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Appendices

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Appendix A

Linear curve fit results with latitude excluded as variable

The first figure in each case is a plot of the actual pCO 2 values, together with the mean values of the estimated pCO 2 from the 100 curve fits. The 95% confidence interval is also shown for the 100 curve fits in each case. The second figure in each case is the actual pCO 2 values plotted against the estimated pCO 2 values. The 10% error lines indicate the number of points that fall outside the 10% error. The third figure in each case is a histogram of the errors on the estimation of the pCO 2 , showing the distribution of the errors for each case.

It can be seen from the figures that as the number of points sampled from the data set increases, the fit of the data is improved and less points falls outside the 10% error interval. Also, from the figures it can be seen that the D-optimal sampling yields a smaller 95% confidence interval than the random sampling. The histograms generally show that the error on the estimation has a normal distribution around zero.

The mean coefficients for the linear equation for all 100 curve fits are given in Table A.1. The standard deviation for the coefficients are given in Table A.2. It can be seen from the table of the standard deviations that the coefficients for the D-optimal sampling have a smaller standard deviation than the coefficients for the random sampling.

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Table A.1: Coefficients of linear curve fitting

Sampling Random Random Random Random D-optim D-optim D-optim D-optim

No of

points k 10 50 100 200 10 50 100 200

β 1 (×10 2 ) 4.047 4.055 4.065 4.054 3.987 3.898 3.881 3.908 β 2 (×10 0 ) -2.115 -2.316 -2.314 -2.288 -1.545 -1.210 -1.116 -1.205 β 3 (×10 −2 ) 5.227 5.794 4.835 5.448 4.252 8.970 9.751 8.005 β 4 (×10 1 ) -3.128 -2.958 -3.010 -2.950 -2.501 -2.362 -2.321 -2.361

Table A.2: Standard deviation of coefficients of linear curve fitting

Sampling Random Random Random Random D-optim D-optim D-optim D-optim

No of

points k 10 50 100 200 10 50 100 200

β 1 (×10 1 ) 3.774 1.456 1.062 0.6111 3.528 1.826 1.183 0.7855 β 2 (×10 0 ) 1.499 0.4221 0.3099 0.1862 1.073 0.5695 0.4112 0.2514 β 3 (×10 −1 ) 4.156 1.381 1.015 0.6179 2.450 1.176 0.7197 0.5076 β 4 (×10 1 ) 1.750 0.3990 0.2916 0.1726 0.6322 0.3036 0.2066 0.1350

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure A.1: Linear curve fit results. 10 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure A.2: Linear curve fit results with 10 random points sampled. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure A.3: Linear curve fit results. Histogram of errors for all 100 runs and 10 random points sampled.

181

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure A.4: Linear curve fit results. 10 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure A.5: Linear curve fit results with 10 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure A.6: Linear curve fit results. Histogram of errors for all 100 runs and 10 D-optimal sampled points.

182

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure A.7: Linear curve fit results. 50 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure A.8: Linear curve fit results with 50 random points sampled. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure A.9: Linear curve fit results. Histogram of error for all 100 runs and 50 random points sampled.

183

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure A.10: Linear curve fit results. 50 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure A.11: Linear curve fit results with 50 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure A.12: Linear curve fit results. Histogram of errors for all 100 runs and 50 D-optimal sampled points.

184

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure A.13: Linear curve fit results. 100 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure A.14: Linear curve fit results with 100 random points sampled. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure A.15: Linear curve fit results. Histogram of error for all 100 runs and 100 random points sampled.

185

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure A.16: Linear curve fit results. 100 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure A.17: Linear curve fit results with 100 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure A.18: Linear curve fit results. Histogram of errors for all 100 runs and 100 D-optimal sampled points.

186

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure A.19: Linear curve fit results. 200 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure A.20: Linear curve fit results with 200 random points sampled. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure A.21: Linear curve fit results. Histogram of error for all 100 runs and 200 random points sampled.

187

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure A.22: Linear curve fit results. 200 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure A.23: Linear curve fit results with 200 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure A.24: Linear curve fit results. Histogram of errors for all 100 runs and 200 D-optimal sampled points.

