Patent Application
20240118076
The present disclosure is related to ocean forecasting, and more specifically to, but not limited to, the use of ocean acoustic pressure observations for ocean model analysis and forecasting.
Previous methods have focused on two distinct approaches: acoustic tomography and variational retrievals. The use of acoustic tomography for measuring ocean properties was postulated by Munk and Wunsch (1979) and examined in numerous works since then. The basic idea behind acoustic tomography is to conduct an inversion for environmental sound speed by minimizing the difference between modeled and measured acoustic travel times. Cornuelle et al. (1985) conducted an experiment comparing the tomography-derived sound speed environments to in-situ measurements and found strong correlation between them. Skarsoulis and Send (2000) examined the use of acoustic tomography in the presence of strong nonlinear dependency between sound speed and acoustic travel time variations. And a more recent study by Dushaw (2019) examined the impact of ocean acoustic tomography in combination with Argo floats. This method has the disadvantage that it is computationally intensive and can provide inaccurate final results.
The other method, variational retrievals, relies on acoustic pressure data assimilation. Acoustic pressure assimilation is different than acoustic tomography in that rather than examining the difference in acoustic travel times between modeled and measured values, acoustic pressure assimilation involves differencing the recorded pressure (as captured by hydrophones) with modeled values. Typically, acoustic pressure is modeled using some form of a parabolic equation based approach, such as that used in the Navy's Range-dependent Acoustic Model (RAM; Collins et al., 1996). These models use information regarding the sediment conditions, the surface wind and wave state, and the environmental sound speed. The use of variational techniques in assimilating acoustic pressure observations has been examined in previous studies. Most notably by Hursky et al. (2004) where the tangent linear and adjoint of a parabolic equation model is derived and used to assimilate acoustic pressure measurements in order to adjust the assumed sound speed profile. This method aims to produce a retrieval of environmental temperature and salinity profiles, which in turn can be assimilated by an ocean model in order to update the ocean state estimate. While a better approach than tomography, this method has the disadvantage of a double-assimilation, where the assumed errors in the observations are counted twice, once during the retrieval and again during the assimilation of the retrieved profile. This can substantially reduce the impact of the acoustic pressure observation on the ocean model analysis.
A very recent effort by Storto et al. (2021) aimed to update the ocean model directly via the variational assimilation of acoustic pressure observations. In their approach they utilized an observation operator built into a three-dimensional variational (3DVAR) ocean data assimilation system in order update the ocean model analysis with acoustic pressure observations. Their observation operator is based on an artificial intelligence method known as a neural network (NN). This method, while functional, has the disadvantage of the limitations of the training method/period, is vulnerable to large error in cases that deviate from the training set, and result in poor improvement to the ocean model temperature and salinity fields.
This summary is intended to introduce, in simplified form, a selection of concepts that are further described in the Detailed Description. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter. Instead, it is merely presented as a brief overview of the subject matter described and claimed herein.
Disclosed aspects provide the ability to use measured acoustic pressure observations to update an ocean model analysis (three-dimensional temperature, salinity, pressure fields, and/or the like) for the purposes of state estimation and/or ocean forecasting via advanced data assimilation systems and methods in accordance with one or more disclosed aspects, which can be shown in
An ocean analysis equation, forward acoustic model, and adjoint acoustic model can be shown in
The advantages one or more disclosed embodiments include:
This allows for more impact from the acoustic pressure observations on the final ocean state analysis.
For variational assimilation, the tangent linear and adjoint operators must be derived from the original nonlinear model. The RAM tangent linear and adjoint operators are then interfaced with the larger NCOM-4DVAR or 3DVAR systems (Ngodock and Carrier, 2014a; Ngodock and Carrier, 2014b) as observations operators. NCOM-4DVAR is a weak-constraint analysis system based on the indirect representer method of Bennett (1992, 2002) and Chua and Bennett (2001). The system is derived from the incremental formulation of the variational cost function (Courtier 1997), where the time dimension is omitted here for simplicity:
where δx is the increment to the state variable, B is the background error covariance, H is the observation operator, and R is the observation error covariance. d is the set of innovations defined as:
d=y−Hx
b
where y is the observation set and xb is the model background. When taking the derivative
and setting it equal to zero one can find the minimum of the cost function. After some arithmetic, the analysis equation can be derived from this equality as
δxa=BHT(HBHT+R)−1d
For 4DVAR, the background error covariance can be expanded to include the operation of
the tangent linear and adjoint operators to form a four-dimensional background error covariance such that this equation is expanded to:
δxa=MΣCΣTMTHT(HMΣCΣTMTHT+R)−1d
where M is the tangent linear ocean model operator, MT is the adjoint model, and ΣCΣT is a static error correlation (C) symmetrically multiplied by the error standard deviation (Σ) that describes the initial condition error or the model error. The RAM tangent linear and adjoint operators are linked to NCOM-4DVAR as additional observation operators contained within H and HT. In this way the ocean variables of temperature and salinity can be mapped to the acoustic observation space (to be compared to acoustic observations) via H, and acoustic observation information can be passed to the ocean adjoint model via HT. In addition to the tangent linear and adjoint models of RAM, the H and HT operators also contain the tangent linear and adjoint of the sound speed equation. In this case, the sound speed equation used is known in the community as “Chen-Millero-Li” and is described collectively in Chen and Millero (1977) and Millero and Li (1994).
