PARAMETER OPTIMIZATION AND DATA ASSIMILATION TO IMPROVE THE TIDAL PREDICTION OF THE SINGAPORE REGIONAL MODEL
S.K. Ooi(1), H. Gerritsen(2), A. Kurniawan(1), and D.J. Twigt(2).
1. Singapore-Delft Water Alliance, The National University of Singapore. Singapore 2. Deltares, Delft, Netherlands.
S.K. Ooi, Singapore-Delft Water Alliance, The National University of Singapore, Singapore, Singapore.
Tel: + 65-65165865, email:sk.ooi@nus.edu.sg
ABSTRACT
Hydrodynamic modelling is a major tool in the MustHave Box study that focuses on a better understanding of sea level anomalies (SLA) and current anomalies (CA) in the Singapore region. Accurate hydrodynamic modelling is complicated by the tidal flow interactions between the Indian Ocean and the South China Sea, the presence of numerous islands and small channels, seasonal monsoons and short-term weather phenomena. Tidal variation is one of the easier and more important components to assess the accuracy of a hydrodynamic model as it is a deterministic process. The presence of numerous water level observation stations in the region suggests that data assimilation may be a useful tool to improve the tidal prediction of the hydrodynamic model. The steps that have already been taken and that need to be completed for improving the tidal predictions of the model are detailed in this paper. This paper focuses on the initial stages of the data assimilation process particularly the use of single parameter optimization to assess the sensitivity of the tidal constituents at the Java Sea (JS) and South China Sea (SCS) boundaries. This technique is useful and critical in the final data assimilation stage as it provides the modeller a guide to assess results of the data assimilation technique. The results obtained from parameter optimization are discussed. A generic data assimilation and calibration modelling environment called OpenDA which can be used in combination with any model that describes the time evolution of physical processes is described and preliminary verification results are shown.
Keywords: Parameter estimation, data assimilation, tidal constituents, hydrodynamic modelling,
1. INTRODUCTION
The water levels and currents in the Malacca Straits and Singapore Straits are the product of various phenomena including the complex tidal interactions between the Indian and Pacific Oceans, seasonal monsoons and shorter time-scale weather features. The MustHave Box study (Gerritsen et al., 2009) seeks to understand the individual and combined contributions of these phenomena to the residual (non-tidal) currents or current anomalies (CA) and associated sea level anomalies (SLA) in these heavily-trafficked waters as it is of economic and scientific relevance. Depth-integrated hydrodynamic modelling is a practical means to verify and quantify the various phenomena that contribute to the currents and water levels in the Singapore region. The model used for this purpose is the Singapore Regional Model (SRM) developed by Zijl and Kernkamp (2004), which stretches from the Andaman Sea to the coast of Borneo. While the large SRM domain allows for dynamic interactions between Singapore waters and its western and eastern approaches, its main objective of a balanced overall tidal representation around Singapore may not lead to maximum tidal accuracy in all its sub-regions. Ooi et al. (2009) used a selectively refined grid variant of the SRM to show that it is possible to further improve the original SRM’s tidal prediction locally around Singapore at the cost of significantly increased computational time. Newer studies of the South China Sea (for example Zu et al., 2008) are also available which suggest that the overall tidal representation of the SRM could be improved through parameter optimization and data assimilation. This paper details the steps taken and to be taken to improve the overall tidal representation of the SRM with a special focus on parameter optimization of the tidal constituents on the boundaries.
2. SRM – SINGAPORE REGIONAL MODEL
The Singapore Regional Model (SRM) was developed by Zijl and Kernkamp (2004) using the Delft3D-Flow system to simulate tidal flow for the whole water body between the Andaman Sea (AS) in the west and the South China Sea (SCS) and Java Sea (JS) in the east. The tides at the western and eastern boundaries represent the influence of the Indian and Pacific Oceans, respectively, while the tidal interaction which is very complex, is resolved in the interior model domain. The SRM features a boundary-fitted curvilinear-spherical orthogonal grid and was originally calibrated to provide hydrodynamic information for Singapore waters with regards to the coastline geometry of 1999. The model has a total of 38,500 boundary-fitted curvilinear grid cells with a varying resolution tailored to the regions and spatial scales of interest; from 15 km at the open sea boundaries to 200 m around Singapore. The bathymetry in the SRM is based on Admiralty charts giving a maximum depth in the model of about 2000 m in the AS and a maximum depth in the Singapore Strait of approximately 160 m. The tidal range varies from about 2.8 m in the West of Singapore to about 1.5 m to the East of Singapore. It has three open boundaries, located in AS, SCS and JS with total of 17 boundary support points (Figure 1).
Figure 1: Extent of the SRM with its boundary support points (shown by big blue open
circles; where tidal and mean level forcing are prescribed and adjusted) and location of
in-situ measurement stations in the western (a), central (b) and eastern (c) regions.
2.1 TIDAL BOUNDARY CONDITIONS
The tides in the SCS have a (mainly) diurnal character, while it is (mainly) semi-diurnal in the AS. To this end, a total of 8 tidal constituents (Q1, O1, P1, K1, N2, M2, S2, and K2) were originally prescribed at/along the open sea boundaries using information obtained from the FES99 ocean tide model (Lefevre et al., 2002) as a first guess. In Zijl and Kernkamp (2004) these boundary conditions were subsequently improved based on extensive calibration using in-situ tidal data in Singapore. Since then new studies have assessed the local tides in more detail. For example, Zu et al., 2008 have carried out a more detailed tidal analysis of the SCS including parts of the JS by assimilating Topex/Poseidon altimetry data into a barotropic ocean tide model for the eight major constituents (M2, S2, K1, O1, N2, K2, P1 and Q1) using a tidal data inversion scheme. The results of the study by Zu et al. 2008 show differences with the results obtained through the FES ocean tide model. For the present study Zu et al.’s results were used as a starting point to begin the data assimilation process to calibrate the SRM for better tidal representation in the entire model domain.
