o LU CD C C 10 3 0 Data window (days) (e) Figure 5.2e Chapter 5. Determining Model Initial Conditions 84 1 _ 0 . 8 1 0.6 0.4 0.2 0 •0 .2 11* 3 w a1&2 0 w a1 0 w a 2 • h a1&2 • h a1 S h a2 1 1 J I 1 0 21 3 0 Figure 5.2f Data window (days) (f) I 9 0 Chapter 5. Determining Model Initial Conditions 85 temporally denser wind data are required to achieve an overall retrieval quality for the three oceanic initial conditions (h0, u0, and v 0) similar to that obtained by assimilating SLH data. Of the three initial condition, /i0was less sensitive to the temporal sparsity of the wind data and the SLH data than u0 and v0. Without prior information about the first-guesses, temporal resolution of 10 days for the wind or 30 days for the SLH are required for a good retrieval of the oceanic initial conditions in our system. Next, experiments were conducted to examine the effects of combined temporal and spatial sparsity of the wind and the SLH data on retrieving the initial ocean conditions. For these experiments, wind and SLH data were sampled from area 1 with the TAO array spatial resolution of 2° latitude by 15° longitude and once every 1, 10, 30 or 90 days. Zero first guesses were again used. The wind data in area 1 were found to be relatively more important than the SLH data in area 1 in determining h0 (Fig.5.3a and b), i.e., without the wind data in area 2, the quality of the h0 retrieval was about the same as that in the corresponding experiments (Fig.5.2a and b) with the wind data in area 2; the growth of the RMSE and the decrease of the correlation were minor, and no substantial degradation of the retrieval occurred in area 2. However, without the SLH data in area 2, the RMSE degradation in the h0 retrieval was remarkable (by 1-2 orders of magnitude larger, with degradation in both area 1 and area 2). Though the RMSEs were comparable between the experiments assimilating the wind data in area 1 and the SLH data in area 1, the correlation for the experiments assimilating the SLH data showed more apparent degradation, caused mainly by the degradation of the h0 retrieval in area 2. Without the wind or the SLH data in area 2 the information provided was generally insufficient for retrieving the current fields. Compared with the experiments assimilating the wind everywhere in space, the RMSE of the w0 retrieval increased by about one order of magnitude (cf. Fig.5.2c and Fig.5.3c), and the correlation was lower than 0.5 when the window was longer than one day (cf. Fig.5.2d and Fig.5.3d). However, the low overall correlation was caused by the low correlation in area 2, as the correlation for area 1 was Chapter 5. Determining Model Initial Conditions 86 1 0 3 0 data window (days) (a) Figure 5.3 Same as Fig. 5.2 except for assimilating data at every 2nd grid in the meridional direction and every 15th grid point in the zonal direction in area 1. Chapter 5. Determining Model Initial Conditions 8 7 Figure 5.3b (b) Chapter 5. Determining Model Initial Conditions 88 1 0 u O .9 LU 1 0 2 ' CO io"° u 1 0" i i l l S w a1&2 0 w a1 Q w a 2 • h a1&2 • h a1 S h a2 10 3 0 Data window (days) 9 0 (c) Figure 5.3c Chapter 5. Determining Model Initial Conditions 89 o 0-8 O = 0.6 CO .2 0.4 _co CD 5 0.2 0 -0 .2 • w a1&2 • h a1&2 • w a1 h a1 • w a2 m h a2 10 30 Data window (days) (d) 90 Figure 5.3d Chapter 5. Determining Model Initial Conditions 90 ^1 • S w a1&2 O w a1 Q w a 2 • h a1&2 • h a1 S h a2 10 3 0 Data window (days) (e) 9 0 Figure 5.3e Chapter 5. Determining Model Initial Conditions 91 0.8 o = 0.6 CO .1 0.4 05 CD 8 0.2 0 - 0 . 2 I 1 B w a1&2 • h a1&2 w a1 h a1 • w a2 m h a2 2 1 I 1 0 3 0 9 0 Figure 5.3f Data window (days) (f) Chapter 5. Determining Model Initial Conditions 92 insensitive to the increasing wind sparsity and was acceptable (about 0.6). Compared with the experiments assimilating the SLH data everywhere in space, the RMSEs for assimilating the SLH once 1 or 10 days were about two orders of magnitude larger, while the correlations were very low for all the experiments except for having the SLH data once a day. This indicated that without some information from area 2, the available information was insufficient for retrieving the zonal current field. Without wind data or SLH data in area 2, the information insufficiency was even more remarkable for v0 retrieval (Fig.5.3e and f). The retrieval failed because the correlations were generally lower than 0.4 except for the experiment where the SLH data were available once a day. We concluded that the initial sea surface height h0 of our model was more easily estimated and less sensitive to the assimilated data type (wind and SLH), and the temporal and spatial sparsity of data, than the initial currents (u0 and v0). Wind data in area 1 seemed more helpful than the SLH data in determining the oceanic initial conditions. Observations (wind or SLH) in area 1 was enough for estimating h0, but not for the current fields, especially in area 2. 