Typically, the inversion in the above equation is solved using a linear solution method such as one of the conjugate gradient approaches. This requires multiple applications of the tangent linear and adjoint model over a number of iterations until the conjugate gradient converges to a solution. Depending on the size of the problem (model domain size, horizontal and vertical resolution, and length of the assimilation window) the 4DVAR can be computationally expensive. The above equation can be simplified by reducing the tangent linear and adjoint of the ocean model to identity matrices such that it becomes a 3DVAR method:
δxa=ΣCΣTMTHT(HΣCΣTHT+R)−1(y−Hxb)
The inverse in this equation is faster to solve as the covariance ΣCΣT can be applied in an efficient manner. In the case of the data assimilation algorithms used in this demonstration, the correlation matrix, C, is provided by an operator based on the implicit solution to a diffusion equation (Weaver and Courtier, 2001; Carrier and Ngodock, 2010) and is computationally efficient. There is no account for the time variability of the observations on the solution in this equation, however, as that is a necessary consequence of using a simpler method than 4DVAR. The dynamical balance relationships are also missing in this equation, but they can be approximated by adding additional operators. Following the work of Weaver et al. (2005), a linearized “balance operator” can be constructed based on geostrophy and hydrostatics. Doing so would alter this equation so that the embedded background error covariance would have the additional operators:
δxa=KbΣCΣTKbTHT(HKbΣCΣTKbTHT+R)−1(y−Hxb)
where Kb is the linear balance operator (and KbT its adjoint). This equation is the base form of the DVAR used in this approach, with the software derived from the larger 4DVAR system. These models can be applied using an Observing System Simulation Experiment (OSSE) methodology.
The present disclosure provides for a method of determining ocean state. The method may include receiving, by a processing device, data associated with a prior ocean forecast state, and receiving, by the processing device, data associated with a first set of ocean temperature and salinity observations. The method may include receiving, by the processing device, data associated with a first set of ocean acoustic pressure observations. The method may include determining, by the processing device, a correction to the prior ocean forecast state based on a forward acoustic model, on an adjoint acoustic model, on the data associated with a first set of ocean temperature and ocean salinity observations, and on the data associated with a first set of ocean acoustic pressure observations, and generating, by the processing device, a current ocean state based on the determined correction.
The present disclosure provides for a system for determining an ocean state. The system may include a processing device, and a memory device operably coupled to the processing device, the memory device storing computer-readable instructions that, when executed, cause the processing device to perform a method. The method may include receiving data associated with a prior ocean forecast state, and receiving data associated with a first set of ocean temperature and salinity observations. The method may include receiving data associated with a first set of ocean acoustic pressure observations. The method may include determining a correction to the prior ocean forecast state based on a forward acoustic model, an adjoint acoustic model, the data associated with a first set of ocean temperature and ocean salinity observations, and the data associated with a first set of ocean acoustic pressure observations, and generating a current ocean state based on the determined correction.
The present disclosure provides for a non-transitory computer readable medium comprising computer-readable instructions, the computer-readable instructions, when executed, cause a processing device to perform a method. The method may include receiving data associated with a prior ocean forecast state, and receiving data associated with a first set of ocean temperature and salinity observations. The method may include receiving data associated with a first set of ocean acoustic pressure observations. The method may include determining a correction to the prior ocean forecast state based on a forward acoustic model, an adjoint acoustic model, the data associated with a first set of ocean temperature and ocean salinity observations, and the data associated with a first set of ocean acoustic pressure observations, and generating a current ocean state based on the determined correction.
The aspects and features of the present aspects summarized above can be embodied in various forms. The following description shows, by way of illustration, combinations and configurations in which the aspects and features can be put into practice. It is understood that the described aspects, features, and/or embodiments are merely examples, and that one skilled in the art may utilize other aspects, features, and/or embodiments or make structural and functional modifications without departing from the scope of the present disclosure.
One or more aspects described herein may be used to facilitate the use of ocean acoustic pressure observations for ocean model analysis and forecasting.
Disclosed embodiments provide for one or more aspects to improve the ocean state estimation and model forecast via the assimilation of acoustic observations by using the adjoint and tangent linear of the Navy's operational acoustic model, the Range-dependent Acoustic Model (RAM) and integrating them as observation operators within the larger Navy Coastal Ocean Model (NCOM) four dimensional variational (4DVAR) and 3DVAR assimilation systems.
Accurate ocean state estimation is vital to producing useful ocean model predictions for both near and long-term forecasts. These ocean state estimates, or analyses, rely on both ocean observations and advanced data assimilation methods, such as the four-dimensional variational (4DVAR) technique. Ocean observations, however, are scarce, especially in the sub-surface. Most in-situ sub-surface ocean measurements are obtained through the Argo profiling float program (Roemmich et al., 2001) with nearly 4,000 floats as of June 2022, and localized short-term glider deployments. The remaining observations are mainly surface in nature, i.e. sea surface temperature (SST) and height (SSH) as measured from satellites. There exists another sub-surface ocean measurement type that is not widely used in the ocean modeling community through data assimilation, e.g. the ocean acoustic pressure observation. Acoustic pressure observations are obtained via hydrophone and can potentially provide useful information regarding the ocean environment, specifically temperature and salinity; this is because sound propagation through the ocean is sensitive to these ocean variables through sound speed. Acoustic pressure observations are particularly powerful because these act not as pointwise measurements, but as an integrated measurement over distance (potentially tens of kilometers). And, if the sound source is a moving target (such as a ship of opportunity) the area of the ocean that can be observed using one vertical line array of hydrophones could be expansive.
One or more aspects described herein detail the work accomplished under R&D from software development, to sensitivity experiments, up to ocean data assimilation tests. One or more aspects described herein can be organized as follows: (1) how the acoustic data assimilation components are coupled to the larger ocean data assimilation software and a discussion of a series of sensitivity studies and their potential impact on real data assimilation experiments; (2) a description of an ocean data assimilation experiment where acoustic pressure observations are used to update the ocean model analysis and short-term forecast; and (3) a comparison of two competing methods of data assimilation when utilizing acoustic pressure observations.
Aspects described here are organized in the following manner: section 2 provides a description of the coupled ocean-acoustic adjoint model; section 3 provides a description regarding the adjoint sensitivity experiment setup; section 4 provides a description examining the results of the experiments; and a concluding analysis follow in section 5.
