Appendices
177
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.
179
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
180
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
2estimation 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
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
2estimation 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
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
2estimation 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
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
2estimation 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
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
2estimation 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
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
2estimation 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
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
2estimation 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
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
2estimation 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.
188
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
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
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
2estimation 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
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
2estimation 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
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
2estimation 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
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
2estimation 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
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
2estimation 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
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
2estimation 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
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
2estimation 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
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
2estimation 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
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
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
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
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
2estimation 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
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
2estimation 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
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
2estimation 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
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
2estimation 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.
205
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
2estimation 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.
206
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
2estimation 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
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
2estimation 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
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
2estimation 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.
209
210
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.
211
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
212
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
213
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
214
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
2estimation 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
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
2estimation 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