5.2.2 Impact of initial guess on the initialization ENSO is generally believed to be a quasi-periodic phenomenon with the time scale set by the transit times of equatorial Kelvin or Rossby waves. Its predictability depends on our being able to identify where the coupled system is on the ENSO cycle. An ocean spun-up by wind stress may contain errors in the intensity and phase of the cycle as represented by the oceanic anomaly. Using such a spun-up ocean as the first-guess for the adjoint data assimilation could affect the convergence of the minimization procedure. To study this, we conducted four experiments where the first-guesses were from a spun-up ocean with defects: (a) The ocean has the correct phase but incorrect intensity. For generating the first guesses, the magnitude of the initial h in (3.11) was first systematically scaled up by 10% from the control. The first-guess for case 1 was then generated in the same way as the true initial conditions but by using the perturbed h. For case 2, 10% random noise was added to the "true" initial conditions from the control run. Chapter 5. Determining Model Initial Conditions 93 M A X / M I N 0 . 4 5 / - 0 . 7 2 C M 100W 50 0 50 1 0 0 E (a) Figure 5.4 Difference in the initial (a) h0, (b) u0 and (c) v0 perturbations between case 3 and the control run. Chapter 5. Determining Model Initial Conditions 94 M A X / M I N 2.9/-3.7 C M / S (b) Figure 5.4b Chapter 5. Determining Model Initial Conditions 95 M A X / M I N 3 . 2 / - 3 . 2 C M / S 100W 50 0 50 1 00E (c) Figure 5.4c Chapter 5. Determining Model Initial Conditions 96 (b) The ocean has both the phase and the intensity incorrect: The first-guesses for case 3 and case 4 were generated by relaxing the perturbed h located initially to the left (centered at grid 50 along the equator), and to the right (grid 150 along the equator), respectively, of the location in the control run (grid 100 along the equator). Fig.5.4 shows the difference of case 3 from the true initial conditions. Instead of describing a warm event developing in the central equatorial Pacific, case 3 described a warm event developing in the western equatorial Pacific, together with a strong westward current, while case 4 described a warm event which had propagated to the eastern equatorial Pacific accompanied by a strong eastward current (not shown). To compare with the results from previous studies using zero initial guess, experiments in this section were also conducted with data available once every 1, 10, 30, and 90 days only in area 1 with TAO array spatial resolution. With either the wind data or the SLH data, the retrievals of the three initial states from case 1 were very good, showing little sensitivity to the increasing temporal sparsity. Fig.5.5 showed the RMSE and the correlation for the retrievals with the wind data or the SLH data available only at day 90. Though the data were far fewer than the unknowns for these two experiments, the RMSEs were very small and the correlations were very high, especially for the current fields, e.g., with only the wind data, the RMSE was of the order of 10"4for u0 and 10-5 for v0, much smaller than those found earlier by zero initial guess (cf. Fig.5.5a, Fig.5.3c and Fig.5.3e). The correlation for the current fields were also improved in this case ( Fig.5.5b, Fig.5.3d and Fig.5.3f). The quality of the retrievals for area 2 was also very good (not shown). Case 2 was found to be more difficult to retrieve than case 1 as the RMSE and the correlation skill in retrieving the initial conditions from noisy initial guess deteriorated with increasing temporal sparsity of the wind or the SLH. However, even with data only at day 90, the retrievals were still better than those obtained by zero initial guess, especially for the current field retrieval (cf. Fig.5.5 and Fig.5.3). With data only in area 1, retrievals for area 2 were as good as in area 1 (not shown). Since our model is linear, the first guesses of the in-phase case should have the same spatial distributions as the true initial conditions. This might explain why Chapter. 5. Determining Model Initial Conditions 97 1 0 u 1 0" io" y LU 1 0" 1 0" 1 0" 1 0" w i n d w i n d I SLH I S L H & wind SLH • h LTJ u V e a s e l c a s e 2 e a s e l c a s e 2 c a s e 3 c a s e 4 (a) Figure 5.5 (a) RMSE and (b) correlation from retrieving three initial conditions from case 1 and case 2 by assimilating wind or SLH data at day 90 in area 1, and from case 3 and case 4 by assimilating once every day both wind and SLH data in area 1. Chapter 5. Determining Model Initial Conditions 98 Figure 5.5b (b) Chapter 5. Determining Model Initial Conditions 99 the retrievals for case 1 needed few data and the information propagated readily from the later time to the starting time to determine the initial conditions. More information might be required to establish the phase of the initial ocean because phase restructuring implies energy redistribution. As the noise in the initial guess is harmful to retrieving the initial conditions, the noise level could be an important factor in the retrievals. The results from the two out-of-phase ( i.e. anomaly in wrong location) cases supported the above arguments. Having only the wind data or the SLH data in area 1 was generally insufficient for retrieving the initial conditions in both case 3 and case 4. With both wind and SLH data, the retrievals were still very poor. From Fig.5.5 where both wind and SLH data were assimilated once every day, we observed