GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus PublicationsGöttingen, Germany10.5194/gmd-12-195-2019Reanalysis of the PacIOOS Hawaiian Island Ocean Forecast System, an
implementation of the Regional Ocean Modeling System v3.6Hawaiian Island Ocean Forecast SystemPartridgeDaleFriedrichTobiashttps://orcid.org/0000-0001-7324-4100PowellBrian S.powellb@hawaii.eduUniversity of Hawai`i at Mānoa, Department of Oceanography, Marine
Sciences Building, 1000 Pope Road, Honolulu, Hawai`i 96822, USABrian S. Powell (powellb@hawaii.edu)9January20191211952137April20184July20181November20188November2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://gmd.copernicus.org/articles/12/195/2019/gmd-12-195-2019.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/12/195/2019/gmd-12-195-2019.pdf
A 10-year reanalysis of the PacIOOS Hawaiian Island Ocean Forecast System was
produced using an incremental strong-constraint 4-D variational data
assimilation with the Regional Ocean Modeling System (ROMS v3.6).
Observations were assimilated from a range of sources: satellite-derived sea
surface temperature (SST), salinity (SSS), and height anomalies (SSHAs); depth
profiles of temperature and salinity from Argo floats, autonomous Seagliders,
and
shipboard conductivity–temperature–depth (CTD); and surface velocity
measurements from high-frequency radar (HFR). The performance of the
state estimate is examined against a forecast showing an improved
representation of the observations, especially the realization of HFR surface
currents. EOFs of the increments made during the assimilation to the initial
conditions and atmospheric forcing components are computed, revealing the
variables that are influential in producing the state-estimate solution and
the spatial structure the increments form.
Introduction
The Pacific Integrated Ocean Observing System has produced
daily forecasts of the ocean state surrounding the Hawaiian Islands since
2009. To facilitate the forecasts a data assimilation procedure is used to
incorporate recent observational data into the model to produce the optimal
initial state from which to forecast. A number of modeling studies have been
performed with older versions of this model to examine various features of
the modeling framework, such as the state estimation ,
nested models , and the vorticity budget
. In this work, we perform an extended reanalysis from 2007
to 2017 in order to produce a consistent dataset for further studies of the
dynamics around Hawai`i.
The PacIOOS forecast system uses the time-dependent incremental strong-constraint
four-dimensional variational data assimilation (I4D-Var) scheme
within the Regional Ocean Modeling System
(ROMS) to best reduce the
residuals between the model and observations, while maintaining a physically
consistent solution. The class of methods known as 4D-Var are
state-estimation techniques that create a quadratic cost function to be
minimized over a defined time window utilizing observations at the time they
occur in a physically consistent manner to adjust the initial state, boundary
conditions, and atmospheric forcing to represent the measurements. The
I4D-Var scheme is used in operational centers around the world and solves for
increments to the model state, boundary conditions, and atmospheric forcing
using the model physics as a constraint. The combination of I4D-Var within
ROMS has been used in previous studies of various regions
. The details
of the model and the observations used within this study are provided in
Sect. .
Our model domain covers the Hawaiian Island Archipelago
(Fig. ), a dynamically active region for both the ocean and
atmosphere. The North Equatorial Current (NEC), flowing from the east, splits
upon encountering the island of Hawai`i, with the bulk transport traveling
around the south of the island and continuing west, while the North Hawaiian
Ridge Current (NHRC) follows the ridge of the other islands in the chain to
the north. In the atmosphere, there are persistent trade winds from the
northeast that, combined with steep mountainous terrain on the islands, cause
wind wakes in lee of the peaks, particularly on the islands of Hawai`i and
Maui. This introduces strong temperature gradients, increases the seasonal
variability , and drives currents such as the Hawaiian Lee
Countercurrent (HLCC) .
Model domain and bathymetry, with mean currents labeled from
.
There are two main objectives to this study: to assess the skill and
performance of the state-estimation model and to analyze the increments made
to the initial, boundary, and atmospheric forcing terms. For the first
objective, we compare the state-estimate solution with a free-running
forecast over the decadal time period and examine how the performance changes
over time utilizing observations derived from satellites and it situ
measurements. In addition, PacIOOS operates seven high-frequency radar
stations sites across the Hawaiian Islands. The first station was constructed
in 2010, with the remaining six becoming operational over the period from
2011 to 2015. These instruments produce high-resolution (both spatially and
temporally) surface current velocities in the vicinity of the islands of
O`ahu and Hawai`i. The use of HFR observations within a state-estimation
scheme has been shown to produce a significantly improved representation of
surface currents . The impact of the radar
stations will be a key focal point. The performance assessment is achieved
through the statistics produced by the state estimation in
Sect. , followed by a comparison with observations in
Sect. . The forecast skill, a measure of the accuracy for a
forecast system, is computed with reference to a persistence assumption
(Sect. ).