3. DATA ASSIMILATION PROCESS
As the propagation of the tidal wave through the model domain is dependent on both an accurate bathymetry and coastal geometry representation, a reassessment of the model’s performance in representing the tides is necessary given the changes in coastal geometry and associated bathymetry changes in the Singapore region since the creation of the model by Zijn and Kernkamp (2004).
The steps required in the data assimilation (parameter estimation) process after updating the land boundaries are as follows:
1. Evaluate the sensitivity of the SRM to changes in tidal constituents forcing at boundary conditions through single and multi-parameter optimization techniques, that is by changing only one or multiple constituents at one boundary using updated information (e.g. SCS or JS). This step is critical because in a non-linear system such as the SRM, “directional local methods” as used in OpenDA may encounter a local minimum and not the global minimum. Thus it is necessary to separately assess the results of the semi-automated parameter estimation methods.
2. Use a data assimilation tool such as OpenDA to evaluate and improve the overall response of the model by simultaneously varying all the tidal constituents forcing constituents at all the boundaries including boundaries without updated information.
3. Use OpenDA to evaluate and improve local tidal characteristics at particular stations i.e. through varying local bed friction or depth.
Presently step 1 has been completed while the preliminary evaluation of steps 2 and 3 through “twin tests” of a simpler schematized model has been completed satisfactorily.
3.1 OPENDA
OpenDA is a portable interface for enabling data assimilation and calibration tools to be used with different flow models as per the OpenMI concepts (http://openmi.org). It is a continuation of the DATools package which was used successfully by El Serafy et al. (2007) for data assimilation of current and salinity profiles. Since then DATools has been renamed to OpenDA and updated with newer functionalities.
For the data assimilation toolset three semi-automated parameter estimation methods are available in Open-DA: DUD, Powell and Simplex. All three methods can be applied with or without user-defined constraints on the parameter range. By logically varying the parameters that need to be optimised, a Goodness-of-Fit (GoF) criterion can be minimised. A typical GoF criterion could read
N n n n n
w
GoF
1 2)
(
2
1
)
(
(1)where wn is a weight, n is an integral parameter (typically water level), n( )is the model prediction, n is the corresponding measured value and are model parameters (typically tidal amplitudes and phases, local depth, local friction).
The preferred parameter estimation method is the DUD method first described by Ralston and Jennrich (1978) as this method has been shown to be computationally more efficient than the other two methods and is designed specifically for non-linear problems and is designed especially for a GOF criterion written as a least square function as in Eq. 1.
4. PARAMETER OPTIMIZATION RESULTS
The sensitivity of the model was evaluated through predominantly single parameter optimization of the tidal boundaries with values obtained for the four constituents presented by Zu et al., 2008 (i.e. M2, S2, O1 and K1). As the AS has not been covered by Zu et al., 2008, only the amplitude and phases in the SCS and JS have been used. Table 1 summarizes the averaged values across each boundary for the original tidal constituent forcing conditions, the adjusted tidal constituent forcing conditions, the relative change in amplitude (% = adjusted – original / original) between the original and the adjusted amplitudes given in percentage; and the relative change in phase in degrees between the original and adjusted phases ( = adjusted – original).
original adjusted original adjusted (deg.) original adjusted original adjusted (deg.) originaladjusted original adjusted (deg.) original adjusted original adjusted (deg.) South China Sea 0.313 0.285 -8.92 330.296 306.221 -24.076 0.348 0.395 13.41 8.921 354.477 -14.444 0.304 0.257 -15.47 160.256 151.873 -8.383 0.097 0.090 -6.76 183.542 199.030 15.488 Java Sea 0.209 0.178 -14.77 131.223 112.808 -18.414 0.324 0.353 9.03 154.023 153.237 -0.786 0.086 0.047 -46.00 168.170 169.113 0.943 0.039 0.051 31.47 169.487 118.260 -51.226 Boundary Definition Tidal Constituent O1 K1 M2 S2 Amplitude Phase Phase Amplitude Phase Amplitude Phase Amplitude
Table 1: SRM averaged tidal constituent forcing values at the JS and SCS; original (Zijl & Kernkamp, 2004) and adjusted (extracted from Zu et al., 2008 adjusted). % = adjusted –
original / original; = adjusted - original
Twenty three full year simulations for the year 2004 were carried out. The simulations were divided as follows: one simulation with the original boundary conditions, one simulation with all the phases at all the boundaries changed, one simulation with all the phases and amplitudes at all the boundaries changed, one simulation with all the phases at the JS boundary are changed, one simulation with all the phases and amplitudes at the JS boundary changed, one simulation with all the phases at the SCS boundary changed, one simulation with all the phases and amplitudes at the SCS boundary changed, four simulations with each simulation changing only one constituent’s phase is varied for each simulation at the SCS boundary, four simulations with each simulation changing only one constituent’s phase and amplitudes at the SCS boundary, four with each simulation changing only one constituent’s phase separately at the JS boundary, four simulations with each simulation changing only one constituent’s phase and amplitudes at the JS boundary.