Operational ocean modeling relies heavily upon the assimilation of observations to make reasonably accurate forecasts. The ocean is significantly under-sampled, especially in the sub-surface where the observations are limited to an array of profiling floats (Roemmich et al., 2001) or gliders. There is one observation that is taken, however, that can provide sub-surface information regarding the temperature and salinity that is not routinely used in ocean modeling: ocean acoustic observations. These observations are collected in the form of acoustic pressure via hydrophones, typically arranged along vertical lines from the surface down to some depth. The modeling of acoustic pressure relies on environmental information in the form of sound speed profiles (along with sediment conditions and surface wind and wave state) in order to simulate the propagation of acoustic energy through the ocean. Using inverse methods, one can use an acoustic model and observations to recover information regarding the ocean environment and use this information to update the ocean model simulation. Prior to conducting such experiments, however, it is important to characterize the sensitivity of the acoustic pressure simulation to the environmental input; this helps to properly design the data assimilation system and provide guidance on the expected results and impacts. Traditionally, sensitivity information had been obtained in a time-consuming manner where one perturbs the model initial condition variable-by-variable and grid point-by-grid point while recording the changes to the model solution after each run. A more computationally efficient approach is to use the adjoint of the model in question in order to determine the sensitivity of the output to each input variable.
The adjoint of a model is the linear transposition of its tangent linear counterpart; and the tangent linear model is found by linearizing the nonlinear model around its background state (Li et al., 1993; Ngodock et al., 2017). Whereas the tangent linear model describes the evolution of an initial perturbation to the nonlinear model over time and space, the adjoint integrates backwards providing the initial perturbation required to produce the final perturbed state that the adjoint is initialized with. In doing so, the adjoint determines the input variables that the final solution is most sensitive to. In the case of the ocean-acoustics, this inverse method can be used to determine which ocean variables, locations, and times the acoustic propagation is most sensitive to (for a given source depth, location, and frequency). In this present effort, the adjoint of an acoustic model has been developed in order to examine the sensitivity of acoustic pressure to environmental profiles provided by the ocean forecast model.
Similar adjoint-based work with ocean acoustic models has been done within the community in the past. Hermand et al. (2006) used the adjoint of a similar parabolic equation model to examine geoacoustic inversion problems, as did Le Gac et al. (2004). Thode and Kim (2004) used the adjoint model to examine the derivatives of a waveguide field with respect to sound speed, density, and frequency. Meyer and Hermand (2005) used the adjoint method with a wide-angle parabolic equation model to invert for the bottom properties in their model. Finally, Li et al. (2014) used the adjoint method to examine the inversion of an internal wave-perturbed sound speed field through acoustic data assimilation (with both sound speed and acoustic pressure observations). This present work is distinct, however, in that the adjoint of the US Navy's acoustic model, the Range-dependent Acoustic Model (RAM; Collins et al., 1996), is coupled directly to the adjoint of the Navy Coastal Ocean Model (NCOM; Barron et al., 2006), effectively making the adjoint of RAM an “observation operator” of the adjoint of NCOM. In this way, the coupled adjoint sensitivity can be examined to the input variables of temperature and salinity through space and time.
The adjoint sensitivity of acoustic pressure to the ocean model temperature and salinity is examined for a Mid-Atlantic Bight NCOM domain for four source locations, each at a depth of 50 m and frequencies of 50 and 300 Hz (for short ranges, ˜2000 m); a short examination of a mid-range (12 km) and a long-range (30 km) case is provided as well. This work provides some key generalizations that have implications for both adaptive ocean sampling (to aid accurate acoustic modeling) and acoustic observation assimilation via a coupled ocean-acoustic assimilation system.
RAM is a parabolic equation model solved using finite difference methods as a split-step Padé algorithm (Collins et al., 1996). The model is solved on a two-dimensional range-depth plane, with r as the horizontal range from some source and z as the depth from the surface of the ocean. From Collins et al. (1996), the spreading factor r−1/2 is removed from the complex pressure field, p, and the derivation of RAM begins with the far-field version of the reduced wave equation,
where k is the wavenumber. Factoring the operator in Eq. (1) gives
where ko=ω/co(ω is the angular frequency and co is a representative phase speed) and the operator X is given by
If the outgoing energy dominates the backscattered energy, Eq. (2) becomes
Then, the formal solution of Eq. (4) is given by
p(r+αr,z)=exp(ikoΔr(I+X)1/2)p(r,z) (5)
where Δr is the range step. Per Collins et al. (1996), the exponential of the operator square root that appears in Eq. (5) is replaced by a rational approximation to obtain the final split-step Padé algorithm,
where I is the identity operator and αj,n and βj,n are precomputed coefficients of the split-step Padé algorithm for solving the original wave equation implicitly.
For variational assimilation, the tangent linear and adjoint operators must be derived from
the original nonlinear model; this forms the basis operators for the Variational Range-dependent Acoustic Model (VRAM) data assimilation system. In the case of RAM, the algorithm is linear in terms of acoustic pressure (p), but nonlinear in the differential operator, X, which depends upon the wavenumber k in a nonlinear fashion (Ngodock et al., 2017). The original VRAM tangent linear and adjoint operators were derived directly from the tangent linear and adjoint of Eq. (6), see Ngodock et al. (2017). VRAM was recently updated, however, to take advantage of the wrapper software provided by the NSPE program suite, which handles setting various aspects of the acoustic modeling. The adjoint and tangent linear operators of this new version of VRAM uses the adjoint of finite difference method (AFD) to derive the operators directly from the numerical code of RAM itself. The new adjoint of RAM still allows for updating the sound speed profile via the inversion of acoustic pressure as in Ngodock et al. (2017). In order to invert this to environmental temperature and salinity, the tangent linear and adjoint of the sound speed equation was derived and linked to VRAM. In this case, the sound speed equation used is known in the community as “Chen-Millero-Li” and is described collectively in Chen and Millero (1977) and Millero and Li (1994).
