151.2374290470.8987706233

113.282639109.879804709911.57928060842395.1044850447

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38.21735939.36862440191.3254109465.3242451939

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52.11190052.3052028350.037366132174.5597220813

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61.48043651.547059601698.671976561988.2273293287

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79.16923677.14054990034.1155666451262.4578170733

46.11712841.477773334421.5236103207378.7782667846

61.45060563.50457741294.21880350116.577057307

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39.90350140.38184972640.2288172832422.6375426727

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100.731749100.33229503760.15956379811551.7529083333

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64.85088363.88727559520.92853841278.6864334339

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64.41312066.83564221165.868613942534.7585970871

40.25512940.42418924930.0285812783420.8984907601

62.24957562.59243848810.11755505132.730552957

69.92894068.10954298383.310205624151.4023465962

87.21714364.176122685530.888613668410.4724900326

49.72688544.930476410623.005539263256.3048455594

54.85145655.34568922010.244266053631.2963131018

40.63549141.07228354870.1907876901394.7261569895

61.02869459.68576895251.80344685931.5730955206

67.44252067.82499092290.14628428247.4031000089

45.30889943.69079737932.6182526925297.5349910511

77.23115574.61507165126.8438906391187.0075846662

61.14921045.6809288626239.2677300121232.8392519763

70.63907067.391005074410.549925382341.6154664701

77.80737474.73249082629.4549074067190.2328031918

59.53273658.5033147661.05970905775.9374349296

49.15922847.58042634782.492613681178.478208169

52.04964847.591902238119.8714967837178.17171386

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49.07405045.278774522214.4041189216245.2739834657