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Appendix B

Quadratic curve fit results with latitude excluded as variable

The first figure in each case is a plot of the actual pCO 2 values, together with the mean values of the estimated pCO 2 from the 100 curve fits. The 95% confidence interval is also shown for the 100 curve fits in each case. The second figure in each case is the actual pCO 2 values plotted against the estimated pCO 2 values. The 10% error lines indicate the number of points that fall outside the 10% error. The third figure in each case is a histogram of the errors on the estimation of the pCO 2 , showing the distribution of the errors for each case.

It can be seen from the figures that as the number of points sampled from the data set increases, the fit of the data is improved and less points falls outside the 10% error interval. Also, from the figures it can be seen that the D-optimal sampling yields a smaller 95% confidence interval than the random sampling. The histograms generally show that the error on the estimation has a normal distribution around zero.

The mean coefficients for the quadratic equation for all 100 curve fits are given in Table B.1. The standard deviation for the coefficients are given in Table B.2. It can be seen from the table of the standard deviations that the coefficients for the D-optimal sampling have a smaller standard deviation than the coefficients for the random sampling.

189

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Table B.1: Coefficients for quadratic curve fitting

Sam-pling Random Random Random Random D-optim D-optim D-optim D-optim

No of

Points k 25 50 100 200 25 50 100 200

β 1 (×10 2 ) 4.049 4.020 4.094 4.038 3.621 3.501 3.469 3.489 β 2 (×10 0 ) 0.4693 1.235 -0.2634 0.1997 1.661 3.170 3.653 3.244 β 3 (×10 0 ) 0.4697 0.4988 0.4101 0.5037 1.217 1.306 1.339 1.334 β 4 (×10 1 ) -7.827 -7.053 -7.168 -6.967 -4.353 -3.943 -3.721 -3.811 β 5 (×10 −2 ) -9.673 -7.606 -3.196 -3.604 -2.262 -6.140 -7.604 -6.447 β 6 (×10 −3 ) -3.973 -3.257 -2.796 -3.148 -5.142 -5.333 -5.450 -5.495 β 7 (×10 0 ) 9.757 7.704 7.462 7.290 4.047 3.512 3.330 3.443 β 8 (×10 −2 ) -1.326 -3.073 -2.597 -3.077 -5.572 -6.144 -6.272 -6.125 β 9 (×10 0 ) -1.420 -1.433 -0.6603 -0.7012 0.4747 0.3131 0.1749 0.2396 β 10 (×10 −1 ) 4.176 3.241 3.060 2.964 0.01082 -0.03852 -0.2300 -0.2318

Table B.2: Standard deviation of coefficients for quadratic curve fitting

Sam-pling Random Random Random Random D-optim D-optim D-optim D-optim

No of

Points k 25 50 100 200 25 50 100 200

β 1 (×10 1 ) 8.202 4.978 3.647 2.352 6.317 3.727 2.628 1.637 β 2 (×10 1 ) 1.047 0.5636 0.3478 0.2492 0.5322 0.3366 0.2238 0.1481 β 3 (×10 0 ) 1.829 1.122 0.8478 0.5389 1.179 0.6322 0.4834 0.2800 β 4 (×10 1 ) 6.042 2.8461 1.960 1.168 2.583 1.5041 1.164 0.7535 β 5 (×10 −1 ) 3.854 1.887 1.165 0.8020 1.470 0.9291 0.6079 0.4123 β 6 (×10 −2 ) 1.080 0.6229 0.4588 0.2804 0.5689 0.2985 0.2350 0.1355 β 7 (×10 0 ) 10.14 3.593 2.023 1.218 2.900 1.760 1.315 0.8837 β 8 (×10 −2 ) 5.621 2.876 1.853 1.186 2.799 1.738 1.212 0.7297 β 9 (×10 0 ) 6.913 3.365 2.189 1.425 1.397 0.7856 0.5409 0.3487 β 10 (×10 −1 ) 8.368 3.778 2.714 1.706 2.751 1.426 1.174 0.6920

190

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure B.1: Quadratic curve fit results. 25 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure B.2: Quadratic curve fit results with 25 random points sampled. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure B.3: Quadratic curve fit results. Histogram of errors for all 100 runs and 25 random points sampled.

191

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure B.4: Quadratic curve fit results. 25 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure B.5: Quadratic curve fit results with 25 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure B.6: Quadratic curve fit results. Histogram of errors for all 100 runs and 25 D-optimal sampled points.