The simulation results were then compared to the observed tidal constituents in order to assess the model results. The results were evaluated by carrying out tidal analysis of 4 major tidal components (M2, S2, K1 and O1) and comparing the model results to observation stations for three distinct regions in the SRM: the west region (Set A) with around 30 stations, center region (Set B) with around 64 stations and Set C in the east with approximately 61 stations in east region (Set C), see Figure 1a-c respectively.
4.1 RESULTS
The differences between modelled and measured tidal constituents are quantified in the tables below in terms of Root Mean Square Error (RMSE) differences of amplitudes and phases (Hc and Gc for computed amplitudes and phases, Hobs and Gobs for observed amplitudes and phases respectively) and percentage of improvement with regards to the original boundary conditions of Zijl and Kernkamp (2004) which is considered the base line case. The error signal is also quantified as the Mean Summed
Vector Difference (MSVD) which represents the sum of the maximum error that normalised with number of stations. The RMSE and MSVD are determined over all 28 stations in the West region (except for O1 which is only 24 stations), 63 stations in the Center and 60 stations in the East region (except for M2 and S2 which are only 59 and 57 stations). These stations are located in the selected area where effects of the tide generating forces are observed.
4.1.1. VARYING PHASE ONLY AT THE BOUNDARIES
The effects of single parameter variation for the 4 tidal constituents prescribed at the open sea boundaries for when only phase was varied are summarized in Tables 2 to 4 and Figure 2. Figure 2 shows the % improvement of the MSVD for the 4 tidal constituents when parameters are varied at the boundaries for each sub-region in the SRM. Parameter variation is shown from left to right with the % improvement of the MSVD for each tidal constituent over the region shown by the bars when compared to the baseline case (positive being improvement and negative being deterioration). Figure 2 shows that for all three regions single parameter variation gives mixed results. For example when just changing the phase of O1 on the SCS boundary, in the west the MSVD improves for K1, while in the center region all constituents improve and in the west region only K1 and O1 improve with this change on the SCS boundary. The results in Tables 2 to 4 show that the predominant reason for all the improvements are due to improvements in phase predictions for all the constituents that improve except for K1 in the west where the improvement is due to an improvement in the amplitude prediction as shown in Table 2. Thus although only the phases were adjusted the complex interaction of tides in this region results in phase prediction changes as well due to the adjustment of phases at the boundaries.
Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Original BC 0.015 16.889 0.000 0.000 0.018 0.029 8.446 0.000 0.000 0.033 0.171 14.646 0.000 0.000 0.208 0.095 16.013 0.000 0.000 0.119 K1 changed on SCS 0.015 17.139 0.000 -1.480 0.018 0.035 8.838 -20.690 -4.641 0.038 0.178 15.187 -4.094 -3.694 0.212 0.098 16.291 -3.158 -1.736 0.120 O1 changed on SCS 0.018 19.621 -20.000 -16.176 0.020 0.025 8.747 13.793 -3.564 0.030 0.178 15.171 -4.094 -3.585 0.211 0.098 16.329 -3.158 -1.973 0.120 M2 changed on SCS 0.015 17.127 0.000 -1.409 0.018 0.025 8.700 13.793 -3.007 0.030 0.178 15.203 -4.094 -3.803 0.212 0.098 16.303 -3.158 -1.811 0.119 S2 changed on SCS 0.015 17.019 0.000 -0.770 0.018 0.025 8.712 13.793 -3.149 0.030 0.178 15.193 -4.094 -3.735 0.211 0.098 16.292 -3.158 -1.742 0.119 All 4 Changed on SCS 0.018 19.517 -20.000 -15.560 0.020 0.036 8.748 -24.138 -3.576 0.039 0.178 15.169 -4.094 -3.571 0.212 0.099 16.293 -4.211 -1.749 0.119 K1 changed on JS 0.014 17.131 6.667 -1.433 0.018 0.024 15.563 17.241 -84.265 0.033 0.178 15.212 -4.094 -3.865 0.211 0.098 16.335 -3.158 -2.011 0.120 O1 changed on JS 0.014 20.573 6.667 -21.813 0.020 0.025 8.761 13.793 -3.730 0.030 0.178 15.209 -4.094 -3.844 0.212 0.098 16.290 -3.158 -1.730 0.120 M2 changed on JS 0.015 17.048 0.000 -0.941 0.018 0.025 8.703 13.793 -3.043 0.030 0.178 15.195 -4.094 -3.748 0.212 0.098 16.309 -3.158 -1.848 0.120 S2 changed on JS 0.015 17.052 0.000 -0.965 0.018 0.025 8.699 13.793 -2.996 0.030 0.178 15.195 -4.094 -3.748 0.212 0.098 16.326 -3.158 -1.955 0.120 All 4 changed on JS 0.015 21.300 0.000 -26.118 0.020 0.025 15.748 13.793 -86.455 0.034 0.178 15.222 -4.094 -3.933 0.212 0.098 16.339 -3.158 -2.036 0.120 All new BC on SCS and JS 0.015 16.643 0.000 1.457 0.018 0.020 10.091 31.034 -19.477 0.028 0.171 14.665 0.000 -0.130 0.208 0.095 16.024 0.000 -0.069 0.119