61.06216660.89830151720.02685171490.0017387635

74.89684873.16289051543.0066096975149.3990525524

92.96890688.737257238817.9068477438772.6875100014

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66.838024

42.769908

60.788170

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35.737484

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41.645112

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50.929580

89.459004

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50.190818

Nitrate 1

4.989

5.2510.025

4.799

5.196

3.482

4.995

5.618

5.615

5.103

5.478

4.86

4.301

4.079

5.409

5.69

5.555

5.002

5.074

5.447

5.503

5.504

4.412

5.143

4.525

4.292

4.27

5.256

5.531

5.039

5.395

4.264

3.965

5.451

4.485

5.455

5.109

5.313

5.469

4.858

5.316

5.422

5.426

5.396

5.353

5.207

4.722

4.989

5.13

5.177

5.182

4.671

4.108

4.875

4.152

5.049

3.026

3.941

3.897

5.281

5.433

4.546

5.346

3.885

4.766

5.49

5.354

4.666

5.335

5.304

4.963

4.248

5.092

5.665

4.673

5.055

5.848

6.008

6.04

5.925

5.37

3.938

4.277

6.12

6.387

6.499

5.383

4.811

5.744

4.596

5.649

6.6

6.755

5.581

6.611

6.264

4.939

5.919

5.832

5.978

6.945

6.851

7.054

7.079

7.042

6.965

5.421

6.496

6.694

6.864

6.86

4.112

6.366

5.446

7.18

6.717

7.784

7.959

7.92

7.802

7.831

7.484

5.389

6.023

7.461

7.341

7.806

8.093

8.056

8.028

7.918

7.756

7.561

7.341

7.107

6.866

6.631

6.196

6.313

6.304

6.099

6.227

6.23

6.163

6.06

5.942

5.817

5.702

5.615

5.55

5.502

5.467

5.441

5.141

7.342

8.469

8.979

9.095

9.069

8.642

8.117

8.534

8.611

8.505

8.296

8.027

7.12

6.631

8.934

10.14

10.79

10.11

9.255

10.4

10.55

8.92

9.382

10.79

11.3

11.39

11.27

11.03

10.7

10.31

9.549

8.712

9.491

9.349

9.805

9.893

9.506

8.952

9.677

9.862

9.981

9.615

9.735

9.638

9.412

9.108

8.764

8.369

7.666

9.211

9.876

9.769

9.729

10.46

10.93

11.03

10.89

8.6

9.124

13.12

14.17

12.98

13.25

14.04

14.23

14.16

13.94

13.17

12.95

12.76

12.39

12.04

11.61

11.14

9.917

10.65

9.957

11.82

12.23

13.23

8.269

14.1

12.58

14.35

14.18

15.28

14.13

15.17

15.65

15.73

15.63

15.42

11.32

13.44

13.33

11.49

13.6

14.4

13.03

12.89

13

11.02

13.84

14.58

11.28

13.96

14.78

14.55

10.25

12.31

14.27

14.8

14.9

14.04

14.28

14.27

14.13

13.9

13.59

10.4

11.07

8.912

13.25

13.72

14.11

14.44

14.31

14.31

14.19

13.99

13.74

13.43

13.09

12.71

10.43

11.42

11.84

11.98

7.647

10.56

11.81

9.412

13.31

13.8

12.8

13.23

13.54

13.58

10.27

9.367

12.08

12.94

10.16

11.67

9.048

11.8

12.85

12.69

9.851

8.541

11.25

10.51

8.899

8.597

10.03

11.97

10.18

10.18

9.638

11.37

11.39

11.81

11.62

11.74

11.73

11.69

11.62

11.54

11.45

11.34

11.21

11.07

6.835

7.881

7.921

8.448

8.165

6.336

10.54

10.6

7.541

7.846

10.22

10.26

10.24

10

9.335

6.698

8.342

9.632

9.886

9.908

9.893

9.745

9.77

6.929

6.354

6.647

7.506

6.266

8.513

9.136

9.103

9.257

9.269

7.481

8.024

7.456

8.13

8.327

8.287

5.694

8.358

8.228

8.584

8.63

7.7

5.772

5.633

7.409

6.21

8.298

5.99

8.077

8.402

8.459