With this inversion to temperature and salinity in place, the VRAM tangent linear and adjoint operators are able to be interfaced with the larger NCOM Four-Dimensional Variational (4DVAR) system (Ngodock and Carrier, 2014a; Ngodock and Carrier, 2014b) as observations operators. NCOM-4DVAR is a weak-constraint analysis system based on the indirect representer method of Bennett (1992, 2002) and Chua and Bennett (2001). The system is derived from the incremental formulation of the variational cost function (Courtier 1997), where the time dimension is omitted here for simplicity:
where δx is the increment to the state variable, B is the background error covariance, H is the observation operator, and R is the observation error covariance. d is the set of innovations defined as
d=y−Hx
b (8)
where y is the observation set and xb is the model background. When taking the derivative of Eq. (7) and setting it equal to zero one can find the minimum of the cost function. After some arithmetic, the analysis equation can be derived from this equality as
δxa=BHT(HBHT+R)−1d. (9)
For 4DVAR, the background error covariance in Eq. (9) can be expanded to include the operation of the tangent linear and adjoint operators to form a four-dimensional background error covariance such that Eq. (9) is expanded to:
δxa=MΣCΣTMTHT(HMΣCΣTMTHT+R)−1d (10)
where M is the tangent linear ocean model operator, MT is the adjoint model, and ΣCΣT is a static error correlation (C) symmetrically multiplied by the error standard deviation (Σ) that describes the initial condition error or the model error. The VRAM tangent linear and adjoint operators are linked to NCOM-4DVAR as additional observation operators contained within H and HT. In this way the ocean variables of temperature and salinity can be mapped to the acoustic observation space (to be compared to acoustic observations) via H, and acoustic observation information can be passed to the ocean adjoint model via HT. In this present work, no observation assimilation is done; rather the coupled NCOM/VRAM adjoint operator is used to examine the coupled adjoint sensitivity and its implications on data assimilation and adaptive sampling.
The purpose of the sensitivity study is to determine, for low frequency (50 Hz) and mid-frequency (300 Hz), at what vertical levels the acoustic propagation is most sensitive to the ocean environment temperature and salinity for both long- and short-range cases. This could be determined with VRAM alone; however, in order to quantify the sensitivity to input temperature and salinity either at the location of the environmental profile used by RAM or within the larger space/time ocean model domain, then a fully-coupled adjoint model is required. This sensitivity study also provides insight as to what ocean variables, locations, and times the acoustic propagation is most sensitive to within the larger ocean model domain. This type of information may be helpful in designing an adaptive ocean observation sampling scheme for future applications.
For this study a mid-Atlantic Bight NCOM domain has been configured (
Four short-range acoustic source locations are set within the model domain along the continental shelf break. Each source location has a depth of 50 m with one radial each extending due East for about 2000 m. An additional source location is configured for use as both a mid-range and long-range adjoint sensitivity location. For the long-range radial, the source depth is 50 m and the radial extends due South for 30 km. The NCOM domain with surface temperature on 1 May, 2021 along with the acoustic source locations are shown in
Prior to examining the adjoint sensitivity results, one can look first at the background state of the model.
The sensitivity to the input temperature and salinity profiles can also be examined. Here, due to the range-independent nature of the short-range configuration, the acoustic adjoint sensitivity is integrated back to the source location where the adjoint model produces the adjoint sensitivity of acoustic pressure to sound speed. This is further propagated to sensitivity to temperature and salinity via the adjoint of the linearized sound speed equation (Ngodock et al., 2017).
The spatial extent of the adjoint sensitivity of acoustic pressure to the ocean model temperature and salinity can be examined by mapping the sensitivity (in temperature and salinity) across the NCOM domain at some representative depths, e.g. the surface and 50 m.
The behavior of the acoustic pressure and its sensitivity to acoustic pressure in range as well as temperature and salinity for long-range cases can be explored. To examine this, a separate coupled ocean-acoustic adjoint model is run using the aforementioned fifth source location (furthest East position in
Examining the acoustic pressure sensitivity to temperature and salinity at selected locations in range will help to determine if there is a strong range and depth dependence in these fields as well.
There is a fair amount of research that has been devoted over the years to utilizing ocean acoustic measurements, of one form or another, to gather information on ocean temperature and salinity. A great deal of this work has focused on acoustic tomography. The use of acoustic tomography for measuring ocean properties was postulated by Munk and Wunsch (1979) and examined in numerous works since then. The basic idea behind acoustic tomography is to conduct an inversion for environmental sound speed by minimizing the difference between modeled and measured acoustic travel times. Cornuelle et al. (1985) conducted an experiment comparing the tomography-derived sound speed environments to in-situ measurements and found strong correlation between them. Skarsoulis and Send (2000) examined the use of acoustic tomography in the presence of strong nonlinear dependency between sound speed and acoustic travel time variations. And a more recent study by Dushaw (2019) examined the impact of ocean acoustic tomography in combination with Argo floats. Dushaw found that each observing type, when utilized in isolation, is capable of reducing the uncertainty in large-scale monthly average temperature by about 50%; however, when used together the uncertainty is reduced by about 75%.
Acoustic pressure assimilation, however, is different than acoustic tomography in that rather than examining the difference in acoustic travel times between modeled and measured values, acoustic pressure assimilation involves differencing the recorded pressure (as captured by hydrophones) with modeled values. Typically, acoustic pressure is modeled using some form of a parabolic equation-based approach, such as that used in the Navy's Range-dependent Acoustic Model (RAM; Collins et al., 1996). These models use information regarding the sediment conditions, the surface wind and wave state, and the environmental sound speed. The use of variational techniques in assimilating acoustic pressure observations has been examined in previous studies. Most notably by Hursky et al. (2004) where the tangent linear and adjoint of a parabolic equation model is derived and used to assimilate acoustic pressure measurements in order to adjust the assumed sound speed profile. And yet other published studies examine various aspects of acoustic pressure assimilation (Hermand et al., 2006; Le Gac et al., 2004; Charpentier and Roux, 2004). A more recent study by Storto et al. (2021) used a neural-network based observation operator in order to perform coupled ocean-acoustic data assimilation with simulated observations. They found that both methods employed (canonical correlation analysis and neural-networks) were able to improve the skill score of the assimilative model. This present work takes a similar approach where the goal is to improve the ocean state estimation and short-term model forecast via the assimilation of acoustic observations; the novelty is that the adjoint and tangent linear of RAM (referred to as Variational RAM, or VRAM) are integrated as observation operators within the larger Navy Coastal Ocean Model (NCOM) 4DVAR assimilation system. With that, the impact of acoustic pressure observations on the ocean analysis and forecast can be examined.