the same characteristics of information insufficiency found in the experiments with a zero first guess or with very sparse data, i.e., in retrieving the h0, the RMSEs were large and the correlations were high, while the retrievals for current fields were poor. Compared with Fig.5.3, the retrievals for case 3 and case 4 were poorer than those assimilating wind or SLH data with zero initial guess. Fig.5.6 depicts the difference between the retrieved initial current from case 3 and the true initial currents after assimilating both wind and SLH data in area 1 once every day. The initial strong eastward current in Fig.5.4 has been substantially reduced after data assimilation (cf. Fig.5.6b and 5.4b), while the meridional current structure has not (Fig.5.6c and 5.4c). Hence the efficiency of the adjoint method and the information sufficiency depend on the original guess, especially, the phase (i.e. location) of the warm water anomaly. With only wind and SLH data in the equatorial wave guide area, observations once per day will probably be insufficient for estimating the true ocean if the initial first guess for the ocean contains a large phase discrepancy to the true anomaly. 5.2.3 Impact of noisy data on initialization Noise in the data is detrimental to a successful retrieval of initial conditions, as it tends to Chapter 5. Determining Model Initial Conditions 100 M A X / M I N 0 . 3 8 / - 0 . 2 4 C M 100W 50 0 50 1 0 0 E (a) Figure 5.6 (a) RMSE and (b) correlation from retrieving three initial conditions (h0, u0 and v0) for a 90-day assimilation with noisy wind or both noisy wind and SLH data available once every 1 or 10 days at every 2nd grid in the zonal direction and every 15th grid in the meridional direction in area 1. Chapter 5. Determining Model Initial Conditions 101 MAX/MIN 1.4/-2.6 CM/S 100W 50 0 50 1 0 0 E (b) Figure 5.6b Chapter 5. Determining Model Initial Conditions 102 M A X / M I N 3 . 1 / - 3 . 1 100W 50 0 50 1 0 0 E (c) Figure 5.6c Chapter 5. Determining Model Initial Conditions 103 cause ill-conditioning of the problem (Thacker and Long, 1988). We conducted experiments to test the effect of assimilating noisy wind and SLH data in area 1. Normally distributed random noise with zero mean and standard deviation r (which was 10% of the maximum value of each state variable at day 90 of the control run) was added at every time step. The noisy initial conditions in the last section (case 2) were used as first guess. The retrievals of the initial conditions were less sensitive to the noise in wind data than noise in the SLH data. For example, with noisy wind data, the RMSE had the same order of magnitude as that estimated by assimilating perfect data, and showed little degradation to the increasing wind sparsity (Fig.5.7). However, the noise in the SLH was very detrimental to the retrievals of the initial conditions (not shown). Increasing the amount of SLH data (e.g. by decreasing the data window from 10 days to 1 day) actually brought more harm to the retrievals. By assimilating both wind and SLH data (Fig.5.7), the wind did not improve the retrievals, and the results were much poorer than those from assimilating wind data only. The RMSEs from using data once every 10 days was ironically smaller those from using data once every day. The correlation was also very low for the SLH and the zonal current. Hence noisy SLH data were most harmful to the retrievals of the ocean initial conditions. 5.3 Conclusion In this chapter, the sensitivity in retrieving the three initial conditions (SLH, two current components) to data type (wind and SLH), data sparsity, first guess of the initial conditions, and data noise were investigated. Usefulness of data type was tested by retrieving the initial conditions with first guess taken to be zero. Wind data assimilation was found to be useful in retrieving the initial oceanic conditions, with the overall retrieval quality sensitive to the temporal sparsity of the wind data. Information about the SLH seemed to be more effective than the wind in retrieving the three initial fields. Among the three, the retrieval of the initial SLH was least sensitive to data type and Chapter 5. Determining Model Initial Conditions 104 LU Data window (days) (a) Figure 5.7 (a) RMSE and (b) correlation from retrieving three initial conditions (h0, w0 and v 0) for a 90-day assimilation with noisy wind or both noisy wind and SLH data available once every 1 or 10 days at every 2nd grid in the zonal direction and every 15th grid in the meridional direction in area 1. Chapter 5. Determining Model Initial Conditions 105 Figure 5.7b Chapter 5. Determining Model Initial Conditions 106 to the temporal sparsity of data assimilated, while the retrieval of the current fields was sensitive to the temporal sparsity of the data assimilated. A temporal resolution of 10 days for the wind data was found necessary for achieving an acceptable retrieval quality. Since the current fields were determined by both wind and the SLH (see 3.13), the better results with the SLH data in retrieving the current fields might be attributed to the fact that (1) the wind was not a state variable of the ocean model, and (2) the oceanic motion had a slower time