5.943

8.054

5.783

7.235

7.322

4.985

6.257

7.743

6.764

7.671

7.825

7.661

7.714

7.781

7.687

6.917

5.022

5.273

6.142

6.085

5.765

7.018

6.963

7.233

4.705

5.838

5.599

5.375

5.93

6.995

7.018

7.083

7.034

6.932

4.159

6.475

6.909

7.073

4.115

6.55

6.461

6.141

6.979

6.787

5.121

6.852

7.305

7.151

4.772

6.656

5.759

6.305

6.84

7.542

7.729

7.816

7.771

7.224

6.409

6.383

4.664

7.013

6.803

7.375

7.818

8.171

8.305

8.387

7.271

7.604

7.925

8.182

8.323

8.405

6.791

7.566

7.862

8.137

8.299

8.395

8.446

8.278

4.841

7.718

8.75

9.227

9.265

5.782

8.248

9.159

9.349

9.246

9.563

9.517

9.68

9.769

9.817

9.833

9.825

7.967

8.695

7.156

8.531

7.544

7.617

9.431

9.383

9.743

10.23

10.38

10.54

10.1

8.802

7.653

9.353

10.16

10.61

9.769

10.27

6.977

9.192

9.619

7.881

9.356

10.93

10.45

10.33

10.42

10.56

10.54

7.509

10.1

10.86

10.66

11.18

11.41

10.96

10.68

11.03

11.21

11.3

11.34

10.61

10.71

9.72

8.097

10.02

10.89

11.35

11.6

11.73

11.79

11.79

11.76

11.7

11.61

11.49

11.32

8.763

10.19

10.91

10.77

10.3

11.17

11.65

11.92

12.07

9.363

11.09

11.94

9.111

9.909

11.73

11.35

12.53

13.33

11.12

12.67

13.26

12.63

13.22

13.52

13.69

13.75

13.67

13.65

13.59

13.5

13.38

13.22

13.02

12.78

12.55

12.33

12.13

11.93

11.75

11.58

11.43

11.23

11.19

11.07

10.96

10.86

10.77

10.69

10.62

10.56

10.5

10.02

10.83

9.4

10.86

11.29

14.32

14.49

15.45

16.24

16.61

14.5

14.14

13.51

10.63

13.22

13.34

10.29

13.84

14.98

15.21

13.51

14.98

15.4

15.54

15.58

14.77

14.04

12.56

13.68

14.26

14.55

14.17

7.871

13.26

13.48

12.75

14.34

14.79

14.93

12.76

12.26

8.263

11.84

13.72

11.63

11.56

12.62

13.69

13.55

13.81

12.77

7.846

12.49

13.3

13.11

13.35

13.13

13.16

11.96

11.64

11.77

9.1

11.27

7.517

8.828

11.22

12.32

12.7

12.12

12.08

10.85

12.06

12.4

12.4

12.46

12.46

9.888

10.88

10.81

11.21

9.085

6.622

11.4

12.11

9.506

6.753

11.66

11.86

11.9

11.89

11.41

11.16

11.5

8.276

8.961

7.872

10.66

10.4

10.48

11.04

11.25

11.2

10.99

8.254

9.187

8.895

7.663

9.938

10.71

10.84

10.84

10.66

10.15

9.158

9.388

8.809

9.575

9.414

6.786

9.398

10.2

6.177

6.601

10.06

9.947

10.47

10.52

10.52

10.5

10.48

10.46

6.08

8.971

9.836

8.443

6.835

9.051

8.768

9.137

6.506

8.009

9.625

7.218

7.97

9.567

9.756

9.773

9.762

9.742

9.718

9.689

9.655

8.219

8.699

8.932

9.046

6.62

4.796

7.19

9.229

9.372

9.381

9.374

9.36

9.341

9.321

9.295

4.398

7.738

8.626

6.751

6.705

7.975

5.033

8.037

8.263

8.416

8.416

8.407

7.069

7.232

6.643

7.941

7.098

7.913

7.28

5.459

7.568

7.988

8.096

8.127

8.142

8.157

8.167

8.171

8.167

8.155

8.133

8.102

8.061

8.012

7.955

4.247

6.996

8.279

8.652

8.805

5.131

8.222

8.825

9.007

8.774

8.978

9.079

9.134

9.165

9.177

9.174

9.158

9.128

9.083

9.026

8.957

8.877

8.749

8.626

8.509

8.399

8.296

8.2

8.112

8.032

7.956

7.884

7.819

7.761

7.708

7.66

7.616

5.471

7.345

9.203

7.135

8.103

7.058

9.694

10.74

10.97

11.04

11.09

11.12

11.12

5.23

