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Topographies A and B. Note that the abscissa and ordinate axes are labeled with zonal index and meridional index, respectively.
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Based on an isopycnic-coordinate internal tidal model with the adjoint method, the inversion of spatially varying vertical eddy viscosity coefficient (VEVC) is studied in two groups of numerical experiments. In Group One, the influences of independent point schemes (IPSs) exerting on parameter inversion are discussed. Results demonstrate that the V...
Citations
... coefficient (Zhang et al., 2015), and time-varying vertical eddy viscosity coefficients in oceanic Ekman layer models, and the spatially varying settling velocity (Zhang et al., 2018) in SSTMs. In several other studies, the CG (Alekseev et al., 2009;Navon and Legler, 1987) and L-BFGS methods (Jin et al., 2015;Zhang and Wang, 2014;Zou et al., 1993) had the best performance. Thus, they should be compared to select the best algorithm. ...
In this paper, an improved numerical scheme is developed to estimate the initial conditions (ICs) based on a three-dimensional suspended sediment transport model (3D SSTM) with adjoint data assimilation, and the method is then applied to Hangzhou Bay as an example. Specifically, the ICs are estimated by assimilating artificial observations in twin experiments and suspended sediment concentrations (SSCs) retrieved from the Geostationary Ocean Color Imager (GOCI) in practical experiments. In the twin experiments, the sensitivity of the estimated ICs to several influential factors is discussed. The results demonstrate that the conjugate descent algorithm of Fletcher is proven to be better than the steepest descent, finite memory BFGS, and five other conjugate gradient algorithms in estimating the ICs; the estimated ICs are sensitive to initial guess values, and appropriate initial values are necessary for improving the efficiency of convergence and obtaining good results. Additionally, the errors of observations can significantly influence the estimated results. In contrast, the estimated results are not very sensitive to cloud coverage, errors in the background flow field, and length of the assimilation time window. In practical experiments, according to the conclusions of twin experiments, an improved 3D SSTM with the adjoint method is developed for Hangzhou Bay, and the surface GOCI-retrieved SSCs during typical neap and spring tidal cycles are assimilated to estimate the practical ICs. The experimental results imply that with the present estimation method, more accurate ICs can be obtained, which indicates that the adjoint method is effective in the estimation of the ICs in SSTMs. Furthermore, this study verifies that accurate ICs are critical for the numerical modeling of SSCs on the tidal cycle scale. This study is not only useful for further improving the accuracy of ICs in SSTMs but also suggestive for the initialization schemes of other matter transport models.
... However, in many cases, the observed tides are modulated by some non-tidal processes, such as the changes in bed friction, surface slope, vegetation may alter tidal properties (Horsburgh and Wilson, 2007;Jay, 2009;Jay et al., 2010). This phenomenon has also been noted by some scholars, and related research has been conducted to improve the classical harmonic analysis (Jin et al., 2015;Liu et al., 2010;Ramp et al., 2010;Zhao et al., 2010;Xu et al., 2011;Zhang et al., 2011;Guo et al., 2012;Huan et al., 2012;Xu et al., 2013;Gao et al., 2015;Gao et al., 2017;Xu et al., 2016). Jin et al. (2018) proposed EHA to optimize the classical harmonic analysis. ...