One or more aspects described herein examine the coupled ocean-acoustic adjoint sensitivity from four short-distance range/depth planes (or radials) and one long-distance radial within an NCOM domain of the Mid-Atlantic Bight in May, 2021. The lessons learned from that study have helped to form this first proof-of-concept test of the assimilation capabilities of the coupled ocean-acoustic 4DVAR using NCOM and RAM. In this present work, simulated acoustic observations collected near the end of each of the four short-distance radials presented (at 50 Hz) are assimilated and the impact of these observations is examined both at the analysis time and also within the subsequent 24-hr forecast. The choice of short radials and observations simulated at 50 Hz is made based on the findings of the aforementioned adjoint sensitivity analysis, which determined that the sensitivity of the acoustic pressure to input ocean temperature and salinity is vertically broad and nearly uniform within the water column. This suggests the likelihood of strong correction of the ocean model using this configuration.
The presentation of the models (NCOM and RAM) and the data assimilation algorithm (NCOM-4DVAR and VRAM) are described.
The OSSE performed for this work utilizes the NCOM model to generate both the nature run (NR, the proxy for the “true” ocean state) and the simulated ocean. The NR is configured for a 1 km horizontal resolution with 100 vertical levels. The model domain extends from 71.4°-76.7° W and 35.7°-40.0° N using a spherical coordinate projection. The NR model is initialized on 1 Mar. 2021 and run through 31 May, 2021; in this case the first two months (March and April) are used in order to spin-up the model state and observations are sampled using output from May only. The simulated ocean, on the other hand, uses a 3 km horizontal resolution with only 50 vertical levels. This degradation in both horizontal and vertical resolution is meant to capture some of the representation error that ocean models typically suffer from due to computational limitations. Both the NR and simulated ocean models are initialized from the Navy's Global Ocean Forecasting System (GOFS v3.1). The NR uses GOFS output from 1 Mar. 2021 for its initialization, whereas the simulated ocean model uses GOFS output from 1 Apr. 2021. In order to enhance the differences between the two models, however, the simulated ocean model initial condition field is positioned at 15 Apr., 2021 (a time shift) in order to increase error relative to the NR. Both models, however, use lateral boundary conditions from GOFS (with no adjustment) and surface forcing, such as wind stress, atmospheric pressure, and surface heat flux, from the COAMPS© model at hourly intervals (Hodur 1997). The models are compared to each other, through time, from 1 to 31 May, 2021; with the simulated ocean model either as a free-run (FR, no data assimilation) model or as a data assimilative run (AR).
In order to sample the NR for acoustic observations, four short-distance radials are configured using four acoustic source locations within the NR domain.
For the experiments presented here, the NCOM-4DVAR is configured with a 24-hr assimilation window using the strong-constraint option. Initial condition errors are generated by the Navy Coupled Ocean Data Assimilation (NCODA) system (Cummings and Smedstad 2013) using the Generalized Digital Environmental Model (GDEM4) climatological database (Carnes et al., 2010). The 4DVAR analysis is run every 24-hrs, and a 24-hr forecast is generated from each analysis. Acoustic pressure observations are sampled from the nature run at the time of the analysis (at the end of each assimilation window at Oz) during each analysis/forecast cycle.
As mentioned previously, the four acoustic radials used in this study are short-range (˜2 km each) and range-independent. Due to this, the correction to the ocean model temperature and salinity at the location of the environmental profile used by RAM and the VRAM operators can be examined. In this case, each profile from the AR at 0-hr (i.e. the analysis) and 24-hr forecasts during each analysis/forecast cycle is compared to the NR profiles. In the case of the 24-hr forecast, the first profile comparison (1 May, 2021) is an uncorrected forecast state. All subsequent 24-hr forecast profiles, however, are the forecast from the data assimilative analysis; therefore, we can compare these fields to the FR state as well. This comparison will show whether the acoustic observations are having a positive impact on the ocean model profiles at these locations and whether this improvement is maintained into the forecast step.
Examining the results at radial 1 (
The absolute difference at radial 2 is shown in
The results are once again largely positive when examining the next radial, radial 3 (
It is likely that the spatial extent of the correction from acoustic observations during the analysis period extends well beyond the environmental profile used at each source location, given that the improvement gained in the analysis step is maintained during the first 24-hrs of the forecast. To examine this, the two-dimensional map of the analysis increment on 1 May, 2021 is shown in
An analysis of the results centers on the impact of the acoustic observation assimilation in the region surrounding each source location. As shown in the sensitivity results and also the analysis increments shown in
This section describes the use of variational methods based on the four dimensional variational (4DVAR) technique to assimilate acoustic pressure observations in an Observing System Simulation Experiment (OSSE) using the adjoint of the Range-dependent Acoustic Model (RAM) [9] and the Navy Coastal Ocean Model (NCOM) 4DVAR analysis system. The results showed good correction of the ocean environment in the vicinity of the acoustic source location and that correction was propagated from the analysis to the short-term forecast and persisted through all 31 days of the experiment. The 4DVAR method was chosen for that study due to the complexity of the four-dimensional background error covariance provided by 4DVAR, as it was expected this would provide the best use of the acoustic observations for ocean state analysis and forecasting. The 4DVAR method, however, is relatively computationally expensive and can be difficult to derive as it requires the careful construction of the tangent linear and adjoint of an entire ocean modeling system in order to be used. There are several competing analysis methods, such as the simpler three-dimensional variational (3DVAR) approach [35], as well as the various flavors of the Kalman filter, see [36], [37], [38], that are simpler to construct and are far more computationally efficient. The question is, can one of these simpler methods have similar success with the assimilation of acoustic observations as was demonstrated with the 4DVAR approach? This current study aims to assess the performance of the adjoint method for acoustic data assimilation, but with the simpler 3DVAR approach rather than 4DVAR. Here, the adjoint and tangent linear of RAM are used as observation operators within a 3DVAR ocean analysis system. The results are compared to the more complex 4DVAR approach in order to examine the performance relative to the more complex method.