192

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure B.7: Quadratic curve fit results. 50 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure B.8: Quadratic curve fit results with 50 random points sampled. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure B.9: Quadratic curve fit results. Histogram of errors for all 100 runs and 50 random points sampled.

193

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure B.10: Quadratic curve fit results. 50 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure B.11: Quadratic curve fit results with 50 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure B.12: Quadratic curve fit results. Histogram of errors for all 100 runs and 50 D-optimal sampled points.

194

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure B.13: Quadratic curve fit results. 100 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure B.14: Quadratic curve fit results with 100 random points sampled. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure B.15: Quadratic curve fit results. Histogram of errors for all 100 runs and 100 random points sampled.

195

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure B.16: Quadratic curve fit results. 100 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure B.17: Quadratic curve fit results with 100 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure B.18: Quadratic curve fit results. Histogram of errors for all 100 runs and 100 D-optimal sampled points.

196

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure B.19: Quadratic curve fit results. 200 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure B.20: Quadratic curve fit results with 200 randomly sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure B.21: Quadratic curve fit results. Histogram of errors for all 100 runs and 200 random points sampled.

197

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure B.22: Quadratic curve fit results. 200 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure B.23: Quadratic curve fit results with 300 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure B.24: Quadratic curve fit results. Histogram of errors for all 100 runs and 200 D-optimal sampled points.

198

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Appendix C

Cubic curve fit results with latitude excluded as variable

The first figure in each case is a plot of the actual pCO 2 values, together with the mean values of the estimated pCO 2 from the 100 curve fits. The 95% confidence interval is also shown for the 100 curve fits in each case. The second figure in each case is the actual pCO 2 values plotted against the estimated pCO 2 values. The 10% error lines indicate the number of points that fall outside the 10% error. The third figure in each case is a histogram of the errors on the estimation of the pCO 2 , showing the distribution of the errors for each case.

It can be seen from the figures that as the number of points sampled from the data set increases, the fit of the data is improved and less points falls outside the 10% error interval. Also, from the figures it can be seen that the D-optimal sampling yields a smaller 95% confidence interval than the random sampling. The histograms generally show that the error on the estimation has a normal distribution around zero.

The mean coefficients for the cubic equation for all 100 curve fits are given in Table C.1. The standard deviation for the coefficients are given in Table C.2. It can be seen from the table of the standard deviations that the coefficients for the D-optimal sampling have a smaller standard deviation than the coefficients for the random sampling.

199

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Table C.1: Coefficients of Cubic curve fitting

Sam-pling Random Random Random Random D-optim D-optim D-optim D-optim

No of

Points k 50 55 100 200 50 55 100 200

β 1 (×10 2 ) 2.189 2.543 2.819 2.588 2.060 1.960 2.100 2.086 β 2 (×10 1 ) 7.762 6.153 5.338 5.404 6.931 6.935 6.875 6.888 β 3 (×10 0 ) 2.543 2.448 2.119 2.921 4.855 5.127 4.666 4.786 β 4 (×10 1 ) 4.629 -0.03724 -2.289 -0.3086 -5.991 -3.832 -5.511 -5.695 β 5 (×10 0 ) -4.796 -4.234 -3.800 -3.663 -4.674 -4.698 -4.712 -4.734 β 6 (×10 −2 ) -2.071 -3.014 -2.850 -3.322 -4.748 -5.052 -4.624 -4.833 β 7 (×10 1 ) -1.733 -0.7125 0.01918 -0.1074 3.824 3.058 3.596 3.690 β 8 (×10 −1 ) -5.569 -3.390 -2.838 -3.472 -5.884 -5.762 -5.573 -5.576 β 9 (×10 1 ) -6.989 -5.606 -4.734 -4.493 -3.613 -3.685 -3.625 -3.592 β 10 (×10 0 ) 0.6692 1.445 1.400 0.4969 -0.7155 -1.014 -0.7351 -0.7432 β 11 (×10 −2 ) 8.768 8.877 7.893 7.294 8.843 8.942 9.047 9.079 β 12 (×10 −4 ) 0.5948 1.198 1.155 1.083 1.154 1.273 1.165 1.275 β 13 (×10 0 ) 1.112 0.8320 0.3650 0.1812 -4.456 -3.784 -4.266 -4.385 β 14 (×10 −2 ) 1.187 0.5292 0.5801 0.7319 1.633 1.593 1.557 1.577 β 15 (×10 0 ) 2.288 2.021 1.748 1.614 1.441 1.472 1.491 1.491 β 16 (×10 −1 ) 2.384 1.490 1.082 1.111 0.7274 0.7067 0.6056 0.5758 β 17 (×10 −3 ) 1.593 1.218 0.9052 1.116 1.600 1.568 1.501 1.482 β 18 (×10 −2 ) -0.4556 -0.8714 -0.5361 0.1690 1.921 2.014 1.889 1.896 β 19 (×10 1 ) 1.157 0.8970 0.7568 0.6975 0.4350 0.4562 0.4415 0.4349 β 20 (×10 −1 ) -0.9076 -1.919 -2.389 -1.535 -1.645 -1.103 -1.461 -1.444