scale and therefore a longer memory to retain the information obtained from the SLH data, while the faster atmospheric motion has a shorter memory to carry the information obtained from the wind data. More information to retrieve the current fields was needed as the structure of the currents was more complicated, with smaller spatial scales than the SLH, especially for the regions outside the equatorial wave guide area and the regions along the boundary. Wind and SLH in the equatorial wave guide area were used to test the sensitivity of retrieving the initial oceanic fields to the combined (temporal and spatial) sparsity of data, and the sufficiency of the TAO array observing system in retrieving the initial conditions of simple models. Assuming no prior information was available for the ocean (i.e. a zero initial guess for the oceanic fields), wind and SLH data were found to be sufficient for retrieving the initial SLH, but insufficient for retrieving the current fields outside the equatorial wave guide area. The current fields in the equatorial wave guide area could not be retrieved well if the temporal resolution of the observations was longer than one day. The first guess in adjoint data assimilation plays the important role of providing initial background information on the ocean — a good first guess can provide useful information for retrieving the initial conditions, while a poor guess can be detrimental. The data sufficiency for retrieving the initial fields is therefore associated with the quality of the first guess. Two points can be made concerning the first guess effect: (1) Systematic bias in the magnitude of the initial conditions used for the first guess was easily overcome, without the need of a large amount of data as temporally very sparse wind or SLH data in the equatorial wave guide area were enough to retrieve the initial conditions well for the whole model domain. (2) Noisy perturbations in the Chapter 5. Determining Model Initial Conditions 107 magnitude of the initial conditions used for the first guess, however, were more difficult to handle. The retrieval of the initial conditions was found to be sensitive to the data sparsity for this case, which implied data insufficiency. For these two cases, the overall quality in retrieving the initial conditions was still better than those achieved from first guesses of zero, suggesting that initial ocean fields with error in the magnitude were still useful for initializing simple models. A possible explanation is that for a linear system, perturbing the magnitude of the initial guesses does not change the spatial structure of the true initial conditions, so that the model output and the control run output have the same spatial structure as well. The similarity in the spatial structure of the data (taken from the control run output) and of the model output does not lead to the tremendous demand for information, which arose in case 3 and case 4, where the first guesses had large spatial phase difference from the true initial conditions. The uncertainty of the phase differences between the first guess and the true initial condition could not be resolved easily by assimilating data in the equatorial wave guide area, as both wind and SLH data in the equatorial wave guide area with a temporal interval of one day were found to be insufficient. With current observations available from the TAO system, one can expect improvements in the poor current field retrieval. While SLH data was superior to wind data in retrieving the oceanic initial conditions, it was ironic that noise in the SLH data was far more detrimental to the retrieval than noise in the wind data. The reason is that when there is noise in the wind data, the ocean does not adjust quickly enough to the erroneous wind data, and with the atmosphere mainly a slave to the ocean at low frequencies, the wind soon adjusts to the ocean state — hence little damage is done by the noise in the original wind data. In contrast, when there is noise in the ocean data, the slave atmosphere adjusts rapidly to the erroneous ocean data, hence oceanic noise can cause much more damage to the coupled system. Chapter 6. Summary and Discussions 108 Chapter 6 Summary and Discussions This thesis was motivated by the need to develop efficient data assimilation techniques capable of validating and initializing simple equatorial coupled models with tropical atmospheric and oceanic data. A general procedure, assimilating tropical atmospheric and oceanic data into atmosphere-ocean coupled systems to determine their parameters and initial conditions, was developped. The feasibility and potential of applying this procedure to simple equatorial coupled models were studied with the Philander et al. (1984) simple equatorial coupled atmosphere-ocean model, which has the atmosphere and the ocean each represented by a single-layer linear shallow water model. A series of identical twin experiments was conducted to retrieve the six model parameters and the three initial oceanic conditions from data designed to resemble the available tropical data, which are typically asynchronous, asynoptic, sparse, noisy, and of mixed types. The effects of different data