Section 4 is organized as follows: a description of the models (both oceanographic and acoustic), the data assimilation methods used, and the OSSE configuration and observation sampling are described in section 4.1 while section 4.2 provides and discusses the results of the experiment.
The ocean model used in this study is the Navy Coastal Ocean Model (NCOM) [10]. NCOM is a primitive equation model that uses the hydrostatic and Boussinesq approximations and employs the Mellor-Yamada Level 2.5 turbulence closure parameterization for vertical diffusion, and a third-order upwind advection scheme (that is naturally diffusive in the horizontal). The model can be setup to have a number of different vertical coordinate configurations; for this work, NCOM has free sigma levels (terrain following) over fixed z-levels. NCOM is a regional model and relies on a parent ocean model for lateral boundary conditions. For this study, the Navy's Global Ocean Forecast System (GOFS v3.1) provides lateral boundary forcing at 3-hour intervals, while the surface atmospheric forcing is provided by the Navy's Coupled Ocean/Atmosphere Prediction System (COAMPS©) at hourly intervals [20]. In addition to the ocean model, a parabolic equation model known as RAM is employed to model the ocean acoustic pressure. RAM, described in [9], is configured as a two-dimensional range/depth plane model that is initialized with an initial acoustic pressure profile and forced with ocean sound speed profiles at regular intervals along the range/depth plane to simulate the ocean acoustic pressure. In order to simulate the sound propagation from some source, RAM requires the source geographical location, depth, and frequency. RAM obtains the ocean sound speed profiles from an ocean model (NCOM in this study), and the sound speed is computed from the ocean model temperature and salinity using the Chen-Millero-Li method [16], [17]. Bilinear interpolation is used to map the model temperature and salinity profiles to the locations along the RAM range/depth plane, where the sound speed is then computed. The main data assimilation system used in this work is the NCOM-4DVAR analysis system [13], [14]. NCOM-4DVAR is based on the representer-method of 4DVAR [16], [17], and can be configured to run as a weak-constraint system that attempts to account for both the initial condition as well as the model error. As in any variational assimilation system, the NCOM-4DVAR is derived from the basic form of the variational cost function:
where xb is the model background state vector; y is the observation vector; H is the nonlinear observation operator (that maps the model state to the space of the observations); and B and R are the background and observation error covariances, respectively. After some derivation Eq. (11) can be used to derive what is known as the analysis equation:
δxa=BHT(HBHT+R)−1(y−Hxb) (12)
where δxa is the analysis increment that is added to the model background state to form
the analysis, and H is the linearized observation operator (with HT as the transpose or adjoint operator). In the case of the representer method for 4DVAR, B in Eq. (12) can be expanded to include the tangent linear and adjoint operators of the nonlinear ocean model [2] so that Eq. (12) becomes:
δxa=MΣCΣTMTHT(HMΣCΣTMTHT+R)−1(y−Hxb) (13)
where M is the tangent linear of the ocean model (and MT is its adjoint), τ is the background error standard deviation, and C is the background error correlation matrix. In 4DVAR, the tangent linear and adjoint model are used in concert with one another to provide a number of features to the analysis, including the time variability of the correction (i.e. four-dimensional error covariance) and dynamical balance relationships based on the linearized dynamics of the nonlinear ocean model (cross-covariance terms).
Typically, the inversion in Eq. (13) is solved using a linear solution method such as one of the conjugate gradient approaches. This requires multiple applications of the tangent linear and adjoint model over a number of iterations until the conjugate gradient converges to a solution. Depending on the size of the problem (model domain size, horizontal and vertical resolution, and length of the assimilation window) the 4DVAR can be computationally expensive. Eq. (13) can be simplified by reducing the tangent linear and adjoint of the ocean model to identity matrices such that Eq. (13) becomes a 3DVAR method:
δxa=ΣCΣTMTHT(HΣCΣTHT+R)−1(y−Hxb) (14)
The inverse in Eq. (14) is faster to solve as the covariance ΣCΣT can be applied in an efficient manner. In the case of the data assimilation algorithms used in this work, the correlation matrix, C, is provided by an operator based on the implicit solution to a diffusion equation [39],[40] and is computationally efficient. There is no account for the time variability of the observations on the solution in Eq. (14), however, as that is a necessary consequence of using a simpler method than 4DVAR. The dynamical balance relationships are also missing in Eq. (14), but they can be approximated by adding additional operators. Following the work of [41], a linearized “balance operator” can be constructed based on geostrophy and hydrostatics. Doing so would alter Eq. (14) so that the embedded background error covariance would have the additional operators:
δxa=KbΣCΣTKbTHT(HKbΣCΣTKbTHT+R)−1(y−Hxb) (15)
where Kb is the linear balance operator (and KbT its adjoint). Eq. (15) is the base form of the 3DVAR used in this work, with the software derived from the larger 4DVAR system. Both the 3DVAR and 4DVAR used in this study employ a conjugate gradient solver with a preconditioner based on the work of and explored with the NCOM-4DVAR in [43]. In order to assimilate acoustic pressure observations with the ocean 3DVAR or 4DVAR systems, a suitable linear observation operator must be provided that maps the ocean variables to the space of the acoustic pressure observations (and vice versa). RAM depends nonlinearly on the sound speed [3] and, therefore, a tangent linear operator must be derived. The details of this derivation can be found in [3], as well as the construction of the linear transpose, or adjoint, model of RAM. These operators are known collectively as Variational RAM (VRAM). These VRAM operators are linked to the 3DVAR and 4DVAR algorithms as the embedded H and HT operators in Eqs. (13) and (15).
For the study shown here, the 4DVAR is run in weak constraint mode, with model errors set as 10% of the initial condition error (with an identical spatial distribution); the initial condition errors are generated by the Navy Coupled Ocean Data Assimilation (NCODA) system using the Generalized Digital Environmental Model (GDEM4) climatological database from [34]. The 4DVAR is run with a 48-hr assimilation window and observations are assimilated every 6 hours. For the 3DVAR (e.g., in some embodiments), only the initial condition error is used (identical to that used in 4DVAR) and observations are only assimilated once at the analysis time. The cycling runs shown here conduct an analysis every 24-hrs and a 24-hr forecast is run from each analysis from 1 through 31 May, 2021.