200

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Table C.2: Standard deviation of coefficients of Cubic curve fitting

Sam-pling Random Random Random Random D-optim D-optim D-optim D-optim

No of

Points k 50 55 100 200 50 55 100 200

β 1 (×10 2 ) 2.026 1.970 1.258 0.7352 0.3343 0.2695 0.1526 0.1165 β 2 (×10 1 ) 5.815 4.506 2.599 1.565 0.9255 0.7558 0.4868 0.3612 β 3 (×10 0 ) 5.194 5.410 3.534 2.102 0.9504 0.8018 0.4948 0.3729 β 4 (×10 2 ) 2.896 2.601 1.676 1.058 0.5205 0.5594 0.3571 0.2219 β 5 (×10 0 ) 4.023 2.937 1.636 0.9815 0.7418 0.6186 0.3916 0.2834 β 6 (×10 −2 ) 6.003 5.700 3.908 2.245 1.062 0.9600 0.5704 0.4454 β 7 (×10 1 ) 7.693 7.374 4.276 2.534 1.759 1.891 1.206 0.7917 β 8 (×10 −1 ) 5.947 4.988 2.953 1.860 0.9570 0.7550 0.5143 0.3216 β 9 (×10 1 ) 5.499 4.244 2.529 1.424 0.7094 0.5996 0.3883 0.2286 β 10 (×10 0 ) 6.751 6.131 3.650 2.577 0.8925 0.9127 0.6042 0.3617 β 11 (×10 −2 ) 9.457 6.306 3.635 2.135 1.819 1.510 0.9436 0.6633 β 12 (×10 −4 ) 2.544 2.160 1.511 0.8653 0.4722 0.4695 0.2523 0.1980 β 13 (×10 0 ) 6.582 6.001 3.349 1.868 1.600 1.680 1.081 0.7292 β 14 (×10 −2 ) 1.929 1.558 0.8180 0.5672 0.2817 0.2018 0.1516 0.1016 β 15 (×10 0 ) 2.126 1.549 0.8774 0.5081 0.3678 0.2849 0.1914 0.1166 β 16 (×10 −1 ) 3.984 3.353 1.950 1.172 0.5026 0.4157 0.2714 0.1499 β 17 (×10 −3 ) 1.548 1.595 1.080 0.5338 0.2959 0.2565 0.1697 0.09930 β 18 (×10 −2 ) 4.690 4.128 2.304 1.623 0.5319 0.4905 0.3078 0.2041 β 19 (×10 0 ) 9.136 6.852 4.141 2.062 1.166 1.008 0.6320 0.3627 β 20 (×10 −1 ) 9.491 9.688 4.849 3.360 1.230 1.325 0.8681 0.5316

201

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure C.1: Cubic curve fit results. 50 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure C.2: Cubic curve fit results with 50 random points sampled. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure C.3: Cubic curve fit results. Histogram of errors for all 100 runs and 50 random points sampled.

202

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure C.4: Cubic curve fit results. 50 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure C.5: Cubic curve fit results with 50 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure C.6: Cubic curve fit results. Histogram of errors for all 100 runs and 50 D-optimal sampled points.

203

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure C.7: Cubic curve fit results. 55 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure C.8: Cubic curve fit results with 55 random points sampled. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure C.9: Cubic curve fit results. Histogram of errors for all 100 runs and 55 random points sampled.

