The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems: Part III – Observation impact and observation sensitivity in the California Current System

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Abstract

The critical role played by observations during ocean data assimilation was explored when the Regional Ocean Modeling System (ROMS) 4-dimensional variational (4D-Var) data assimilation system was applied sequentially to the California Current circulation. The adjoint of the 4D-Var gain matrix was used to quantify the impact of individual observations and observation platforms on different aspects of the 4D-Var circulation estimates during both analysis and subsequent forecast cycles. In this study we focus on the alongshore and cross-shore transport of the California Current System associated with wind-induced coastal upwelling along the central California coast. The majority of the observations available during any given analysis cycle are from satellite platforms in the form of SST and SSH, and on average these data exert the largest controlling influence on the analysis increments and forecast skill of coastal transport. However, subsurface in situ observations from Argo floats, CTDs, XBTs and tagged marine mammals often have a considerable impact on analyses and forecasts of coastal transport, even though these observations represent a relatively small fraction of the available data at any particular time.

During 4D-Var the observations are used to correct for uncertainties in the model control variables, namely the initial conditions, surface forcing, and open boundary conditions. It is found that correcting for uncertainties in both the initial conditions and surface forcing has the largest impact on the analysis increments in alongshore transport, while the cross-shore transport is controlled mainly by the surface forcing. The memory of the circulation associated with the control variable increments was also explored in relation to 7 day forecasts of the coastal circulation. Despite the importance of correcting for surface forcing uncertainties during analysis cycles, the coastal transport during forecast cycles initialized from the analyses has less memory of the surface forcing corrections, and is controlled primarily by the analysis initial conditions.

Using the adjoint of the entire 4D-Var system we have also explored the sensitivity of the coastal transport to changes in the observations and the observation array. A single integration of the adjoint of 4D-Var can be used to predict the change that occurs when observations from different platforms are omitted from the 4D-Var analysis. Thus observing system experiments can be performed for each data assimilation cycle at a fraction of the computational cost that would be required to repeat the 4D-Var analyses when observations are withheld. This is the third part of a three part series describing the ROMS 4D-Var systems.

Highlights

► The impact of observations on California Current circulation estimates is assessed. ► All observations are found to be important in controlling coastal transports. ► In situ observations have significant impact despite their relatively small number. ► The adjoint of 4D-Var can be used reliably for observing system experiments. ► The impact of uncertainties in the observations on coastal transports is assessed.

Introduction

Critical components of any ocean data assimilation system are the observations, and a rigorous quantitative assessment of the impact of the observations on ocean circulation estimates derived from data assimilation should form an essential part of all modern data assimilation systems. In meteorology, the continuous monitoring of observation impacts on atmospheric analyses and forecasts has become common-place, and has demonstrated the importance of such activities both for data quality control and for assessing the efficiency of current observational networks and satellite platforms (e.g. Cardinali, 2009). With the rapid expansion of ocean observing systems in recent years, similar monitoring efforts will doubtless prove valuable in oceanography.

In Moore et al. (2011), hereafter referred to as Part I, we described in detail a suite of 4-dimensional variational (4D-Var) data assimilation systems that have been developed for the Regional Ocean Modeling System (ROMS), a widely used community ocean model (Haidvogel et al., 2008). The performance of the ROMS 4D-Var system was demonstrated in Moore et al. (2011), hereafter referred to as Part II, in connection with the California Current System (CCS). In this third companion paper, we will demonstrate two additional capabilities of the ROMS 4D-Var system, namely observation impact and observation sensitivity. Observation impact calculations can be used to quantify the contribution of each individual datum to the difference between the background (or prior) and the analysis (or posterior) of some aspect of the ocean circulation. Observation sensitivity calculations, on the other hand, quantify the change that will occur in the circulation estimate as a result of changes in the observations or observation array. In addition to routine monitoring, observation impact calculations can provide quantitative information about systematic errors in the model in places or regions where the model is unable to reliably fit the observations. Similarly, observation impact calculations can demonstrate the influence of initialization shocks on the circulation that are associated with each individual datum. Observation sensitivity calculations, on the other hand, provide important information about the influence of data error and uncertainties on the circulation estimates, and are an efficient tool for observing system experiments (OSEs), offering also the potential for the design of observation arrays and adaptive sampling strategies.

The observation impact and observation sensitivity calculations presented here are based on the adjoint approach, and utilize the property of adjoint operators for identifying the subspace of the model state-vector that is activated by the observations. While adjoint-based methods have been used previously in oceanography in attempts to identify optimal observing locations and observation types (e.g. Köhl and Stammer, 2004, Zhang et al., 2010), our focus here is on the impact of existing observations on estimates of the ocean circulation, and we follow the approach originally developed in meteorology by Langland and Baker (2004) in support of numerical weather prediction, further developed by Daescu, 2008, Zhu and Gelaro, 2008, Gelaro and Zhu, 2009.