% Improvement MSVD
Tidal Constituent of S2 RMSE % Improvement
MSVD
*Only 24 stations have been considered here, no observed O1 at some stations
Tidal Constituent of K1 % Improvement
MSVD
Tidal Constituent of M2 Single Parameter Variation
(Phase adjusted) RMSE % Improvement MSVD RMSE Tidal Constituent of O1*
RMSE
Table 2: Summary of tidal model quality in West region of SRM for phase varying
simulations (averaged for 28 stations)
Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Original BC 0.049 12.602 0.000 0.000 0.066 0.092 12.901 0.000 0.000 0.087 0.111 17.786 0.000 0.000 0.214 0.066 14.511 0.000 0.000 0.082 K1 changed on SCS 0.050 12.883 -2.041 -2.230 0.067 0.092 15.016 0.000 -16.394 0.091 0.109 17.624 1.802 0.911 0.212 0.065 14.165 1.515 2.384 0.081 O1 changed on SCS 0.048 11.139 2.041 11.609 0.058 0.088 11.670 4.348 9.542 0.084 0.108 17.372 2.703 2.328 0.207 0.065 14.076 1.515 2.998 0.080 M2 changed on SCS 0.049 13.223 0.000 -4.928 0.068 0.088 11.683 4.348 9.441 0.083 0.113 17.304 -1.802 2.710 0.209 0.065 14.218 1.515 2.019 0.081 S2 changed on SCS 0.049 12.982 0.000 -3.015 0.067 0.089 11.610 3.261 10.007 0.083 0.108 17.501 2.703 1.602 0.210 0.066 16.745 0.000 -15.395 0.092 All 4 Changed on SCS 0.049 11.209 0.000 11.054 0.058 0.091 15.189 1.087 -17.735 0.092 0.114 17.311 -2.703 2.671 0.209 0.066 16.866 0.000 -16.229 0.092 K1 changed on JS 0.048 13.699 2.041 -8.705 0.069 0.096 15.591 -4.348 -20.851 0.109 0.106 17.308 4.505 2.688 0.206 0.065 14.077 1.515 2.991 0.080 O1 changed on JS 0.053 17.052 -8.163 -35.312 0.087 0.088 11.513 4.348 10.759 0.083 0.108 17.517 2.703 1.512 0.211 0.065 14.152 1.515 2.474 0.081 M2 changed on JS 0.049 13.022 0.000 -3.333 0.067 0.089 11.609 3.261 10.015 0.083 0.108 17.511 2.703 1.546 0.210 0.065 14.106 1.515 2.791 0.080 S2 changed on JS 0.050 13.018 -2.041 -3.301 0.067 0.089 11.613 3.261 9.984 0.083 0.108 17.501 2.703 1.602 0.210 0.065 13.169 1.515 9.248 0.078 All 4 changed on JS 0.052 18.068 -6.122 -43.374 0.090 0.096 15.776 -4.348 -22.285 0.110 0.106 17.319 4.505 2.626 0.207 0.065 13.076 1.515 9.889 0.077 All new BC on SCS and JS 0.048 12.647 2.041 -0.357 0.066 0.084 11.521 8.696 10.697 0.083 0.114 17.519 -2.703 1.501 0.211 0.066 15.032 0.000 -3.590 0.082
% Improvement Single Parameter Variation
(Phase adjusted)
Tidal Constituent of O1 Tidal Constituent of K1 Tidal Constituent of M2 Tidal Constituent of S2 RMSE % Improvement
MSVD RMSE % Improvement MSVD RMSE % Improvement MSVD RMSE MSVD
Table 3: Summary of tidal model quality in Center region of SRM for phase varying
simulations (averaged for 63 stations)
Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Original BC 0.061 30.049 0.000 0.000 0.117 0.140 19.631 0.000 0.000 0.172 0.106 45.776 0.000 0.000 0.116 0.067 51.782 0.000 0.000 0.066 K1 changed on SCS 0.059 30.243 3.279 -0.646 0.117 0.139 20.242 0.714 -3.112 0.170 0.108 46.263 -1.887 -1.064 0.119 0.068 54.199 -1.493 -4.668 0.066 O1 changed on SCS 0.062 28.546 -1.639 5.002 0.104 0.131 19.489 6.429 0.723 0.162 0.108 45.689 -1.887 0.190 0.118 0.067 54.739 0.000 -5.710 0.066 M2 changed on SCS 0.059 30.031 3.279 0.060 0.116 0.131 19.523 6.429 0.550 0.162 0.110 44.771 -3.774 2.195 0.096 0.068 54.480 -1.493 -5.210 0.066 S2 changed on SCS 0.059 30.041 3.279 0.027 0.116 0.131 19.552 6.429 0.402 0.162 0.108 45.850 -1.887 -0.162 0.119 0.066 47.739 1.493 7.808 0.065 All 4 Changed on SCS 0.062 28.687 -1.639 4.533 0.105 0.139 20.146 0.714 -2.623 0.170 0.110 46.023 -3.774 -0.540 0.097 0.066 47.655 1.493 7.970 0.065 K1 changed on JS 0.057 29.708 6.557 1.135 0.113 0.101 21.351 27.857 -8.762 0.130 0.106 45.090 0.000 1.499 0.117 0.067 52.653 0.000 -1.682 0.066 O1 changed