204

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure C.10: Cubic curve fit results. 55 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure C.11: Cubic curve fit results with 55 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure C.12: Cubic curve fit results. Histogram of errors for all 100 runs and 55 D-optimal sampled points.

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure C.13: Cubic curve fit results. 100 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure C.14: Cubic curve fit results with 100 random points sampled. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure C.15: Cubic curve fit results. Histogram of errors for all 100 runs and 100 random points sampled.

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure C.16: Cubic curve fit results. 100 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure C.17: Cubic curve fit results with 100 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure C.18: Cubic curve fit results. Histogram of errors for all 100 runs and 100 D-optimal sampled points.

207

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure C.19: Cubic curve fit results. 200 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure C.20: Cubic curve fit results with 200 random points sampled. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure C.21: Cubic curve fit results. Histogram of errors for all 100 runs and 200 random points sampled.

208

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure C.22: Cubic curve fit results. 200 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure C.23: Cubic curve fit results with 200 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure C.24: Cubic curve fit results. Histogram of errors for all 100 runs and 200 D-optimal sampled points.

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210

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Appendix D

Fourth order curve fit with latitude excluded as variable

The first figure in each case is a plot of the actual pCO 2 values, together with the mean values of the estimated pCO 2 from the 100 curve fits. The 95% confidence interval is also shown for the 100 curve fits in each case. The second figure in each case is the actual pCO 2 values plotted against the estimated pCO 2 values. The 10% error lines indicate the number of points that fall outside the 10% error. The third figure in each case is a histogram of the errors on the estimation of the pCO 2 , showing the distribution of the errors for each case.

It can be seen from the figures that as the number of points sampled from the data set increases, the fit of the data is improved and less points falls outside the 10% error interval. Also, from the figures it can be seen that the D-optimal sampling yields a smaller 95% confidence interval than the random sampling. The histograms generally show that the error on the estimation has a normal distribution around zero.

The mean coefficients for the fourth order equation for all 100 curve fits are given in Table D.1.

The standard deviation for the coefficients are given in Table D.2. It can be seen from the table of the standard deviations that the coefficients for the D-optimal sampling have a smaller standard deviation than the coefficients for the random sampling.

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Table D.1: Coefficients of Fourth Order curve fitting

Sampling Random Random Random Random D-optim D-optim D-optim D-optim

No of

Points 100 120 200 400 100 120 200 400

β 1 (×10 2 ) -4.116 -4.422 -4.517 -3.835 0.4143 0.5537 0.42111 0.5590 β 2 (×10 2 ) 3.464 3.524 3.417 3.266 1.643 1.629 1.646 1.615 β 3 (×10 1 ) 1.504 1.595 1.754 1.535 1.088 1.041 1.091 1.042 β 4 (×10 2 ) 8.600 9.220 9.528 8.353 0.3807 0.1579 0.4305 0.2182 β 5 (×10 1 ) -2.605 -2.760 -2.690 -2.634 -1.128 -1.123 -1.150 -1.118 β 6 (×10 −1 ) -1.440 -1.498 -1.872 -1.622 -1.366 -1.283 -1.363 -1.304 β 7 (×10 2 ) -3.626 -3.906 -3.984 -3.509 0.3477 0.4680 0.3566 0.4393 β 8 (×10 0 ) -4.874 -4.848 -4.667 -4.318 -3.111 -3.107 -3.105 -3.053 β 9 (×10 2 ) -4.034 -4.151 -4.055 -3.860 -1.499 -1.470 -1.499 -1.464 β 10 (×10 1 ) -0.8107 -1.057 -0.1293 -0.9682 -0.2467 -0.2279 -0.3032 -0.2416 β 11 (×10 −1 ) 6.165 7.281 7.375 7.494 2.481 2.523 2.708 2.544 β 12 (×10 −3 ) 1.020 0.9444 1.178 1.047 0.7732 0.6980 0.7503 0.7315 β 13 (×10 1 ) 7.130 7.324 7.577 6.770 -0.3617 -0.6089 -0.4356 -0.5699 β 14 (×10 −1 ) 2.793 2.771 2.517 2.337 1.377 1.367 1.355 1.342 β 15 (×10 1 ) 2.030 2.203 2.240 2.217 0.8357 0.8225 0.8512 0.8233 β 16 (×10 0 ) 3.972 4.107 3.956 3.613 2.049 2.052 2.058 2.033 β 17 (×10 −2 ) 1.973 1.911 1.972 1.811 1.747 1.757 1.767 1.713 β 18 (×10 −1 ) 0.3071 0.6059 1.052 0.7381 0.6909 0.6998 0.7695 0.7000 β 19 (×10 2 ) 1.164 1.213 1.167 1.107 0.33971 0.3280 0.3366 0.3257 β 20 (×10 0 ) 0.6154 1.605 1.964 1.207 -1.323 -1.388 -1.192 -1.330 β 21 (×10 0 ) -4.164 -4.255 -4.711 -4.294 -0.1492 0.007981 -0.08673 -0.01166 β 22 (×10 −1 ) -2.766 -3.100 -3.023 -2.265 1.096 1.204 1.079 1.170 β 23 (×10 1 ) -1.103 -1.101 -1.041 -0.9772 -0.2176 -0.2029 -0.2137 -0.2014 β 24 (×10 −2 ) 2.684 1.885 1.193 1.353 0.8578 0.8248 0.7069 0.7927 β 25 (×10 −1 ) -4.668 -5.685 -5.010 -4.552 -2.208 -2.225 -2.194 -2.192 β 26 (×10 0 ) -2.762 -3.129 -3.445 -3.456 -1.451 -1.429 -1.450 -1.411 β 27 (×10 −4 ) -5.550 -6.740 -8.138 -6.686 -5.774 -5.851 -5.919 -5.698 β 28 (×10 −3 ) -6.446 -5.431 -7.319 -6.573 -6.957 -7.058 -7.299 -7.131