7.947

10.82

10.98

11.06

11.12

11.17

11.21

11.25

11.29

11.29

11.28

11.25

11.23

11.19

11.13

11.06

10.97

10.86

10.74

10.59

10.4

10.23

10.06

9.9

9.751

9.606

9.472

9.345

9.227

9.117

9.016

8.924

8.84

8.764

8.694

8.631

8.574

8.523

8.477

8.433

8.393

8.356

8.324

8.296

8.272

8.252

8.236

8.224

8.214

8.206

8.201

8.199

8.198

8.201

8.205

8.21

8.217

8.222

8.228

8.237

8.246

8.252

8.255

8.256

8.257

8.261

8.266

8.274

8.282

8.292

8.304

8.316

8.328

8.342

8.359

8.376

8.395

8.414

8.436

8.457

8.479

8.5

8.521

8.54

8.559

8.58

8.601

8.623

8.644

8.666

8.688

8.711

8.735

8.76

8.785

8.812

8.84

8.855

8.872

8.889

8.906

8.916

8.926

8.938

8.951

8.965

8.981

8.999

9.017

9.037

9.057

9.066

9.073

9.076

9.08

9.082

9.085

9.085

9.085

9.087

9.091

9.096

9.101

9.104

9.105

9.104

9.102

9.099

9.096

9.095

9.094

9.094

9.095

9.096

9.098

9.101

9.104

9.107

9.112

9.116

9.122

9.127

9.133

9.138

9.144

9.149

9.154

9.16

9.164

9.168

9.164

9.161

9.158

9.155

9.152

9.141

9.131

9.119

9.108

9.098

9.088

9.079

9.071

9.059

9.048

9.038

9.02

9

8.975

8.952

8.927

8.904

8.879

8.855

8.831

8.808

8.783

8.757

6.501

7.833

10.28

13.39

13.56

13.57

13.54

10.19

8.627

11.26

8.579

10.01

12.62

11.69

12.55

12.63

6.989

11.77

10.35

12

11.56

8.883

11.58

11.9

6.593

11.41

10.36

11.47

11.58

7.5

8.569

11.09

11.28

8.728

9.962

10.8

8.602

9.419

5.032

10.14

10.13

10.11

10.12

10.09

10.04

10.06

9.861

9.265

9.323

9.528

9.558

9.669

6.254

4.77

9.292

9.491

9.489

9.398

9.362

9.409

8.974

8.163

7.151

8.03

7.423

5.751

6.825

8.844

7.33

8.155

8.657

8.453

8.381

8.745

7.888

8.437

8.645

5.812

7.47

8.296

8.576

8.68

8.717

8.077

7.473

7.541

7.92

8.111

8.208

7.876

6.868

6.798

7.024

7.239

7.662

7.884

8.001

8.058

8.077

7.686

4.412

7.311

8.064

7.307

8.076

6.861

4.56

6.339

7.298

7.156

7.903

8.096

7.436

4.826

5.026

7.107

5.228

5.443

7.189

6.344

5.66

5.308

6.637

6.021

5.61

5.587

4.64

5.805

6.47

6.524

6.491

6.609

6.348

6.684

6.625

6.801

6.894

6.96

6.052

6.498

5.105

6.087

5.653

4.77

6.511

7.138

7.368

5.128

5.409

6.772

7.164

5.47

6.295

7.379

7.707

7.844

7.932

7.99

6.707

7.19

6.136

6.311

7.342

7.888

8.146

8.286

8.372

8.421

8.447

8.453

8.256

8.281

8.28

8.259

5.995

7.205

6.946

7.219

8.2

7.104

8.349

8.921

9.219

6.368

8.374

9.241

9.633

9.84

9.955

5.979

6.392

7.854

8.859

7.91

9.54

8.847

8.425

8.09

8.05

8.847

9.851

8.313

5.687

9.428

8.645

7.493

8.879

9.766

9.981

10.07

10.13

8.997

9.743

10.05

10.2

10.28

10.34

10.36

10.36

9.383

9.666

9.826

6.091

8.657

9.227

10.34

10.83

8.75

5.155

8.789

8.138

10.12

9.119

10.09

10.09

9.671

6.956

8.668

9.719

8.999

9.995

10.24

8.314

9.253

7.364

7.829

9.675

5.992

9.118

7.164

9.45

10.11

10.22

10.27

10.32

10.36

10.39

10.42

10.44

6.436

7.462

9.638

10.13

10.37

10.48

10.53

10.56

10.57

10.57

10.54

6.11

7.312

9.89

9.519

9.675

10.26

9.597

8.916

9.569

10.33

10.65

10.8

10.88

10.91

10.92

10.9