on JS 0.063 27.676 -3.279 7.897 0.117 0.129 19.660 7.857 -0.148 0.162 0.107 45.727 -0.943 0.107 0.119 0.068 54.372 -1.493 -5.002 0.066 M2 changed on JS 0.059 30.044 3.279 0.017 0.115 0.131 19.562 6.429 0.351 0.162 0.107 45.072 -0.943 1.538 0.117 0.068 54.758 -1.493 -5.747 0.066 S2 changed on JS 0.059 30.051 3.279 -0.007 0.116 0.131 19.555 6.429 0.387 0.162 0.108 45.899 -1.887 -0.269 0.119 0.054 36.279 19.403 29.939 0.053 All 4 changed on JS 0.063 27.437 -3.279 8.692 0.116 0.101 21.542 27.857 -9.735 0.131 0.105 44.670 0.943 2.416 0.117 0.054 34.205 19.403 33.944 0.052 All new BC on SCS and JS 0.065 25.867 -6.557 13.917 0.102 0.109 21.742 22.143 -10.753 0.147 0.107 36.134 -0.943 21.063 0.091 0.051 32.645 23.881 36.957 0.050
RMSE % Improvement MSVD RMSE % Improvement
*Only 59 stations have been considered here, no observed M2 at one station *Only 57 stations have been considered here, no observed S2 at some stations
Single Parameter Variation (Phase adjusted)
Tidal Constituent of O1 Tidal Constituent of K1 Tidal Constituent of M2* Tidal Constituent of S2**
RMSE % Improvement MSVD RMSE % Improvement MSVD MSVD
Table 4: Summary of tidal model quality in East region of SRM for phase varying
simulations
a) K1 changed on SCS O1 changed on SCS M2 changed on SCS S2 changed on SCS All 4 Changed on SCS K1 changed on JS O1 changed on JS M2 changed on JS S2 changed on JS All 4 changed on JS All new BC on SCS and JS O1 0.907 -9.977 0.227 0.680 -9.524 0.000 -7.256 0.454 0.680 -10.658 1.134 K1 -16.126 8.550 8.874 9.307 -16.991 -0.325 10.173 9.199 8.983 -1.840 16.558 M2 -1.839 -1.702 -2.063 -1.788 -1.960 -1.736 -1.908 -1.839 -1.839 -1.839 -0.275 S2 -0.571 -0.601 -0.450 -0.450 -0.270 -0.661 -0.601 -0.601 -0.781 -0.841 0.180 -20.000 -15.000 -10.000 -5.000 0.000 5.000 10.000 15.000 20.000 % MSVDPercentage MSVD Improvement in West Region
b) K1 changed on SCS O1 changed on SCS M2 changed on SCS S2 changed on SCS All 4 Changed on SCS K1 changed on JS O1 changed on JS M2 changed on JS S2 changed on JS All 4 changed on JS All new BC on SCS and JS O1 -2.543 11.770 -3.754 -2.180 11.044 -5.377 -32.768 -2.422 -2.567 -37.830 -1.090 K1 -4.320 3.700 4.320 4.211 -5.231 -24.936 4.776 4.284 4.265 -26.340 4.885 M2 0.942 3.138 2.218 1.980 2.188 3.909 1.216 1.706 1.958 3.130 1.372 S2 1.089 1.789 0.817 -12.094 -12.580 2.217 1.322 1.575 4.375 5.328 -0.992 -50.000 -40.000 -30.000 -20.000 -10.000 0.000 10.000 20.000 % MSVD
Percentage MSVD Improvement in Center Region
c) K1 changed on SCS O1 changed on SCS M2 changed on SCS S2 changed on SCS All 4 Changed on SCS K1 changed on JS O1 changed on JS M2 changed on JS S2 changed on JS All 4 changed on JS All new BC on SCS and JS O1 0.596 11.685 1.235 1.491 10.706 3.464 0.284 1.619 1.533 1.136 13.389 K1 0.875 5.364 5.617 5.481 1.069 24.373 5.724 5.423 5.452 23.800 14.052 M2 -2.242 -1.427 17.179 -1.820 16.887 -0.699 -1.834 -0.728 -1.951 -0.466 21.648 S2 -0.186 0.080 0.159 1.222 1.116 0.425 0.159 0.053 20.212 21.116 24.489 -5.000 0.000 5.000 10.000 15.000 20.000 25.000 30.000 % MSVD
Percentage MSVD Improvement in East Region
Figure 2: Percentage MSVD Improvement of SRM which is only phases adjusted in a)
West Region, b) Center region and c) East region
.The best overall scenario when only phase is varied is clearly shown to be when all the phases of all the constituents at all the boundaries are changed. In the important center region this scenario results in an average improvement in the MSVD of 1% with slight deterioration of the predictions for the O1 and S2 constituents (<1.1% MSVD) but with a similar improvement magnitude (>1.4% MSVD) for the other two constituents in the center region. For this same scenario all constituents in the west region improve except for M2 which regresses slightly (<0.3% MSVD) resulting in an average improvement in % MSVD for all the constituents of 4.4%. In the east region all the tidal constituents have significantly improved (>13% MSVD) resulting in an average improvement of 18.4% for all the four constituents. For this scenario Tables 2 to 4 illustrates that a combination of phase and amplitude improvements results in the MSVD improvements for the different constituents in all the regions.