These models are applied using an OSSE methodology. An OSSE is a powerful tool that can be used to examine either a new data assimilation system, observation type or platform, or both [44], [25]. Typically, an OSSE experiment will consist of two component models, a proxy for the “real” ocean (known as the nature run) and a model that will assimilate observations (known as the assimilative run). The nature run must be of sufficiently high spatial resolution so as to resolve smaller-scale features and phenomenon that exist in the real ocean and that are not always captured by lower-resolution operational ocean models. The assimilative run, on the other hand, should suffer from sources of error that are typical for an ocean model (and a regional model in the case of this study). These sources of error include initial condition error (due imperfect analysis of the starting ocean state), imperfect boundary conditions, and errors due to resolution (i.e. unresolved features). These errors can be included, for example, by providing the assimilative model a different initial condition than is used by the nature run, time-shifting the boundary conditions, and running the assimilative model at lower resolution than the nature run. One can also use entirely different ocean models for the nature and assimilative model runs to truly uncouple the runs from one another.
In this study, the nature and assimilative runs are both conducted using NCOM, known as a fraternal twin experiment. Here both models cover the same geographical area, with the domain extending from 71.4° to 76.7° W longitude and 35.7° to 40.0° N latitude. The nature run is configured for 100 vertical levels with a horizontal resolution of 1 km. The assimilative model, on the other hand, has only 50 vertical levels with a horizontal resolution of 3 km. The nature run is initialized on 1 Mar. 2021 and is run through 31 May, 2021. The months of March and April are used as a spin-up period leaving the month of May for the study period shown in this work. The assimilative model is initialized on 1 Apr. 2021, but using an initial condition derived from GOFS on 15 Apr. 2021. This time-shift provides additional error to the initial condition and produces a model solution in the assimilative run that differs substantially from the nature run. No additional source of error is included in the assimilative model as the initial condition time shift and resolution differences produce a model solution during the month of May that is of sufficient difference.
The focus of this study is to compare and contrast the assimilation of simulated acoustic pressure observations using a 3DVAR and 4DVAR assimilation method; as such, acoustic pressure observations must be sampled from the nature run in this OSSE. To do this, RAM is run along specific range/depth planes positioned within the ocean domain, and the ocean nature run provides environmental profiles of temperature and salinity to be used to compute the sound speed as required by RAM. For this work two such range/depth planes are configured to provide acoustic pressure “observations” for assimilation. Each plane is roughly 2000 m in length from source to the location of the simulated receivers. Due to the resolution of the assimilative model, this short distance effectively makes the RAM model range-independent as only one model profile is used to provide sound speed for the entire plane (e.g., in some embodiments). At the end of each plane the simulated acoustic pressure from the nature run is sampled at depths from 15 m to 105 m, every 15 m; this simulates the sampling one might get from a typical vertical line array. The location of each acoustic source is fixed in time (no moving targets) and is shown in
Prior to examining the results of the assimilation experiments it is useful to investigate the performance of the free-run model against the nature run in order to gauge the impact of the observations on the model's performance. In this case, we can examine the performance in two ways, the acoustic model performance and the ocean model performance. For the acoustic model, let us examine the acoustic pressure simulation from the free-run model versus that from the nature run at the end of radial 0 and radial 1.
We can now examine both the analysis and 24-hr forecast error, from both the 3DVAR and 4DVAR experiments, by comparing the root mean square error (RMSE) of the source temperature and salinity profiles against the corresponding profiles from the NR.
Again, when compared to
Section 2 of this report details the configuration of the coupled ocean-acoustic data assimilation software and an application of an adjoint sensitivity study is presented. The study shown here mainly focuses on short-range cases (2000 m), however results from a mid-range and a long-range case are also shown. This study shows that acoustic pressure at the end of each radial is sensitive to ocean temperature and salinity in a spatially-broad manner at 50 Hz for the short-range cases, but exhibits a more small-spatial-scale sensitivity to temperature and salinity at 300 Hz. This suggests that, given a data assimilation application, a more large-scale correction could be made using lower frequency acoustic pressure observations, while smaller-scale adjustments to the ocean profiles can be made using higher frequency acoustic pressure observations. The sensitivities exhibit a great deal of overlap, however, which precludes the possibility of simultaneous assimilation. This is akin to a multiscale problem, where observations contain both large and small scale information. In this case, a multiscale data assimilation approach has shown promise in previous works (Li et al., 2015; Carrier et al., 2019; Souopgui et al., 2020) and the general principle may be applied to the multifrequency data assimilation problem, where lower frequency observations are assimilated in the first analysis step while higher frequency observations are assimilated in a subsequent analysis step.
The sensitivity of acoustic pressure to the full four-dimensional ocean model state is also examined here. For this the coupled adjoint model is integrated for 120 hours and the sensitivity of acoustic pressure at the end of each radial is examined at two depths (0 and 50 m) across the entire ocean model domain. These results show significant displacement of the high sensitivity areas away from the radial locations themselves upstream the flow within the domain. This suggest that the acoustic pressure observations could have a substantial impact on the ocean model (spatially) given a sufficiently long assimilation window. Also, this indicates that coupled ocean-acoustic adjoint sensitivity information may be useful within an adaptive sampling application; this could be examined using an Observing System Simulation Experiment (OSSE; Zeng et al., 2020) study. A mid-range and long-range case is examined to investigate the possible impact of acoustic observations over longer ranges past 2 km. These results indicate that the impact to the ocean model from possible acoustic observation assimilation will be highly range and depth dependent based on the waveguide.