distributions and noise on the retrieval quality of the six parameters and the effects of different data distributions and noise, and the initial guesses of control variables on the retrieval quality of the three initial oceanic conditions were investigated. The retrieval quality was quantified by the relative estimation error between the estimated parameters and the true parameters, the root mean square error (RMSE), and the correlation between the retrieved initial conditions and the true initial conditions. My main purpose was to identify the usefulness of the wind and the SLH data in retrieving the six parameters and the three initial oceanic conditions, and to test the efficiency of information transfer between the atmosphere and the ocean. With both wind and the SLH data available everywhere in space and time, the six model parameters were easily retrieved by the adjoint method, though the six guessed parameters were a factor of five larger than their true values, with the corresponding model output describing a Chapter 6. Summary and Discussions 109 much stronger warm event than the control run. Winds were found to be very important in parameter retrieval as the relative estimation errors of all parameters tended to grow when the time interval between available wind data was increased. The temporal density of SLH observations had less effect on the estimates than that of the wind data. Winds were able to retrieve the initial oceanic conditions but the overall retrieval quality for the initial conditions was sensitive to the temporal sparsity of the wind data. The temporal density of SLH observations again had less effect on the retrieval of the initial conditions than that of the wind data. Since the atmosphere and its adjustment to the external forcing vary much faster than the ocean, the information required for determining the coupled system must therefore be more demanding for the atmospheric part than for the oceanic part. The information transfer within the coupled system seemed to work well only when long assimilation periods were used. Spatial sparsity was found to be less influential than temporal sparsity to both parameter estimation and initialization. Spatially sparse winds (TAO array resolution in area 1 and very sparse resolution in area 2) can cause degradation of parameter estimations, while the same spatially sparse SLH had little effect on parameter estimations. SLH observations at a few points in area 1 were found to be enough to retrieve the six parameters if temporally dense wind data were provided. Wind and SLH in the equatorial wave guide area were found to be sufficient for retrieving the initial SLH, but not enough for retrieving the current fields outside the equatorial wave guide area. The current fields in the equatorial wave guide area could not be retrieved well if the temporal resolution of the observations was longer than one day. More data were required to retrieve the six parameters than to determine the three initial oceanic fields, despite the fact that the number of parameters was far fewer than the number of unknown initial conditions. Though the data amount required for both problems was comparable, the estimated error growth for the six parameters was as sensitive as those for initializing the three oceanic fields, to the increasing data sparsity in time and space. The number of minimization iterations for retrieving the six parameters was far fewer than that for retrieving the three initial conditions. Chapter 6. Summary and Discussions 110 Fitting model to spatially sparse data allows the small scale structures of the atmospheric and oceanic motions to be ignored. This is why the SLH data were found to be almost redundant in the parameter estimations. However, if data were too sparse, two problems could arise: (1) the cost and the gradient of the problem are so small that the minimization cannot be carried out, and (2) the cost and the gradient can be reduced to very small values, however the retrieval quality of the problem can be low. Our studies showed that the coupling parameters a and y and the atmospheric Rayleigh damping coefficient A were well determined in most situations, while the oceanic damping coefficients a and b and the atmospheric Newtonian cooling coefficient B were more sensitive to the wind and the SLH sparsity. The relatively good estimation for a and y (compared to the damping parameters) might be attributed to the combined contributions of information from both parts of the coupled system, or to a more direct constraint by the two parts of the coupled system. Noisy data greatly influenced parameter estimation. The degrading effect on the estimates was smaller for having noise in the SLH data than having noise in the wind or in both the wind and SLH. The coefficients a and A were relatively unaffected by both atmospheric and oceanic noise, while coefficients B and a were most sensitive to combined noise. A longer assimilation period (window) was found to improve the estimation performance. However, it was found that the most preferable window for the wind data was not the most preferable for the SLH data, which suggested the need for a careful selection of the most appropriate