4.1.2. VARYING TIDAL PHASE AND AMPLITUDE SIMULTANEOUSLY AT THE
BOUNDARIES
The results for parameter variation for the 4 tidal constituents prescribed at the open sea boundaries for when amplitude and phase were varied simultaneously are summarized in Tables 5 to 7 and Figure 3. Similarly to the previous section in which only the phase is varied, the results in Figure 3 show that for all three regions parameter variation give mixed improvement outcomes for all the different constituents. Unlike in the previous section, the best overall improvement in MSVD is obtained when only the phase and amplitude of S2 are changed simultaneously on the JS boundary. This scenario results in an average improvement in MSVD in the western region of 1.8% with K1 and O1 predictions improving while M2 and S2 deteriorate. In the important central region the average improvement is 2% with only K1 deteriorating (2.54%). In the eastern region the average improvement is 6.62% with only M2 predictions deteriorating. Similar to the results in the previous section, the results of such a general improvement over the entire model is due not wholly to the improvement in phase or amplitude but to a combination of both.
Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Original BC 0.015 16.889 0.000 0.000 0.018 0.029 8.446 0.000 0.000 0.033 0.171 14.646 0.000 0.000 0.208 0.095 16.013 0.000 0.000 0.119 K1 changed on SCS 0.015 17.987 0.000 -6.501 0.020 0.045 9.462 -55.172 -12.029 0.045 0.178 15.194 -4.094 -3.742 0.211 0.098 16.260 -3.158 -1.542 0.120 O1 changed on SCS 0.016 21.500 -6.667 -27.302 0.020 0.025 8.830 13.793 -4.547 0.031 0.178 15.161 -4.094 -3.516 0.211 0.098 16.323 -3.158 -1.936 0.120 M2 changed on SCS 0.015 17.090 0.000 -1.190 0.018 0.026 8.658 10.345 -2.510 0.030 0.178 15.185 -4.094 -3.680 0.211 0.098 16.313 -3.158 -1.873 0.120 S2 changed on SCS 0.015 17.014 0.000 -0.740 0.018 0.025 8.723 13.793 -3.280 0.030 0.178 15.191 -4.094 -3.721 0.211 0.099 16.258 -4.211 -1.530 0.119 All 4 Changed on SCS 0.016 20.180 -6.667 -19.486 0.019 0.047 9.570 -62.069 -13.308 0.047 0.178 15.142 -4.094 -3.387 0.210 0.099 16.231 -4.211 -1.361 0.119 K1 changed on JS 0.014 17.131 6.667 -1.433 0.018 0.024 15.665 17.241 -85.472 0.033 0.178 15.213 -4.094 -3.871 0.211 0.098 16.335 -3.158 -2.011 0.120 O1 changed on JS 0.015 20.113 0.000 -19.089 0.021 0.025 8.678 13.793 -2.747 0.030 0.178 15.216 -4.094 -3.892 0.212 0.098 16.295 -3.158 -1.761 0.120 M2 changed on JS 0.015 16.956 0.000 -0.397 0.018 0.025 8.690 13.793 -2.889 0.030 0.178 15.196 -4.094 -3.755 0.212 0.098 16.307 -3.158 -1.836 0.120 S2 changed on JS 0.015 17.052 0.000 -0.965 0.018 0.025 8.701 13.793 -3.019 0.030 0.178 15.195 -4.094 -3.748 0.212 0.098 16.322 -3.158 -1.930 0.120 All 4 changed on JS 0.016 20.707 -6.667 -22.606 0.022 0.024 15.667 17.241 -85.496 0.034 0.178 15.231 -4.094 -3.994 0.212 0.098 16.336 -3.158 -2.017 0.120 All new BC on SCS and JS 0.014 17.562 6.667 -3.985 0.018 0.026 14.634 10.345 -73.265 0.037 0.178 15.193 -4.094 -3.735 0.211 0.098 16.252 -3.158 -1.493 0.119
% Improvement
*Only 24 stations have been considered here, no observed O1 at some stations Single Parameter Variation
(Amplitude&Phase adjusted)
Tidal Constituent of O1* Tidal Constituent of K1 Tidal Constituent of M2 Tidal Constituent of S2 RMSE % Improvement
MSVD RMSE % Improvement MSVD RMSE % Improvement MSVD RMSE MSVD
Table 5: Summary of tidal model quality in West region of SRM for phase and
amplitude varying simulations (averaged for 28 stations)
Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Original BC 0.049 12.602 0.000 0.000 0.066 0.092 12.901 0.000 0.000 0.087 0.111 17.786 0.000 0.000 0.214 0.066 14.511 0.000 0.000 0.082 K1 changed on SCS 0.046 14.081 6.122 -11.736 0.070 0.063 19.613 31.522 -52.027 0.095 0.106 17.244 4.505 3.047 0.206 0.065 13.924 1.515 4.045 0.080 O1 changed on SCS 0.055 11.223 -12.245 10.943 0.064 0.090 11.743 2.174 8.976 0.086 0.107 17.261 3.604 2.952 0.205 0.065 13.937 1.515 3.956 0.080 M2 changed on SCS 0.049 12.983 0.000 -3.023 0.067 0.090 11.667 2.174 9.565 0.084 0.125 17.856 -12.613 -0.394 0.221 0.065 13.935 1.515 3.969 0.079 S2 changed on SCS 0.049 12.960 0.000 -2.841 0.067 0.089 11.602 3.261 10.069 0.083 0.108 17.486 2.703 1.687 0.209 0.066 22.434 0.000 -54.600 0.107 All 4 Changed on SCS 0.049 10.802 0.000 14.283 0.057 0.065 20.348 29.348 -57.724 0.099 0.121 17.360 -9.009 2.395 0.214 0.066 21.874 0.000 -50.741 0.105 K1 changed on JS 0.048 13.706 2.041 -8.761 0.069 0.096 15.649 -4.348 -21.301 0.109 0.106 17.342 4.505 