Our primary focus is the circulation in the vicinity of the central California coast which is characterized by a pronounced seasonal cycle of upwelling that is driven both by equatorward, upwelling favorable winds, and wind stress curl farther offshore. Coastal upwelling is generally most intense during the spring and early summer in this region, and is accompanied by pronounced changes in the ocean circulation (see Checkley and Barth (2009) for a recent review). Good dynamical indicators of the seasonal circulation changes are the alongshore and cross-shore transport near the coast, and we use these indicators here to demonstrate how the available observations collude to modify the coastal circulation via data assimilation during different times of the year.

A brief summary of ROMS 4D-Var and the notation used throughout is presented in Section 2, while a more complete description of the system can be found in Parts I and II. Since the ROMS and 4D-Var configurations used in the experiments presented here are also described in detail in Part II, only a brief overview will be given in Section 3. The impact of uncertainties in the different components of the 4D-Var control vector on coastal transport is analyzed in Section 4 when ROMS is run in both an analysis and forecast mode. Section 5 deals with the impact of each observation on the coastal transports during analysis-forecast cycles. The transport sensitivity to changes in the observations and observing array is explored in Section 6 where we also demonstrate an alternate approach to OSEs. During conventional OSEs, a sequence of data assimilation cycles is run in which selected observations are withheld, with the result that both the prior and posterior state estimates change during each cycle compared to the estimates obtained when all observations are assimilated. Conversely, in the approach considered here, the prior state estimate remains unchanged for each cycle, and the impact of withholding observations is computed by considering perturbations to the observations using the adjoint of the data assimilation system. A summary of important results and conclusions is presented in Section 7.

Section snippets

ROMS 4D-Var: a summary and notation

Following the notation introduced in Part I, we denote by x the ROMS state-vector comprised of all the grid point values of temperature T, salinity S, velocity (u, v), and free surface height ζ.1 Data assimilation seeks to combine in an optimal way a background (or prior) estimate of the circulation, xb(t), with observations which are arranged in the vector yo. The background circulation is a solution of ROMS

Model and 4D-Var configuration

The configuration of ROMS and 4D-Var used in the calculations reported below is described in detail in Part II and by Broquet et al., 2009a, Broquet et al., 2009b, Broquet et al., 2011, so only a brief description will be given here.

The ROMS CCS domain spans the region 134°W to 116°W and 31°N to 48°N, with 10 km horizontal resolution and 42 terrain following σ-levels in the vertical. This is the model referred to in Part II as WC10, and for consistency we shall use the same notation here also.

Sequential strong constraint I4D-Var

To demonstrate the observation impact capabilities of ROMS, I4D-Var was run sequentially for the period July 2002–December 2004. Because of the uncertainty in assigning model error, all experiments presented here assume the strong constraint, although there is nothing to preclude the same computations using the weak constraint instead. In all cases, the control vector δz was comprised of increments to the initial conditions, surface forcing, and open boundary conditions. The background initial

Analysis-forecast cycle observation impacts

As described in Part I (Section 7.2), the impact of each observation on ΔJ in (4) can also be computed during analysis and forecast cycles. Specifically, for the case of a single outer-loop, the change in J(t) due to assimilating the observations is given to first-order by J(xa(t))-J(xb(t))=ΔJ(t)dTKTMbTJ/x|xb, where MbTJ/x|xb denotes the time convolution of J/x|xb with the adjoint of ROMS linearized about the prior circulation; d = y  H(xb(t)) is the innovation vector; and K˜ is the

Observation sensitivity

As described in Part I (Section 7.3), the analysis (posterior) control vector zˆ can be expressed as:zˆ=zb+K(d)where K(d) represents the entire data assimilation procedure, and is a nonlinear function of the innovation vector d by virtue of the conjugate gradient method used to identify the cost function minimum. In addition, K(d)K˜d, and the practical gain matrix K is also a function of d according to the Lanczos vector formulation of the conjugate gradient method that is employed in ROMS

Summary and conclusions

Sequential application of ROMS 4D-Var to the California Current System demonstrates that data assimilation yields substantial changes in the background ocean circulation estimates. The contribution of the different components of the increment control vector and each observation platform to these changes can be assigned using methods established in numerical weather prediction to quantify observation impacts. In addition, the sensitivity of the circulation estimates to changes in the

Acknowledgements

We are grateful for the continued support of the Office of Naval Research (N00014-01-1-0209, N00014-06-1-0406, N00014-08-1-0556, N00014-10-1-0322), and for support from the National Science Foundation (OCE-0628690, OCE-0121176, OCE-0121506) and the National Ocean Partnership Program (NA05NOS4731242). Any opinions, findings, and conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the National Science Foundation. J. Doyle acknowledges

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