window for a coupled system. A priori information for individual parameters as implemented in the cost function was useful in providing information for the size of the parameters and enhancing the convexity of the cost function, but not as a substitute for inadequate data. Among the three oceanic initial conditions, the retrieval of the initial SLH was least sensitive to data type and to the temporal sparsity of the assimilated data, while the retrieval of the current fields was sensitive to the temporal sparsity of the data assimilated. A temporal resolution of 10 days for the wind data was found necessary for achieving an acceptable retrieval quality. Chapter 6. Summary and Discussions 111 Since the current fields were determined by both the wind and the SLH (see 3.13), the better results with the SLH data in retrieving the current fields might be attributed to the fact that (1) the wind was not a state variable of the ocean model, and (2) the oceanic motion had a slower time scale and therefore a longer memory to retain the information obtained from the SLH data, while the faster atmospheric motion had a shorter memory to carry the information obtained from the wind data. The current fields needed more information to retrieve as the structure of the currents was more complicated, with smaller spatial scales than the SLH, especially for the regions outside the equatorial wave guide area and the regions along the boundary. The first guess in adjoint data assimilation plays the important role of providing initial background information on the ocean — a good first guess can provide useful information for retrieving the initial conditions, while a poor guess can be detrimental. The data sufficiency for retrieving the initial fields is therefore associated with the quality of the first guess. Two points can be made concerning the first guess effect: (1) Systematic bias in the magnitude of the initial conditions used for the first guess was easily overcome, without the need of a large amount of data as temporally very sparse wind or SLH data in the equatorial wave guide area were enough to retrieve the initial conditions well for the whole model domain. (2) Noisy perturbations in the magnitude of the initial conditions used for the first guess, however, were more difficult to handle. The retrieval of the initial conditions was found to be sensitive to the data sparsity for this case, which implied data insufficiency. For these two cases, the overall quality in retrieving the initial conditions was still better than those achieved from first zero guesses, suggesting that initial ocean fields with error in the magnitude were still useful for initializing simple models. A possible explanation is that for a linear system, perturbing the magnitude of the initial guesses does not change the spatial structure of the true initial conditions, so that the model output and the control run output have the same spatial structure as well. The similarity in the spatial structure of the data (taken from the control run output) and of the model output does not lead to the tremendous demand for information, which arose in case 3 and case 4, where the first guesses had large spatial phase difference from the true initial conditions. The uncertainty of the phase Chapter 6. Summary and Discussions 112 difference between the first guess and the true initial condition could not be resolved easily by assimilating data in the equatorial wave guide area, as both wind and SLH data in the equatorial wave guide area with a temporal interval of one day were found to be insufficient. With current observations available from the TAO system, one can expect improvements in the poor current field retrieval. While SLH data was superior to wind data in retrieving the oceanic initial conditions, it was ironic that noise in the SLH data was far more detrimental to the retrieval than noise in the wind data. The reason is that when there is noise in the wind data, the ocean does not adjust quickly enough to the erroneous wind data, and with the atmosphere mainly a slave to the ocean at low frequencies, the wind soon adjusts to the ocean state ~ hence little damage is done by the noise in the original wind data. 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The Lagrangian (3.4) for model (3.9-3.10) is given by: , » dua ga dha L = J + ]{ua(— fva + ——Aua) + E dt rcostp dA t b\a , , , ga dha + Va(— + fua + ~ — + Ava) + dt r dtp , ,*rdha , da .dua , C?(vaCOS0) + /?A(-^- + 7(-TT + - A —TT— L ) + Bha + aK) dt rcos» ovl o'cp „ <7W0 go dh , „, , (7? rcostp dA *,dv„ , g0 dh0 , + v0(—-rfu0 + ^ — + aVo-yva) (A-l) otf r o>> ,*.dh0 do ,du0 (9(v o cos0l , , , , dt rcostp dA dtp where ua,va,ha,u0,v0,andh0 are the Lagrange multipliers (or adjoint variables) for the corresponding atmospheric and the oceanic model state variables, respectively. J is defined by (3.2). By integrating (A-l) by parts, L becomes Appendix A . Derivation of the Continuous Adjoint Coupled Model -26 £ = J + j {[(—T —Ua)-fvaUa+ ( — — /?