The nodal cycle significantly affects the variation of sea level, which plays an important role in the design and plan of the coastal engineering, including coastal structures, harbor setting up and bay estuary engineering. Traditionally, the amplitude of the M2 nodal cycle is taken as constant in previous studies, but the amplitude of the M2 nodal cycle at San Francisco is nonstationary and temporally varying, because of the changes in physical properties of ocean and coastal morphology. Furthermore, the temporally varying amplitude of the M2 nodal cycle is difficult to be accurately extracted by using the conventional methods. In this study, a novel approach, namely enhanced harmonic analysis (EHA), is used to estimate the temporal variation in the amplitude of the M2 nodal cycle at San Francisco by analyzing the 116-year sea level observations. In the EHA, the independent point scheme and cubic spline interpolation are used. Considering the signal-to-noise ratios of the M2 nodal cycle and the long-term trend, the optimal number of the independent points in EHA is 10. The estimated results show that the 18.61-year nodal amplitude of the M2 tide have significantly temporal variations, with about 28-year and 42-year periods. In addition, the long-term trend of the M2 tidal amplitude is about 34.63 mm/century. Using the EHA with 10 independent points, the root-mean-square error between the observed and estimated monthly M2 tidal amplitude is 5.85 mm, which is decreased by 29.09% compared with that without considering the temporally varying amplitude of the M2 nodal cycle. The consideration of the temporal variations in the amplitudes of the M2 nodal cycle and the accurate extraction using EHA will have important significance for improving the precision of long-term sea level prediction and providing the helpful information for the design of the coastal engineering.
... Nevertheless, Zhang and Wang (2014) concluded that the efficiency of the L-BFGS was better than that of the GD for simple conditions in two-dimensional tidal models. Jin et al. (2015) stated that inversion results using the L-BFGS were closer to prescribed values than those using GD. However, Lu and Zhang (2006) and Zhang et al. (2011) found that GD was more efficient in the estimation of spatially varying bottom friction coefficient in a tidal model. ...
Temporal vertical eddy viscosity coefficient (VEVC) in an Ekman layer model is estimated using an adjoint method. Twin experiments are carried out to investigate the influences of several factors on inversion results, and the conclusions of twin experiments are 1) the adjoint method is a capable method to estimate different kinds of temporal distributions of VEVCs; 2) the gradient descent algorithm is better than CONMIN and L-BFGS for the present problem, although the posterior two algorithms perform better on convergence efficiency; 3) inversion results are sensitive to initial guesses; 4) the model is applicable to different wind conditions; 5) the inversion result with thick boundary layer depth (BLD) is slightly better than thin BLD; 6) inversion results are more sensitive to observations in upper layers than those in lower layers; 7) inversion results are still acceptable when data noise exists, indicating the method can sustain noise to a certain degree; 8) a regularization method is proved to be useful to improve the results for present problem; and 9) the present method can tolerate the existence of balance errors due to the imperfection of governing equations. The methodology is further validated in practical experiments where Ekman currents are derived from Bermuda Testbed Mooring data and assimilated. Modeled Ekman currents coincide well with observed ones, especially for upper layers. The results demonstrate that the assumptions of depth dependence and time dependence are equally important for VEVCs. The feasibility of the typical Ekman model, the imperfection of Ekman balance equations, and the deficiencies of the present method are discussed. This method provides a potential way to realize the time variations of VEVCs in ocean models.
... As a result, it will lead to continuous harmonic results directly rather than manual interpolations. The philosophy of the independent point scheme (IPS) has been put forward in some studies to deal with the ill-posedness of the inverse problem, for 1D (Zhang and Lu 2010;Gao et al. 2013) and 2D parameters (Lu and Zhang 2006;Jin et al. 2015). Readers who are interested in this issue are referred to Pan et al. (2017) and Guo et al. (2017) for more details. ...
As an effective tool to distinguish different tidal components, classical tidal current harmonic analysis has been widely used to obtain harmonic parameters of internal tidal currents. However, harmonic parameters cannot exactly reveal the motion of internal tides as the irregular temporal variations for internal tides are significant in many regions of the world’s oceans. An enhanced harmonic analysis (EHA) algorithm based on the independent point scheme and cubic spline interpolation is presented in this paper to obtain harmonic parameters with temporal variations for different tidal constituents of internal tides. Moreover, this algorithm is applied to analyze 14-month long current data obtained from a mooring located on the continental shelf of northeastern South China Sea. The obvious irregular temporal variations for the four principle constituents (M2, K1, S2 and O1) of internal tides in this region are indicated. Hopefully, this algorithm might present a brand-new view for researchers to investigate the irregular temporal motions of internal tides.