8.018

9.375

10.04

9.698

10.21

6.248

10.01

11.03

10.96

11.02

11.34

11.5

10.16

9.144

10.37

9.273

9.985

10.23

10.61

8.031

10.38

10.03

11.13

9.976

9.914

8.659

8.716

9.046

10.79

10.09

10.99

11.33

11.47

11.53

10.45

9.645

10.17

9.457

8.514

8.151

10.22

10.92

11.19

11.3

11.34

11.34

11.32

11.27

11.2

11.11

11.01

10.28

10.33

10.21

10.2

10.15

10.07

9.957

9.746

7.91

8.696

7.899

6.342

9.67

10.76

11.1

11.26

8.543

7.659

9.262

10.17

10.58

10.39

10.65

10.75

10.78

10.77

10.74

10.69

10.62

10.53

10.44

10.19

10.1

10

8.765

8.968

8.821

8.135

6.077

8.686

8.132

7.215

8.997

9.716

8.411

9.45

9.858

9.693

9.477

8.015

6.294

7.695

7.699

7.226

8.719

9.503

7.849

8.039

7.805

7.152

8.848

6.543

8.07

9.088

9.21

7.744

8.416

8.921

6.788

8.416

6.083

6.623

6.339

6.254

8.351

7.773

6.937

8.459

8.38

8.575

8.367

4.72

6.036

8.141

8.033

7.832

6.902

5.187

5.968

7.808

7.794

7.876

7.713

6.539

7.428

4.731

5.046

6.053

6.081

7.344

5.758

6.833

5.193

6.915

7.144

6.767

6.863

3.87

6.584

6.775

6.556

5.51

4.599

6.395

6.489

6.121

4.449

5.57

6.425

6.521

6.573

6.579

6.576

6.57

5.408

5.045

5.799

6.142

5.725

5.153

5.717

4.197

5.466

5.666

4.743

4.667

4.855

6.004

5.236

5.952

5.432

5.907

6.139

6.221

6.254

3.8

5.032

5.574

5.933

6.097

6.16

5.672

3.763

5.426

4.559

4.793

5.754

6.159

6.009

3.689

5.384

6.331

6.299

4.516

6.289

5.73

6.494

6.72

6.696

6.616

6.795

6.9

6.865

6.942

6.991

7.02

6.042

6.408

5.273

5.16

6.463

7.021

4.072

6.962

7.327

6.996

4.572

6.809

6.763

7.654

6.695

6.842

5.718

6.757

7.65

7.53

7.63

8.034

8.17

6.588

4.881

7.933

8.381

6.935

7.051

6.841

5.784

7.602

7.87

6.599

8.012

8.544

8.279

6.217

8.219

8.709

8.882

8.984

9.051

7.128

6.523

7.265

8.572

9.032

9.236

6.262

8.283

8.969

9.313

9.484

9.581

9.635

9.664

9.67

8.654

8.854

9.066

9.184

9.247

8.38

7.237

6.075

6.145

6.205

8.745

8.667

9.632

7.749

9.614

9.866

8.49

8.197

9.543

6.449

9.197

6.744

9.284

9.766

6.955

7.237

9.256

9.702

9.904

9.642

9.896

10

10.06

7.945

8.386

9.145

7.364

6.465

6.796

9.581

9.96

10.05

10.02

10.1

10.15

9.689

9.887

9.987

9.917

8.71

9.217

6.898

6.943

5.926

9.797

10.19

10.22

8.119

9.412

9.597

7.175

7.371

9.592

8.318

9.698

10.04

9.765

7.378

7.733

8.366

8.494

8.241

8.714

9.798

8.563

9.474

9.314

8.538

8.376

7.182

9.346

9.85

10.01

8.629

8.782

8.973

8.106

9.232

6.564

9.109

9.874

9.957

7.422

9.275

9.894

6.042

9.39

8.971

9.889

10.2

10.33

10.38

10.4

6.285

6.249

9.729

9.66

9.95

10.01

9.975

7.994

5.816

7.53

9.378

9.424

8.236

9.353

9.569

9.686

9.732

8.809

9.194

9.37

9.452

9.373

9.397

9.387

9.354

9.302

9.235

9.156

8.657

6.922

5.119

8.068

8.991

9.233

7.423

6.94

8.448

8.921

9.131

6.355

7.226

8.532

6.803

7.24

6.877

6.952

5.861

8.095

8.542

8.82

5.253

7.706

8.005

7.52

5.788

8.211

6.417

5.493

7.517

7.912

8.078

8.083

8.076

3.554

6.764

6.934

6.584

7.073

6.51

5.627

6.111

6.341

6.73

5.778

5.586

5.113

6.597