Section focuses on the second objective of the paper, to
examine the increments to the initial state and atmospheric forcing to
determine how the model is adjusted. By evaluating the empirical orthogonal
functions (EOFs) of these increments we determine the spatial patterns in the
variability. Since physical modes are not always independent
, the interpretation of EOF modes must be undertaken with
some caution. As such the resulting modes will not necessarily represent a
physical phenomenon, but will highlight the variable spatial patterns made
over time by the I4D-Var algorithm.
Numerical model and data assimilation systemModel configuration
The Regional Ocean Modeling System (ROMS) version 3.6 is used to simulate the
physical ocean around the Hawaiian Islands. ROMS is a free-surface,
hydrostatic, primitive equation model using a stretched coordinate system in
the vertical to follow the underwater terrain. In order to allow varying time
steps for the barotropic and baroclinic components, ROMS utilizes a
split-explicit time stepping scheme (for more details on ROMS, see
).
The Hawaiian Island domain covers 164–153∘ W longitude and
17–23∘ N latitude, with bathymetry provided by the Hawaiian Mapping
Research Group , shown in Fig. . The grid
has 4 km horizontal resolution with 32 vertical s levels configured to
provide a higher resolution in the more variable upper regions. The
configuration model, including the method for assimilating surface HFRs and
the associated vertical stretching scheme, is identical to the one first
presented in .
Tidal forcing is produced using the OSU Tidal Prediction Software (OTPS)
, which is based on the Laplace tidal equations from
the TOPEX/Poseidon Global Inverse Solution (TPXO). Tidal constituents included in
this simulation are the eight main harmonics, M2, S2, N2, K2,
K1, O1, P1, and Q1, as well as two long-period and one nonlinear
constituent: Mf, Mm, and M4. To avoid any long-term
drifting of the tidal phases related to constituents we do not consider, the
tidal harmonics are updated each year to define the phases in terms of the
start of that year.
Lateral boundary conditions are taken from the HYbrid Coordinate Ocean Model
(HYCOM) and are applied daily. Within ROMs, we apply
the boundary differently for each variable; Chapman
conditions are applied to the free surface, Flather
conditions for transferring momentum from 2-D barotropic energy out of the
domain, and 3-D momentum and tracer variables are clamped to match
HYCOM. A sponge layer of 12 grid cells (48 km) linearly relaxes the
viscosity by a factor of 4 and diffusivity by a factor of 2 close to the
boundary to account for imbalances between HYCOM and ROMS.
From 2007 to 2009, atmospheric forcing fields (excluding the wind) are
provided by the National Center for Environmental Prediction (NCEP)
reanalysis fields . For the wind forcing, a
combination of two different forcings is utilized: (i) a 1/2∘
resolution CORA/NCEP wind product that combines
QuikScat measurements with NCEP wind fields and (ii) the CORA/NCEP winds
blended with the results from a 1/12∘ resolution PSU/NCAR mesoscale
model (MM5; ) of the Hawaiian islands
. The MM5 model was forced at its boundaries with the
global NCEP fields; hence, it is a consistent dynamical downscaling of the
global fields. The MM5 model domain is smaller than the ocean grid domain,
extending only to 160.5∘ W in the lee. Therefore, for (ii), we must
blend the modeled and CORA/NCEP winds to generate a consistent field for the
entire region with 1/12∘ winds where available and 1/2∘
winds everywhere else.
To blend the two, we convert the MM5 winds to anomalies by subtracting a
30-day mean centered about the record of interest. We compute the mean for
the same period from the CORA/NCEP winds. The difference between the two
means provides a bias estimate. The bias is removed from the MM5 anomalies
and the CORA/NCEP mean is added. Within a 1∘ box around the boundary
of the MM5 data, we taper the anomalies to zero with a cosine filter to avoid
abrupt changes to the field. This step ensures that the mean of the CORA/NCEP
field is preserved, while its structure and variability is greatly enhanced by
the MM5.
From July 2009, atmospheric forcing is provided locally by a high-resolution
Weather Regional Forecast (WRF) model . WRF supplies information
about surface air pressure, surface air temperature, longwave and shortwave
radiation, relative humidity, rainfall rate, and 10 m wind speeds. The
ocean model is forced using these data every 6 h, taken from the
atmospheric model with 6 km resolution across the entire domain.