... (3) 以流速观测值和模拟值的差驱动反向模型, 对猜测值进行优化; [37][38][39] ; 不过, 也有研究认为 GD 法反演的结果 更好 [40][41] 可能导致对一部分地转流速与 Ekman 流速发生混叠, 造成对 Ekman 流速的错误估计 [52][53] 。为此, 我们参 考 Roach 等的做法, 将地转剪切也考虑进来, 把地转 流速拆分为两部分 [54] : ...
... The important factors that affect the result of IPS are the selection of independent points and interpolation format. Studies, such as [24,25], have already coupled IPs distribution scheme with Cressman interpolation aiming at capturing more accurate results in the solutions and parameter inversions. Obviously, the optimal IPS distribution scheme matched with different interpolation should be not necessarily the same, due to the different interpolation effect. ...
We propose to apply Piecewise Parabolic Method (PPM), a high order and conservative interpolation, for the parameters estimation in a PM 2.5 transport adjoint model. Numerical experiments are taken to show the accuracy of PPM in space and its ability to increase the well-posedness of the inverse problem. Based on the obtained results, the PPM provides better interpolation quality by employing much fewer independent points. Meanwhile, this method is still well-behaved in the case of insufficient observations. In twin experiments, two prescribed parameters, including the initial condition (IC) and the source and sink (SS), are successfully estimated by the PPM with lower interpolation errors than the Cressman interpolation. In practical experiments, simulation results show good agreement with the observations of the period when the 21th APEC summit took place.
... Several studies have been done regarding the application of the L-BFGS method in adjoint models Jin et al., 2015). Because of the high complexity of cost functions in the control variable space and limitation of observations, sometimes the L-BFGS method fails to be a relatively better gradient related method than the steepest decent method . ...
To simulate the dynamic process of total nitrogen (TN) in seas, numerical modelling combined with the adjoint method is implemented in this study. Because nonpoint source terms (ST) and initial values (IV) of TN are essential but difficult to determine, the adjoint method was applied to a numerical model, and the ST and IV terms of TN were inverted via routine monitoring data in the Bohai Sea. In twin experiments, the adjoint method was capable of inverting the prescribed spatio-temporally distributed ST and the spatial distributed IV. In practical experiments, the results demonstrated that the simulation precision with ST inversion was higher than that with IV inversion and was accurate with joint initial values and source term (IST) inversion. This result indicates that nonpoint source TN is essential for the simulation of TN concentration. Furthermore, the simulated results indicate that the pollution in three bays of the Bohai Sea is rather severe. The model in this study is not specific to the Bohai Sea and can be generalized to other areas, such as the Beibu Gulf. These findings may assist in the development of cost-effective controls for accidental or planned industrial pollutant releases into coastal waters.
... on the BFGS algorithm described by Liu and Nocedal [51] coupled with a quadratic interpolation line search [52]. Generally speaking, this latter has proved to be particularly effective in these recent years in the area of inversion (see535455 to name but a few) due to its effectiveness in minimizing nonlinear problems [56]. The two-dimensional domain is a square of 2 cm length. ...
Diffuse optical tomography problems rely on the solution of an optimization problem for which the dimension of the parameter space is usually large. Thus, gradient-type optimizers are likely to be used, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, along with the adjoint-state method to compute the cost function gradient. Usually, the
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-inner product is chosen within the extraction procedure (i.e., in the definition of the relationship between the cost function gradient and the directional derivative of the cost function) while alternative inner products that act as regularization can be used. This paper presents some results based on space-dependent Sobolev inner products and shows that this method acts as an efficient low-pass filter on the cost function gradient. Numerical results indicate that the use of Sobolev gradients can be particularly attractive in the context of inverse problems, particularly because of the simplicity of this regularization, since a single additional diffusion equation is to be solved, and also because the quality of the solution is smoothly varying with respect to the regularization parameter.