Continued on next page

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Table D.1 – continued from previous page

Sampling Random Random Random Random D-optim D-optim D-optim D-optim

No of

Points k 100 120 200 400 100 120 200 400

β 29 (×10 −1 ) -1.374 -1.469 -1.249 -1.143 -0.4154 -0.4125 -0.4019 -0.3986 β 30 (×10 −1 ) -1.892 -2.399 -3.053 -3.260 -1.437 -1.426 -1.540 -1.464 β 31 (×10 −6 ) -2.092 -1.365 -1.988 -1.884 -1.295 -1.049 -1.226 -1.238 β 32 (×10 −5 ) -1.821 -2.184 -1.798 -1.469 -0.5274 -0.5221 -0.4857 -0.3468 β 33 (×10 −4 ) -7.476 -6.936 -7.117 -6.758 -7.127 -7.152 -7.179 -7.095 β 34 (×10 −3 ) -3.518 -3.536 -2.882 -2.625 -0.3155 -0.2856 -0.2517 -0.2407 β 35 (×10 −3 ) -4.189 -6.559 -7.118 -7.605 -2.470 -2.637 -3.033 -2.732

Table D.2: Standard deviation for coefficients of fourth order curve fitting

Sampling Random Random Random Random D-optim D-optim D-optim D-optim

No of

Points k 100 120 200 400 100 120 200 400

β 1 (×10 2 ) 5.235 4.651 2.712 1.574 0.8560 0.7495 0.5609 0.3497 β 2 (×10 2 ) 2.097 1.641 0.8740 0.5178 0.3641 0.2873 0.2474 0.1468 β 3 (×10 1 ) 1.482 1.488 0.9499 0.5141 0.3173 0.3021 0.2172 0.1720 β 4 (×10 2 ) 9.035 7.952 4.815 2.883 1.405 1.255 0.9287 0.6137 β 5 (×10 1 ) 1.967 1.609 0.7950 0.5039 0.4120 0.3363 0.2955 0.1841 β 6 (×10 −1 ) 2.326 2.246 1.509 0.7513 0.5589 0.5129 0.3710 0.2950 β 7 (×10 2 ) 4.111 3.228 1.805 1.128 0.6252 0.5259 0.4045 0.2575 β 8 (×10 0 ) 3.903 2.951 1.542 0.9661 0.4172 0.3596 0.2781 0.1649 β 9 (×10 2 ) 2.594 2.051 1.137 0.6518 0.3972 0.2932 0.2615 0.1567 β 10 (×10 1 ) 2.615 2.511 1.548 0.9467 0.4839 0.4473 0.3028 0.2382 β 11 (×10 −1 ) 9.104 7.216 3.543 2.474 1.951 1.606 1.435 0.9162 β 12 (×10 −3 ) 1.877 1.620 1.029 0.5262 0.4076 0.3761 0.2595 0.2071 β 13 (×10 1 ) 7.542 5.205 2.726 1.779 1.123 0.8980 0.7066 0.4331 β 14 (×10 −1 ) 2.198 1.684 0.8996 0.5180 0.3029 0.2548 0.1922 0.1139