«) + AuaUa] + i dt dt rcos> dX dX r,dv*aVa dvl , , * , ga ,dvlcos(j)ha dvl cos f . * [(—^ —Va)-fuaVa+ ( T - ^ 7"— k ) + AVaVa ] + or dt rcosc? d(p df r.dKha dK, , da ,duaK dK . , 4 dK cos fva da dhl , ^ T~) + I ^ T i TTM«) + ( 1 T T V « ) dt dt rcosc? oW, <7/i rcosc? (70 r tfc? +Bhafh + ahak,] + T / V ^ u ^ u ^r*u * „ * g0 .dh0u0 du0 , s. , * * n . [( r — U0 ) - fVoUo + (—77 — h0 ) + aUoUo - JUaUo ] + rcosc? dA dA T,„r0,o go ,dVo COS fho (^ Vo COS (f) . * , [(—r r - V p ) - / M 0 V 0 + ° ( — — / l 0 ) + f l V 0 V 0 - y V f l V 0 ] + rcos> df df (A-2) r . J 0 .duoK dhl d0 dhlcosfVo d0 dK rcosc/) C M , o'/l rcosc? df r df +bh0ho]}dadt. Reordering (A-2) by bringing the common terms together gives ^ = J + J uc—;— + —^— + —^—) +c—-— + — ^ — + — ^ — ) J + z dt dt dt dt dt dt ga dhqUq + dvl cos fha ^ + da duaK + dK cos 0V f l ^ rcos> t9A rcosc? dX df go ,dh0Uo dvl cos f'ho. d0 ,du0K dhl cos c?v<, - ( - 7 - 7 - + — ) + r ( — 7 7 — + — )J du0u0 dul dt dt dvlv0 dvl dt dt dhlho dK dt dt rcos>v dX df rcosc? v dX df [ — + jVa - - 7 7 - + AU„ - yU0\Ua + dt rcosc? dX [ dVa j* * da dha A * *, 7 fUa 7 — + Ava ~ JV0 JVa dt r df , r e?/k g« dul dvl cos f * + [ 7 e — ( — + r — J 1 ) + BK ]K + dt rcoscp dX df dul * J 0 <97?o . [—z— + fVo --ZTT-+ au0]Uo + dt rcosc? dX r _ ^ _ / M ; _ ^ + A V : ] V O (A-3) r df i r c?A0* g0 ,C9M^ dvlcosf . , +[ 2 — ( 1 —Z1) + bh0+ aha ]h0 }dodt. dt rcosc/) <9A <70 By specifying w*,v*,/i*,M*,v*,and/i* to be zero along the boundaries, those spatial Appendix A . Derivation of the Continuous Adjoint Coupled Model 127 integration terms in row 2 and row 3 of (A-3) vanish. And if the Lagrange multipliers in the first row of (A-2) are defined to be equal to zero at time T, (A-3) can be rewritten as: L = J - [uaua + v*ava + h*aha + U0U0 + V*0V0 + fhho]t=0 ! rr dua - * da dK . * *, j{[—^- + pa 7 ^ T + A u « ~ + £ dt rcosfp dA [ dva r * da dha . A * .. * i JUa — + AVa ~ JVo \Va dt r dtp ,r 9K ga ,du*a dv*aCOS(p , +[ ( — + r——) + Bha ]ha + dt rcostp dA d(p r du0 r * do dh0 *., [—— + fv0 --zrr + au0]Uo + dt rcos> dA dvl , * d0 dh*0 * [—z--fu0 — + av0 ]v0 (A-4) dt r d dA l~~ K - — ^ + AVa - YvWVa + {8AVa ~ c5yV> f l + dt r dtp r dK ga ,dua , dVaCOSfy D , * n o , , X j n , * U _ I_ [ ( ^ T — + ~) + BK ]Oha + OBhaha + dt rcostp dA dip cfu0 j. * (to [ h JVo — + au0 \oUo + oau0Uo + dt rcostp dA r dv0 r * do dh0 it, ~ . o * [—— - fu0 — + av0 ]ov0 + bav0Vo + (A-5) dt r dna + ~Z— = v dt rcosc/) dA dtp dha dul , d 0 <9/z0* , dD —-~ + /v 0 7^T" + aMo+":r- = 0 dt rcosc? dA du0 dvl » d0 dhl * dD f A-M ——-fuo — + av0 + — = 0 l A °J OT r (7C? aMo g 0 dul , dvlcoscp , D l . , , rt rcosc? dA d
z) / Hcos(/)h^ is the potential vorticity, Uh = HXu and Vh = H*v are the volume transports, H is the equivalent depth, h and z are the latitudes at which the volume transports and the potential vorticity are computed, and (BI) &() = [( W - ( Un]/AX *( ) = [( );+i/2-( )j-m]/A0) r=0 s T=T2 *T„ S7/SZ> = I S(fc>0) 1=0 s 57/5y= I I tH(hd2,s)xua(t2,s) + v*0(tvt2,s)xva(t2,s)l r2=0 s r\=0 where s denotes the spatial variables, T2 is the total days of coupling, and Ta = 90 and T0 = 8 are respectively the number of time steps for the atmospheric model and the oceanic model in one day. Appendix C Verifying Gradient Calculations 131 Appendix C Verifying Gradient Calculations The correctness of the adjoint code and the gradient computations must be verified before they can be used for data assimilation (Talagrand, 1991, Navon et al, 1992). The adjointness is defined as: {a,Mb} = {M*a,b} (CI) where {,} is inner product, a and b any vectors, M an operator (usually a subroutine), and M*is the adjoint operator of M. To check the adjoint model of our forward model, the tangent-linear code of our model must be developed from the forward model by differentiating each subroutine of the model (3.9-3.10). The adjoint code was developed directly from the tangent-linear model by transposing each subroutine of the tangent-linear model. (CI) was then used to check the adjointness of the adjoint model. Before checking (CI), the linearization of the forward model (3.9-3.10) must first be checked, i.e., for every subroutine of the model, AMa-dMa must be of order 5a where AMa = M{a + 8a) - M(a) is the output perturbation in finite difference form resulting from an input perturbation 8a, and 8Ma the perturbation computed from the tangent-linear code 8M. This check must hold to machine accuracy. The accuracy of gradient computation can be tested by the Taylor expansion. Let the Taylor expansion of cost function J be J(x,p + eh) = J(x,p) + ehTVpJ + 0(e2) (C2) where termp is the control vector containing unknowns, e a scalar, h a unit vector given by h = can be rewritten as a function of £ J(x,p + eh)-J(x,p) m e(v,7iv,7iryv,/ l + 0 i e ) If the values of J and the gradient are correctly calculated, the value of (pfe) will linearly Appendix C Verifying Gradient Calculations 132 approach 1 with e decreasing through a wide range of magnitudes. If 0(e) does not approach 1, errors must exist in the gradient calculation. Fig.C shows that our gradient for the six parameters is correct. - 9 - 7 - 5 - 3 1 0 log(e) Figure C Verification of the gradient computation. Note that O(e)=l+O(e) indeed holds until round-off errors become important for very small e. Appendix C Verifying Gradient Calculations Appendix D 133 Summary of Experiments and Results Table 1 A summary of the experiments performed