Prior to the experiment, a 6-year non-assimilative model was run using the
same initial state, boundary conditions, and atmospheric forcing. The
variability of the model is used to produce an estimate of the background
error covariances used within I4D-Var, as well as the mean sea surface height
to use with sea level anomaly observations.
The cost function of the I4D-Var method penalizes for the increments made to
the initial conditions, the boundary conditions, and the forcing and for the
deviations of the model state from the observations. A detailed derivation of
the cost function can be found in
, , ,
, and . To
formulate the solution, we must provide estimates of the uncertainty matrices
in both the model and observations. The model uncertainty matrix,
P, is estimated using the variability of the 6-year run
described above, while the observation uncertainty matrix, R, is
assumed to be diagonal (i.e., observations are independent). The
implementation of I4D-Var in ROMS is covered extensively in
.
Experiment setup
The reanalysis covers a period of 10 years from July 2007 to July 2017. The
period of assimilation for the I4D-Var cycles is 4 days, which corresponds
to the limit of the linearity assumption within the domain
. The atmospheric forcing is adjusted every 6 h,
while the boundaries are every 12 h. An analysis of these adjustments is
performed in Sect. .
During each I4D-Var cycle, a minimization procedure is applied. The
nonlinear model is first integrated forward to estimate the background state
(the first outer loop). Then the tangent-linear and adjoint models
are integrated in multiple inner loops to minimize the cost function
(J). After the last inner loop the nonlinear model is updated (see Fig. 1
of ). Prior methodological experiments yielded that for
our setting a sufficient reduction in J (and an acceptable computational
cost) can be achieved using a single outer loop with 13 inner loops
.
Several 4- and 8-day forecasts are performed from the end of each cycle using
the assimilated state as initial conditions, and the short-range (1–4 days)
and midrange (5–8 days) forecasts are evaluated for skill.
Observations
Observational data used within this study include satellite measurements of
the ocean surface of temperature, height, and salinity, in situ depth
profiles of temperature and salinity, and surface velocities from high-frequency radar. Observations within one Rossby radius (∼ 80 km) of
the domain's boundary are neglected. It should be emphasized that no
observations were withheld from the assimilation for the purpose of
validation. The I4D-Var method seeks to represent the observations by
exploiting the linearized model dynamics. Therefore, all available
observations are used to constrain this representation.
Satellite-derived measurements
Sea surface temperature (SST) observations are available from two sources at
different time periods: initially we used the Global Ocean Data Assimilation
Experiment High Resolution Sea Surface Temperature (GHRSST) level 4 OSTIA
Global Foundation Sea Surface Temperature Analysis , referred to
as OSTIA for this work. The data are distributed by the Physical Oceanography
Distributed Active Archive Center (PO.DAAC) using optimal interpolation to
combine data from the Advanced Very High Resolution Radiometer (AVHRR), the
Advanced Along Track Scanning Radiometer (AATSR), the Spinning Enhanced
Visible and Infrared Imager (SEVIRI), the Advanced Microwave Scanning
Radiometer-EOS (AMSRE), the Tropical Rainfall Measuring Mission Microwave
Imager (TMI), and in situ data. This distribution provides a highly smoothed
daily gridded global dataset at the surface at a 6 km spatial resolution,
accurate between 0.2 and 0.5 ∘C in the domain.
Beginning in April 2008, we switched to using the GHRSST level 4 K10_SST
global 1 m sea surface temperature analysis dataset produced
by the Naval Oceanographic Office and referred to as NAVO for this work.
Also distributed by PO.DAAC, this product combines, in a weighted average,
data from AVHRR, AMSRE, and the Geostationary Operational Environmental
Satellite (GOES) imager. This distribution provides a daily gridded global
dataset at 1 m of depth at a 10 km spatial resolution, accurate to
0.4∘C in the domain.
Sea surface height (SSH) observations are derived using sea level anomaly
data from the Archiving, Validation and Interpretation of Satellite
Oceanographic data (AVISO) delayed-time along-track information. The data
come from multiple altimeter satellites measuring the anomaly with respect
to a 20-year mean SSH, homogenized against one of the missions to ensure
consistency. Each track has approximately 7 km spatial resolution and will
usually make multiple passes through our domain each day. To convert from sea
level anomaly to sea surface height we add the mean SSH field taken from the
6-year model run described above, to which we add the barotropic tidal
prediction from TPXO. The accuracy of the swaths depends on the source
satellite and ranges from 5 to 7 cm. We use the AVISO product that has been
fully filtered and quality controlled until May 2016. At the time of the
experiment, the delayed time data were unavailable beyond May 2016, so the
near-real-time data were used.