Section 3 provides a description of a simulated-observation assimilation case where the impact of the assimilation of acoustic pressure observations on the ocean model analysis and forecast is examined. A Mid-Atlantic Bight NCOM domain is used to assess the impact of acoustic pressure observations on the ocean model analysis and 24-hr forecast using a coupled ocean-acoustic 4DVAR system. This investigation uses an OSSE framework with simulated observations. In this case a high-resolution ocean model is used as the nature run and is used to provide the environmental profiles for a nature run of the acoustic model, RAM, from which simulated acoustic pressure observations are sampled. Four source locations (with a source depth of 50 m at a frequency of 50 Hz) are used with one short-range radial extending from each source by roughly 2 km. The nature run and the assimilative model both treat each radial as range-independent, due to the short nature of each radial relative to each model's horizontal resolution. The simulated acoustic pressure observations are assimilated daily over a 31-day period covering May, 2021. The results of the assimilative model's analysis and 24-hr forecast are compared directly to the nature run at the location of each environmental profile as well as within a 1° by 1° box surrounding each profile. The results of this comparison show that the acoustic pressure observations have a generally positive impact on the assimilative model analysis and 24-hr forecast, with substantially lowered errors at the location of the environmental profile itself, and a decrease of error within the region surrounding each profile relative to the FR. The results shown here indicate that acoustic pressure observations can have a positive impact on ocean state estimation, as well as for improving short-term ocean forecasts.
Section 4 of this report details the relative performance of a 3DVAR approach to the assimilation of acoustic pressure observations relative to the more complex 4DVAR approach as well as a non-assimilative free run model within an OSSE configuration. The results show that the 3DVAR method can properly assimilate acoustic pressure observations, with errors nearly identical to that from a 4DVAR method, when employing the same observation operators derived from an acoustic model (RAM). The 4DVAR method does show some slight improvement in performance in terms of correction to the temperature profile at the source location nearer the surface, as well as capturing the character of the standard deviation from the mean profile over all 31 days of the experiment. The 4DVAR method also shows some slight improvement over the 3DVAR in terms of the acoustic pressure profile at the end of each radial, especially radial 0 from 60-80 m depth. Despite this slightly improved performance from 4DVAR, the results from the 3DVAR experiment are still very competitive and nearly identical in some cases with the 4DVAR. And given the superior computational efficiency of the 3DVAR method (roughly 5× faster in this study), the selection of the 3DVAR method for acoustic observation assimilation certainly produces meaningful and useful results.
According to some aspects, one or more disclosed embodiments may have one or more specific applications. According to some aspects, one or more disclosed aspects may be used to facilitate a water-based operation. For example, ocean forecasts, such as described herein, can be used for drift prediction, search & rescue, and acoustic modeling. According to some aspects, one or more disclosed aspects may be used to develop a mission route plan associated with operating a vessel. In some cases, one or more disclosed aspects may be used to facilitate a strategic operation, which can include a defensive tactical operation or naval operation. One or more aspects described herein may be used to facilitate naval operations.
In one example, Navy Tactical Decision Aids rely heavily on ocean model output for accuracy. Ocean models rely on observations to maintain accuracy, but the ocean is severely under-observed. Navy Tactical Decision Aids rely on accurate ocean model forecasting, and use of ocean acoustic observations as described herein are a source of observation that improves the ocean model accuracy. This may provide personnel (e.g., Navy personnel, etc.) the capability to use an existing observation that is abundant and already collected, but in a new and novel way as described herein.
In some cases one or more aspects may involve the method of data assimilation itself. In this case, acoustic pressure observations could be employed within a method of Kalman Filtering (KF). In this approach, the same observation operators could be used within any flavor of the Kalman approach, where an ensemble of ocean model forecasts provides the ocean background error covariance. The anticipated result would be similar to any comparison of 3DVAR/4DVAR/KF.
According to some aspects, one or more aspects described herein may be applied in a series of simulated-observation studies to greatly reduce ocean model error in both the analysis and 24-hour forecast.
One or more aspects described herein may be implemented on virtually any type of computer regardless of the platform being used. For example, as shown in
Further, those skilled in the art will appreciate that one or more elements of the aforementioned computer system 4400 may be located at a remote location and connected to the other elements over a network. Further, the disclosure may be implemented on a distributed system having a plurality of nodes, where each portion of the disclosure (e.g., real-time instrumentation component, response vehicle(s), data sources, etc.) may be located on a different node within the distributed system. In one embodiment of the disclosure, the node corresponds to a computer system. Alternatively, the node may correspond to a processor with associated physical memory. The node may alternatively correspond to a processor with shared memory and/or resources. Further, software instructions to perform embodiments of the disclosure may be stored on a computer-readable medium (i.e., a non-transitory computer-readable medium) such as a compact disc (CD), a diskette, a tape, a file, or any other computer readable storage device. The present disclosure provides for a non-transitory computer readable medium comprising computer code, the computer code, when executed by a processor, causes the processor to perform aspects disclosed herein. According to some aspects, one or more steps of method 4300 may be performed by the computer system 4400 and/or one or more components of computer system 4400. In some cases, one or more aspects may include causing display of a generated current ocean state forecast, such as via output means 4412.
Embodiments for the use of ocean acoustic pressure observations for ocean model analysis and forecasting been described. Although particular embodiments, aspects, and features have been described and illustrated, one skilled in the art may readily appreciate that the aspects described herein are not limited to only those embodiments, aspects, and features but also contemplates any and all modifications and alternative embodiments that are within the spirit and scope of the underlying aspects described and claimed herein. The present application contemplates any and all modifications within the spirit and scope of the underlying aspects described and claimed herein, and all such modifications and alternative embodiments are deemed to be within the scope and spirit of the present disclosure.
This Application is a nonprovisional application of and claims the benefit of priority under 35 U.S.C. § 119 based on U.S. Provisional Patent Application No. 63/411,604 filed on Sep. 29, 2022. The Provisional Application and all references cited herein are hereby incorporated by reference into the present disclosure in their entirety.
The United States Government has ownership rights in this invention. Licensing inquiries may be directed to Office of Technology Transfer, US Naval Research Laboratory, Code 1004, Washington, DC 20375, USA; +1.202.767.7230; techtran@nrl.navy.mil, referencing Navy Case # 211166.
| Number | Date | Country | |
|---|---|---|---|
| 63411604 | Sep 2022 | US |