Continued on next page

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Table D.2 – continued from previous page

Sampling Random Random Random Random D-optim D-optim D-optim D-optim

No of

Points k 100 120 200 400 100 120 200 400

β 15 (×10 1 ) 1.807 1.458 0.6970 0.4493 0.2461 0.1832 0.1593 0.1072 β 16 (×10 0 ) 3.953 3.240 1.687 1.012 0.44581 0.3731 0.3478 0.2006 β 17 (×10 −2 ) 2.143 1.498 0.8535 0.5114 0.2493 0.2306 0.1591 0.1200 β 18 (×10 −1 ) 3.203 3.162 1.971 1.146 0.6633 0.5770 0.3876 0.3001 β 19 (×10 1 ) 8.108 6.313 3.335 1.896 1.230 0.8788 0.7779 0.4765 β 20 (×10 0 ) 8.364 7.079 3.872 2.544 1.202 1.094 0.7518 0.5898 β 21 (×10 0 ) 5.349 3.590 1.658 1.156 0.8114 0.6238 0.5025 0.3008 β 22 (×10 −1 ) 6.996 5.149 2.809 1.933 0.8630 0.7332 0.5427 0.3934 β 23 (×10 0 ) 7.317 5.743 2.983 1.686 1.009 0.7334 0.6006 0.3934 β 24 (×10 −2 ) 5.715 5.154 2.823 1.629 0.7883 0.7310 0.4585 0.3790 β 25 (×10 −1 ) 7.709 5.656 2.554 1.637 0.8513 0.6893 0.6413 0.3774 β 26 (×10 0 ) 3.947 3.135 1.286 0.8207 0.4243 0.3013 0.2740 0.1766 β 27 (×10 −3 ) 1.390 1.372 0.8153 0.4913 0.3081 0.2432 0.1753 0.1286 β 28 (×10 −3 ) 15.56 13.82 6.487 3.915 1.510 1.331 1.181 0.6622 β 29 (×10 −1 ) 1.134 0.8720 0.4762 0.2666 0.15732 0.1220 0.09276 0.06428 β 30 (×10 −1 ) 5.313 3.953 1.873 1.318 0.5978 0.4533 0.3851 0.2792 β 31 (×10 −6 ) 6.183 4.653 2.641 1.559 1.347 1.243 0.8111 0.6296 β 32 (×10 −5 ) 5.304 3.986 2.375 1.193 0.8086 0.8057 0.5696 0.4289 β 33 (×10 −4 ) 6.914 4.882 2.526 1.695 0.6464 0.6391 0.4631 0.2528 β 34 (×10 −3 ) 3.974 3.349 1.703 0.8663 0.6886 0.5659 0.4152 0.2607 β 35 (×10 −2 ) 1.638 1.210 0.6569 0.4604 0.3431 0.2774 0.2504 0.1608

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure D.1: Fourth order curve fit results. 100 Random points sampled.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure D.2: Fourth order curve fit results with 100 random points sampled. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure D.3: Fourth order curve fit results. Histogram of errors for all 100 runs and 100 random points sampled.

215

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0 1000 2000 3000 4000 5000 6000 200

250 300 350 400 450

Observation number pCO2 (microatmosphere)

Model µpCO2 Model µpCO

2±1.96σ Actual pCO

2

Figure D.4: Fourth order curve fit results. 100 D-optimal sampled points.

250 300 350 400 450

250 300 350 400 450

Actual pCO2 values Model pCO2 values

Figure D.5: Fourth order curve fit results with 100 D-optimal sampled points. The degree to which the modelled points fall in the 10 % error margin is shown. The mean pCO

2

estimation for all 100 runs is shown here.

−2000 −150 −100 −50 0 50 100 150 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 105

Actual pCO 2 − Model pCO

2

No of observations

Figure D.6: Fourth order curve fit results. Histogram of errors for all 100 runs and 100 D-optimal sampled points.

216

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