Sea surface salinity (SSS) data are taken from Aquarius mission daily L3
gridded dataset distributed by PO.DAAC. The satellite uses
a combination of radiometers and scatterometers to estimate the surface
salinity mapped to a coarse 1∘ resolution. Errors for this product
are around 0.2 ppt. Data for this product are available from August 2011
until June 2015.
In situ measurements
Depth profiles of temperature and salinity are obtained from threes sources:
the Hawai`i Ocean Time-Series (HOT) shipboard conductivity–temperature–depth
(CTD) casts, the global network of Argo floats, and autonomous Seagliders
operated by the University of Hawai`i.
The HOT project conducts monthly cruises to the deep water station ALOHA (A
Long-term Oligotrophic Habitat Assessment; located at
23∘45′ N, 158∘00′ W; see Fig. 1) in order to develop
continuous datasets of physical and biochemical ocean parameters. CTD
stations of temperature and salinity are concentrated in the region around
the station, although some are also established along the ship route.
HOT also conducts regular Seaglider missions departing from station ALOHA. In
addition, PacIOOS conducts occasional Seaglider surveys in areas close to the
south coast of O`ahu. The buoyancy-driven autonomous underwater vehicles take
profiles and transects at depth of temperature and salinity.
Observations from the global Argo float network are available from the Argo
array network . The free-drifting floats profile temperature and
salinity during ascension and descension every 10 days of depths down to
2000 m . Argo measurements tend to occur in the model
domain at a rate of about one to two profiles per day.
Representational errors for HOT CTDs, Argo floats, and Seagliders are defined
by the variance of observational data from all available sources across our
domain sorted into depth bins. These profiles resemble a typical
temperature–salinity profile, with a peak temperature error of 0.8 K and
peak salinity error of 0.15 ppt occurring in the mixed layer at a depth
around 100 m.
Composite image of percentage coverage for all radar sites (situated
at green dots) when all are operational. Where two sites overlap the greater
value is taken to indicate the level of coverage at each
point.
Number of observations used within data assimilation run. Note that
there tend to be orders-of-magnitude more satellite or remotely sensed
observations than in situ.
High-frequency radar measurements
HFR measurements of surface currents are available from PacIOOS at seven
sites around the Hawaiian islands: five around the southwest of O`ahu and
two on the east coast of Hawai`i. Data are available from the first site
in October 2010 with the other sites coming online at various times, the most
recent being October 2015. The range for the HFRs on O`ahu extend
approximately 150 km from the coast, while the two Hawai`i sites are
focused on currents around the northeast of the island and have a shorter
range. At the range limits, HFR data are less reliable due to the higher
noise level of the returns. Figure shows the percentage
availability of data in the region. HFR measurements from any return location
that is missing more than 20 % of its data over the 4-day assimilation
period are ignored. Both spatially and temporally, the resolution for all
sites is significantly higher than the model resolution. The HFR data are
low-pass filtered at 3 h to remove the high-frequency signals that may not
be resolved by the model (atmospheric forcing fields are every 6 h). We
then provide the spatial field of data every 3 h. The associated error is
calculated individually for each spatial point as the accuracy of the
measurements is determined by the levels of interference, which increases
with range. For each observation point we calculate the power spectral
density and calculate the noise as per , with a minimum of
7 cm s-1. At the extreme, errors may reach 17 cm s-1.
The numbers of observations for each 4-day cycle from all sources are shown
in Fig. . Sea surface temperature measurements from both OSTIA
and NAVO are consistently the most available observation source, and by the
end of the time period HFR is supplying a similar quantity. In situ
measurements, which include both temperature and salinity for each of the
instruments, provide a smaller amount of data by an order of magnitude.
Time series of percentage reduction in the I4D-Var cost function;
in the left column are pre-HFR observations and in the right column are post-HFR observations, with the mean value
given in parentheses. Dashed lines mark the limit of 0, below which there
is no reduction in the cost function for that variable. (a) Total
cost function reduction for all observations; (b) sea surface height
observations, (c) temperature observations; (d) salinity
observations; (e) HFR observations.
Assimilation statistics
In this section we examine the state estimate to quantify the performance
during our time period.
Cost function reduction
I4D-Var minimizes the residuals between the model and observations over each
4-day cycle. We calculate the percentage reduction between the initial and
final cost function for each cycle to assess how the assimilation performs
over time. Additionally, the I4D-Var algorithm reports the individual
contributions by the state variables considered by the data assimilation to the
total cost function. Hence we can examine the cost function in detail for
those observation types that are most critical for its reduction. However, it
should be noted that for this decomposition we do not distinguish between
observation sources.