to retrieve the three oceanic initial conditions from assimilating wind (w) and/or SLH (h) data. Data assimilation period was 90 days. The retrieval quality for the three oceanic initial conditions is classified as: excellent (excl), good (gd), acceptable (acpt) or poor (pr). The RMSE (r) ranges for the classification are: pr (r>10'), acpt (lO-'-lO-2), gd (10-2-104) , excl (r<104), and the correlation (c) ranges are: pr (c<0.5), acpt (0.50.8). Data in the TAO array were sampled every 15° in the zonal direction and every 2° in the meridional direction. experiments 1 w sparse in time Fig.5.2 2 same as 1 Fig.5.2 3 same as 1 Fig.5.2 4 same as 1 Fig.5.2 5 h sparse in time Fig.5.2 6 same as 5 Fig.5.2 7 same as 5 Fig.5.2 8 same as 5 Fig.5.2 6 -9 w in TAO Fig.5.3 10 h in TAO Fig.5.3 11-14 same as 10 Fig.5.3 15 noisy w Fig.5.7 16 noisy w Fig.5.7 17 noisy w and h Fig.5.7 18 noisy w and h Fig.5.7 Effect of initial guesses 19-20 in-phase guesses 21-22 noisy perturbat. 23-24 out-phase guesses Data w everywhere and once per day w everywhere and once every 10 days w everywhere and once every 30 days w everywhere and once every 90 days h everywhere and once per day h everywhere and once every 10 days h everywhere and once every 30 days h everywhere and once every 90 days w in TAO once every 1, 10, 30, 90 days h in TAO, once per day h in TAO, once every 10, 30, 90 days w in TAO, once everyday w in TAO, once every 10 days w and h in TAO, once everyday w and h in TAO, once every 10 days Fig.5.5 w or h in TAO, once every 90 days w or h in TAO, once every 90 days w and h in TAO, once everyday result appraisal Correlation RMSE h0u0vo excl u0vo h0 acpt h0 u0 exclvo gd u0 gd, h0 vo acpt h0 u0 exclvo pr u0 gd 7. . . 4. vo pr h0, u0, vo excl. h0 vo gd; u0 excl h0 u0 vo excl. h0u0vo gd h0, u0, vo excl h0 u0 gd vo acpt h0 excl; u0 h0 u0 vo acpt; vo pr acpt h0 excl; h0 u0 vo acpt u0 gd;vo pr h0 u0 excl; vo h0u0vo acpt pr h0 excl; h0u0vo acpt u0 vo pr h0 u0 vo excl h0 u0 vo acpt h0 vo excl; h0 u0 vo acpt u0 acpt h u0 pr; h0 u0 vo pr h0 vo u0 pr vo acpt h0 u0 pr; vo acpt h0 u0 vo excl h0 u0 vo excl h0 u0 vo excl h0u0vo acpt h0 u0 vo pr h0 u0 vo acpt Appendix D. Summary of experiments and Results 134 Table 2 A summary of the experiments performed to retrieve the six parameters from assimilating wind (w) and/or SLH (h) data. Unless specifically mentioned, the data assimilation period was 40 days. The retrieval quality is classified as: excellent (excl), good (gd), acceptable (acpt) or poor (pr). The relative estimation error ranges (r) of the classes are: pr (r>10_1), acpt ( lCU-lO 2 ), gd (10-2- lO 4), or excl (r<104). Experiments 0 continuous wind Fig 4.2 1 sparse w in time Fig.4.3a 2 same as 1 Fig.4.3a 3 sparse h in time Fig.4.3b 4 same as 3 Fig.4.3b 5 sparse w in time and space, sparse h in space, Fig.4.4a 6 same as 5 Fig.4.4a 7-8 sparse h in time and space, sparse w in space Fig.4.4b 9 h at two latitudes Fig. 4.5 10 h at three points Fig. 4.5 11 noisy h Fig. 4.6 12 noisy w Fig. 4.6 13 noisy w and h Fig. 4.6 14 5 day assimilation Fig. 4.7a 15 20 day assimilation, Fig. 4.7a 16 40 day assimilation, Fig. 4.7a 17 5 day assimilation Fig. 4.7b 18 20 day assimilation Fig. 4.7b 19 40 day assimilation Fig. 4.7b 20 5 day assimilation, Fig. 4.10 21 20 day assimilation Fig. 4.10 22 40 day assimilation Fig. 4.10 Data distribution in time and space w everywhere and every time step w everywhere and once per day, h everywhere in time and space w everywhere and once every two days, h everywhere in time and space h everywhere and once per day, w everywhere in time and space h everywhere and once every two days, w everywhere in time and space h and w at every 10*10 grids in 10°S -30°S and 10°N-30°, at every 2*15 grids in 10°S-10°N; w once per day, h once every timestep same as 5 but w available once every two days same as 5 but h available once every day, w every time step same as 7 but h available once every two days same as 7 but h once everyday and every 10 grids along 25°N and 25°S same as 9 except h at grid points (30, 100 and 170)at 0.5°N w and h available everywhere, 10 % noise added to h w and h available everywhere, 10 % noise added to w w and h available everywhere, 10 % noise added to both w everywhere and once per day, h everywhere in time and space w everywhere and once per day, h everywhere in time and space w everywhere and once per day, h everywhere in time and space h everywhere and once per day, w everywhere in time and space same as 16 same as 16 same as 9 same as 9 but with a priori information same as 9 but with a priori information result appraisal a, y,A,a,b and B excl a, y and A gd; a, b and B acpt a, y, A, a, b acpt; B pr a, y,A, and b excl a and B gd a excl; y, A, a, b and B gd a, y acpt; B, A, a and b pr or, y acpt; B, A, a and b pr a, y, A, b and B excl a gd a, y, A and b excl a and B gd a, y,A,b and B acpt a pr a, y, A, a, b and B excl a, A and /excl; a, b and B gd a, A and b gd; a, /and B acpt a, A and b gd; a, /and B acpt a, /and A acpt; a, b, B pr a, /and A gd; a and B acpt; b pr oc, y, A and B gd; a and b acpt a and A gd; B, b and y acpt; a pr same as 17 a excl; A and ygd; B and b acpt; a pr a, A, B, b, a and y pr a, /and a acpt; A, B and b pr same as 21