2.496 0.206 0.065 14.070 1.515 3.039 0.080 O1 changed on JS 0.048 17.471 2.041 -38.637 0.086 0.087 11.539 5.435 10.557 0.082 0.109 17.586 1.802 1.124 0.213 0.065 14.187 1.515 2.233 0.081 M2 changed on JS 0.049 12.861 0.000 -2.055 0.066 0.089 11.643 3.261 9.751 0.084 0.109 17.927 1.802 -0.793 0.213 0.065 14.136 1.515 2.584 0.080 S2 changed on JS 0.050 13.015 -2.041 -3.277 0.067 0.089 11.612 3.261 9.991 0.083 0.108 17.498 2.703 1.619 0.210 0.065 13.149 1.515 9.386 0.078 All 4 changed on JS 0.048 18.410 2.041 -46.088 0.089 0.096 15.797 -4.348 -22.448 0.110 0.107 17.797 3.604 -0.062 0.211 0.065 13.110 1.515 9.655 0.078 All new BC on SCS and JS 0.046 13.088 6.122 -3.857 0.065 0.071 14.361 22.826 -11.317 0.086 0.120 17.771 -8.108 0.084 0.215 0.065 19.267 1.515 -32.775 0.095
% Improvement MSVD RMSE % Improvement Single Parameter Variation
(Amplitude&Phase adjusted)
Tidal Constituent of O1 Tidal Constituent of K1 Tidal Constituent of M2 Tidal Constituent of S2
RMSE % Improvement MSVD RMSE % Improvement MSVD RMSE MSVD
Table 6: Summary of tidal model quality in Center region of SRM for phase and
amplitude varying simulations (averaged for 63 stations)
Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Hc-Hobs Gc-Gobs Original BC 0.061 30.049 0.000 0.000 0.117 0.140 19.631 0.000 0.000 0.172 0.106 45.776 0.000 0.000 0.116 0.067 51.782 0.000 0.000 0.066 K1 changed on SCS 0.056 29.839 8.197 0.699 0.114 0.078 19.661 44.286 -0.153 0.126 0.106 45.992 0.000 -0.472 0.116 0.067 53.816 0.000 -3.928 0.066 O1 changed on SCS 0.070 28.239 -14.754 6.023 0.105 0.133 19.406 5.000 1.146 0.163 0.108 45.885 -1.887 -0.238 0.118 0.067 54.665 0.000 -5.568 0.066 M2 changed on SCS 0.059 30.024 3.279 0.083 0.116 0.131 19.463 6.429 0.856 0.162 0.113 44.205 -6.604 3.432 0.107 0.067 53.914 0.000 -4.117 0.066 S2 changed on SCS 0.059 30.049 3.279 0.000 0.116 0.130 19.556 7.143 0.382 0.162 0.108 45.804 -1.887 -0.061 0.118 0.062 50.948 7.463 1.611 0.067 All 4 Changed on SCS 0.062 27.890 -1.639 7.185 0.099 0.079 19.467 43.571 0.835 0.125 0.111 43.765 -4.717 4.393 0.104 0.060 50.850 10.448 1.800 0.067 K1 changed on JS 0.057 29.840 6.557 0.696 0.114 0.108 21.502 22.857 -9.531 0.136 0.106 45.037 0.000 1.614 0.117 0.067 53.231 0.000 -2.798 0.066 O1 changed on JS 0.059 27.116 3.279 9.761 0.107 0.129 19.662 7.857 -0.158 0.162 0.107 45.262 -0.943 1.123 0.118 0.068 54.038 -1.493 -4.357 0.066 M2 changed on JS 0.059 30.105 3.279 -0.186 0.116 0.131 19.508 6.429 0.627 0.162 0.116 51.581 -9.434 -12.681 0.126 0.067 54.565 0.000 -5.374 0.066 S2 changed on JS 0.059 30.056 3.279 -0.023 0.116 0.131 19.557 6.429 0.377 0.162 0.108 45.915 -1.887 -0.304 0.119 0.051 31.176 23.881 39.794 0.052 All 4 changed on JS 0.060 27.011 1.639 10.110 0.106 0.107 21.598 23.571 -10.020 0.136 0.114 49.214 -7.547 -7.510 0.123 0.051 30.694 23.881 40.725 0.052 All new BC on SCS and JS 0.049 24.454 19.672 18.620 0.080 0.089 21.517 36.429 -9.607 0.133 0.119 44.918 -12.264 1.874 0.109 0.048 36.640 28.358 29.242 0.054
% Improvement
MSVD RMSE % Improvement
*Only 59 stations have been considered here, no observed M2 at one station *Only 57 stations have been considered here, no observed S2 at some stations
Single Parameter Variation (Amplitude&Phase adjusted)
Tidal Constituent of O1 Tidal Constituent of K1 Tidal Constituent of M2* Tidal Constituent of S2** RMSE % Improvement
MSVD RMSE % Improvement MSVD RMSE MSVD
Table 7: Summary of tidal model quality in East region of SRM for phase and
amplitude varying simulations
a) K1 changed on SCS O1 changed on SCS M2 changed on SCS S2 changed on SCS All 4 Changed on SCS K1 changed on JS O1 changed on JS M2 changed on JS S2 changed on JS All 4 changed on JS All new BC on SCS and JS O1 -6.349 -8.844 0.227 0.680 -5.215 -0.227 -16.553 0.680 0.680 -20.181 0.907 K1 -35.823 7.251 8.009 9.524 -40.909 -0.649 10.498 8.766 8.983 -1.515 -13.312 M2 -1.719 -1.599 -1.564 -1.771 -1.186 -1.753 -1.960 -1.891 -1.839 -1.943 -1.358 S2 -0.571 -0.661 -0.811 0.120 -0.030 -0.661 -0.571 -0.541 -0.721 -0.691 0.000 -50.000 -40.000 -30.000 -20.000 -10.000 0.000 10.000 20.000 % M S VD
Percentage MSVD Improvement in West Region
b) K1 changed on SCS O1 changed on SCS M2 changed on SCS S2 changed on SCS All 4 Changed on SCS K1 changed on JS O1 changed on JS M2 changed on JS S2 changed on JS All 4 changed on JS All new BC on SCS and JS O1 -7.217 2.955 -1.841 -1.889 12.303 -5.086 -30.564 -1.163 -2.543 -36.014 1.550 K1 -9.369 1.732 3.901 4.120 -13.288 -25.410 5.614 4.010 4.265 -25.993 1.221 M2 3.508 4.035 -3.397 2.136 0.148 3.731 0.630 0.541 1.980 1.454 -0.282 S2 2.333 2.567 2.625 -31.674 -29.224 2.255 1.089 1.400 4.297 4.861 -16.741 -40.000 -30.000 -20.000 -10.000 0.000 10.000 20.000 % M S VD