6.72

6.499

5.991

5.925

5.338

4.861

5.707

5.31

6.169

5.387

5.89

3.699

6.143

6.006

6.188

6.337

6.298

6.34

6.074

6.19

6.241

4.507

4.64

5.353

4.554

5.263

4.521

3.971

5.312

5.91

5.528

5.063

4.389

5.783

5.983

5.819

5.289

5.567

5.059

4.043

3.182

5.55

5.784

4.507

5.479

5.572

3.323

4.889

4.716

4.532

3.579

5.249

5.273

5

5.281

5.273

4.327

4.983

4.581

2.839

3.512

4.743

4.805

3.106

4.632

4.7

4.823

4.83

3.994

2.529

4.263

4.546

4.378

4.474

4.086

4.438

4.523

4.546

4.553

3.932

4.202

4.371

3.813

4.148

4.306

4.39

4.33

4.392

4.433

4.322

3.813

3.767

3.523

2.616

3.377

3.947

2.699

4.088

4.382

4.27

4.318

4.402

4.437

2.688

2.702

3.654

3.398

3.874

3.929

4.491

3.919

4.52

4.276

3.948

4.627

4.617

3.792

4.63

4.951

5.102

5.195

4.572

4.941

4.994

3.683

3.844

4.249

5.13

5.292

5.677

5.847

5.951

3.849

5.357

5.718

5.994

6.187

6.301

6.123

5.083

4.718

5.322

6.089

6.464

5.805

6.297

4.744

4.284

5.832

5.815

6.124

6.987

7.162

6.311

6.526

4.126

6.005

6.639

7.416

7.565

7.641

7.703

7.77

7.824

7.872

7.909

7.936

7.952

7.959

7.954

4.534

6.631

5.614

7.163

5.613

6.384

6.133

6.143

7.517

8.183

8.288

8.378

8.452

8.528

8.594

8.559

6.046

7.646

7.443

7.566

7.577

8.218

6.105

7.241

8.486

8.897

9.079

9.173

5.012

8.176

7.302

5.703

7.81

8.923

8.739

9.106

9.226

8.118

8.042

7.963

8.8

9.056

7.609

8.618

9.05

6.139

8.409

7.596

8.925

9.431

9.65

9.767

5.038

8.837

9.06

9.601

9.816

9.845

8.808

6.367

8.928

9.454

9.779

9.147

9.511

9.741

9.851

9.913

9.94

9.103

9.346

9.481

9.454

9.506

7.041

7.404

8.584

9.477

9.922

10.17

10.31

10.37

10.39

10.39

10.35

10.29

10.21

10.11

9.964

9.816

9.675

9.543

7.901

8.884

9.441

9.641

10.01

10.22

10.33

9.248

9.56

8.977

10.21

9.644

8.007

10.04

11.76

12.42

12.73

12.89

12.98

12.77

9.849

9.672

11.58

12.42

11.59

12.22

12.69

12.94

13.05

13.08

13.05

12.98

8.478

11.06

10.22

11.96

12.72

10.55

11.92

12.65

13

13.17

9.595

11.17

11.37

8.261

7.495

12.34

11.8

8.824

10.82

12.12

12.25

9.25

10.26

8.994

10.04

9.597

10.93

10.84

11.39

11.53

11.39

11.26

11.37

11.4

10.61

10.7

10.83

10.88

10.72

10.73

10.69

10.63

9.347

7.6

9.058

6.1

9.422

10.09

6.988

9.871

10.6

9.435

10.37

10.52

8.879

7.995

7.366

9.543

8.57

6.828

6.828

9.645

9.533

10.08

10.13

10.13

10.12

10.01

10.03

10.02

9.609

9.698

9.727

9.721

7.985

8.583

8.882

9.037

7.156

6.934

7.427

5.936

5.991

6.748

8.944

8.841

7.165

6.086

8.942

8.929

7.324

6.783

7.751

5.612

6.986

7.761

6.003

7.437

8.081

5.664

7.867

6.373

7.763

7.818

7.815

6.009

6.491

5.622

4.752

5.479

6.76

7.162

7.243

7.242

6.545

6.064

6.755

7.051

5.297

6.451

3.988

4.71

6.568

5.482

5.826

6.621

6.376

4.232

6.426

6.426

6.529

6.542

6.539

6.532

6.524

6.515

6.504

6.185

6.26

6.128

6.186

6.215

6.227

4.251

3.53

5.064

4.632

4.394

6.022

5.723

5.847

6.063

5.878

4.615

4.725

3.989

4.068

5.491

4.463

4.44

5.433

5.456

4.541

5.279

4.736

5.061

5.528

4.976

5.398