Percentage MSVD Improvement in Center Region
c) K1 changed on SCS O1 changed on SCS M2 changed on SCS S2 changed on SCS All 4 Changed on SCS K1 changed on JS O1 changed on JS M2 changed on JS S2 changed on JS All 4 changed on JS All new BC on SCS and JS O1 2.598 10.152 1.477 1.434 15.405 3.024 8.647 1.420 1.519 9.641 31.634 K1 26.356 4.908 5.724 5.549 26.851 20.622 5.695 5.345 5.481 20.603 22.731 M2 0.175 -1.208 8.269 -1.776 10.511 -0.917 -1.383 -8.094 -1.965 -5.867 6.435 S2 0.558 0.212 0.425 -1.726 -1.328 0.159 0.159 0.319 21.434 21.673 17.902 -15.000 -10.000 -5.000 0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000 % M S VD
Percentage MSVD Improvement in East Region
Figure 3: Percentage MSVD Improvement of SRM when both phase and amplitude are
varied simultaneously in a) West Region, b) Center region and c) East region
.Overall it can be seen that changing the phases and amplitudes simultaneously using the values from Zu et al., 2008 does not provide as significant an improvement as the case when only the phases are changed. Comparing the overall improvements in all the regions it can be seen that the best overall result is the scenario when only the phases of the tides are changed for all constituents at both the SCS and JS boundaries.
5. VERIFICATION OF DUD IN OPENDA
With parameter estimation results available, it was decided to proceed further in optimizing the tidal calibration of the SRM through data assimilation techniques found in OpenDA. Of these techniques, DUD was found to be the most promising technique given its original design for non-linear problems. To test the effectiveness of DUD, “twin experiments” were carried out using a simpler model than the SRM. The FTI model was smaller (800 grid points), only 2 tidal constituents imposed on the open boundaries (M2 and S2) and a uniform depth of 5 m. A baseline simulation was carried out with the initial boundary conditions as detailed. Water levels were extracted at a few observation points within the model to be used as observed data for the DUD model to compare additional model simulations where certain conditions of the FTI model were changed to reflect incorrect initial conditions which are very likely in the initial phase of setup for any numerical model before calibration or assimilation with actual observation data. OpenDA was able to obtain the correct result within 3 – 4 outer iterations. Table 8 shows the results obtained using OpenDA specifically the DUD model for the twin tests for a tidal model when either depth or tidal boundary conditions were modified. A few more detailed verification runs with OpenDA are currently being carried out but the initial results reported here suggests that OpenDA using DUD is suitable for carrying out the final stages of data assimilation as outlined in Steps 2 and 3 as described in Section 3.
Test Number of iterations Original BC Modified BC OpenDA results Modified tidal BC phase
(global) 3 Phase = 20.000 deg. Phase = 10.000 deg. Phase = 20.000 deg. Modified depth in only a sub
region of the model (local)
3 Depth = 5.000 m. Depth = 4.500 m Depth = 4.999 m
Table 8: Results from OpenDA Verification of DUD method
6. CONCLUSIONS
The results of the initial phase of data assimilation suggest that the phases obtained from Zu et al., 2008 are significant in improving the overall tidal prediction of the SRM with average MSVD improvements of 4.4%, 1.04% and 18.4% for the western, central and eastern regions respectively. The improvements within the centre region are smaller, which was expected given the initial model calibrations of the SRM for Singapore local waters. Results in general indicate that the overall accuracy of tidal predictions by the SRM can be further improved.
The results of the single parameter optimization are useful as they also provide a range of values for which the DUD method in OpenDA can be applied when used for a general multi-parameter optimization. The initial tests using OpenDA suggests that OpenDA and specifically the DUD method will be a useful tool for multi-parameter optimization for both the overall SRM prediction and for particular regions or stations within the SRM. The currently ongoing step is to combine the results of the single-parameter optimization with OpenDA to obtain a better calibrated overall tide model for the entire model domain.