5.31

4.151

4.143

5.18

3.178

4.441

5.131

5.355

5.168

4.023

4.976

5.488

5.162

4.804

4.653

3.901

4.566

5.682

5.894

5.935

5.907

6.047

6.13

6.193

6.242

6.284

6.316

6.341

5.568

5.737

3.225

5.984

6.62

5.497

6.452

6.795

6.873

6.627

5.693

6.272

5.038

6.184

6.909

7.202

7.355

7.446

7.505

7.542

5.925

4.267

4.407

7.013

7.772

7.787

6.884

5.445

7.706

7.139

6.881

6.568

7.837

7.463

4.678

5.751

5.32

7.826

7.958

8.022

8.066

8.175

8.258

8.317

8.378

8.432

8.491

8.547

8.588

8.617

8.63

8.629

8.617

8.592

8.553

8.501

4.2

8.508

8.915

7.392

8.171

9.171

9.434

9.413

9.493

7.458

5.715

8.959

8.853

9.563

9.527

8.606

9.276

9.675

5.089

9.212

9.574

9.141

9.173

9.624

5.734

8.289

9.584

9.064

9.746

9.962

10.07

10.15

10.2

10.23

10.24

8.573

7.478

6.827

9.292

10.08

10.26

10.49

10.6

10.25

10.4

10.49

10.28

10.35

10.37

10.35

6.328

9.24

10.37

8.845

6.588

10.32

11.2

11.46

11.51

11.61

11.67

11.71

11.73

11.73

11.71

11.68

11.63

11.55

10.84

9.999

9.961

10.51

10.82

7.099

9.302

9.469

9.786

12.02

8.821

11.84

12.48

12.69

12.8

12.78

10.24

9.197

11.49

12.3

8.61

8.59

11.75

12.52

10.6

6.552

9.853

8.87

10.04

10.46

11.37

11.49

11.55

9.022

9.637

10.76

11.27

11.35

10.59

9.61

6.484

9.888

10.88

7.428

8.416

8.735

10.41

10.73

10.23

5.168

9.491

10.14

9.845

9.414

9.293

9.9

6.971

9.572

5.715

8.721

8.361

9.057

9.178

7.953

9.185

6.386

7.7

8.64

8.97

9.155

9.239

7.965

8.111

6.546

8.097

7.266

6.147

8.471

4.746

8.37

7.393

8.238

7.992

8.498

8.558

8.561

6.839

7.17

7.765

7.425

7.983

5.169

5.464

5.141

7.88

7.305

7.645

5.74

7.554

7.652

7.377

6.935

7.571

7.72

7.757

4.472

3.772

5.611

6.982

7.016

6.691

4.71

6.347

6.719

6.688

5.119

6.38

6.625

6.149

6.512

6.514

5.432

5.101

5.923

5.205

4.037

5.854

5.427

4.613

5.674

4.711

3.783

5.675

5.151

4.479

4.87

5.56

3.565

4.464

4.911

5.026

4.779

3.572

5.082

5.128

5.145

5.144

5.142

5.139

3.404

4.731

4.998

5.054

3.662

3.899

4.748

4.935

4.989

3.325

4.521

4.819

2.553

Ammonium 1

0.9966

1.047

0.9535

1.029

0.6832

0.9753

1.091

1.086

0.9831

1.051

0.928

0.8125

0.7611

1.002

1.051

1.025

0.9211

0.9307

0.9977

1.01

1.014

0.8043

0.9278

0.8085

0.7525

0.736

0.9003

0.9485

0.8647

0.9263

0.7256

0.6612

0.9035

0.7416

0.901

0.8453

0.8806

0.9121

0.8124

0.8907

0.9174

0.9307

0.9403

0.9488

0.9369

0.8475

0.8895

0.9205

0.9402

0.9542

0.8546

0.715

0.8237

0.684

0.82

0.4675

0.6025

0.5976

0.8121

0.8638

0.7732

0.9629

0.7412

0.9534

1.143

1.154

1.039

1.223

1.248

1.199

1.06

1.304

1.475

1.242

1.375

1.615

1.674

1.69

1.661

1.517

1.162

1.301

1.88

1.969

2.004

1.671

1.518

1.829

1.482

1.839

2.156

2.206

1.83

2.176

2.063

1.646

1.992

1.976

2.039

2.369

2.328

2.379

2.359

2.311

2.245

1.76

2.131

2.18

2.204

2.159

1.388

2.195

1.882

2.474

2.305

2.66

2.693

2.643

2.566

2.535

2.385

1.727

1.949

2.412

2.362

2.497

2.564

2.523

Documents

Extracting Complex Nonlinear Response Surfaces from Deterministic Models