Figure shows the time series of the total reduction and the
percentage reduction in the cost function for each of the variables we
observe: sea surface height, temperature, salinity, and HFR. A value of 0
means the final cost function is the same as the initial and no reduction has
occurred. The plot is split into two distinct time periods, before and after
the HFR observations, in order to assess changes in the relative contributions
of each variable to the overall reduction.
The total cost function of all data (Fig. a) is on average
halved for each cycle, with an improvement from 49 % of the original
value to 55 % when HFR observations are available. Looking at the
breakdown in Fig. b–e, we see that the final cost function
associated with the other observed variables (sea surface height,
temperature, and salinity) is reduced by a smaller percentage than before HFR
was included. Given that the structure of the cost function is determined by
the type and number of observations, this change in contribution to the cost
function reduction can be expected when adding a large number of HFR
measurements to the data assimilation.
Salinity measurements tend to contribute the least improvement, ranging from
34 % (pre-HFR) to 16 % (post-HFR). Salinity data are the least numerous
(Fig. ) and SSS fields taken from Aquarius are subject to high
noise levels (0.2 ppt) and coarse spatial resolution. The mid-2014 drop in
cost function reduction for salinity data coincides with the loss of two
Seagliders. After the cessation of Seaglider missions salinity data were only
available through Aquarius (until mid-2015) and sporadic Argo profiles.
The cost function associated with HFR measurements is reduced by 60 % of
the initial value, meaning the model is closer to the HFR observations after
the assimilation.
(a) Gantt chart of remotely sensed observations used in the
study. (b) Optimality of I4D-Var data assimilation with the dashed
line representing the theoretical minimum.
Optimality
Another measure of performance is the theoretical minimum value of the
cost function (Jmin). For a linear system and assuming that the
error matrices P and R have been determined correctly,
Jmin is a chi-squared variable whose degrees of freedom are given
by the number of assimilated observations (Nobs)
. The expected value of Jmin is then given by
〈Jmin〉=Nobs2.
Using above the equation, an optimality value (γ) can be defined:
γ=2⋅JminNobs,
which should reach a value of 1 with a standard deviation of
2/Nobs.
This optimality value provides a simple representation of how consistently
the error matrices (P and R) are specified, since the
error covariances normalize the cost function. Figure
shows a time series of the calculated optimality value for the model run, in
addition to a timeline of the availability of certain observations for
reference. Over the full time period the mean optimality is 0.95. However,
there are large significant deviations over the course of the time period. In
the pre-HFR period the optimality is low, suggesting that the error bounds on
observations are too wide. Since SST is the dominant source of observations
before HFR, the prescribed errors associated with SST may be too large.
Post-HFR, the optimality value increases, suggesting the errors in this
period are underestimated. A large optimality value arises when the cost
function is large (i.e., large differences between the model and
observations). There were two anomalous cycles in 2011; the first coincides
with the introduction of a second radar site. From 2012 onwards the
optimality value is generally good, if highly variable. The increase in
optimality given the available observations points to an underestimation of
HFR errors or at the least a persistent difference between the model and HFR
observations.
Error consistency
The consistency of the assimilation can be assessed by comparing the error
matrices P and R specified a priori with the
observation and background error covariances determined a posteriori
. Using the difference between the observation j
(yj) and the modeled background value (xb) mapped to the
observation location by the operator Hj,
djob=yj-Hj(xb),
and the difference between xb and the analysis value (xa)
mapped to the observation location,
djab=Hj(xa)-Hj(xb),
one can compute the expected a posteriori background error:
(σib)̃2=1pi∑j=1pi(Hj(xa)-Hj(xb))(yj-Hj(xb)),
where i refers to the observation type and pi is the number of
observations of that type.
Time series of spatially averaged background (blue) and observation
(green) errors, with thick lines showing a priori values and thin lines the
posterior calculated using Eqs. () and
(). (a) Sea surface height; (b) sea
surface temperature; (c) sea surface salinity; and
(d) HFR.
Similarly, using the difference between the observation j and the
modeled analysis value (xa) mapped to the observation,
djoa=yj-Hj(xa),
the expected a posteriori observation error can be calculated:
(σib)̃2=1pi∑j=1pi(yj-Hj(xa))(yj-Hj(xb)).
For a detailed description of the above diagnostics the reader is referred to
. If the variances in P and
R are correctly specified a priori, they will be consistent with
the a posteriori errors defined above. Figure shows both
the a priori and a posteriori errors for the remotely sensed data. The
observation a priori values are calculated as the mean error of the
observations in each cycle, while the background a priori values are defined
as the variability of a free-running nonlinear model. If the a posteriori
errors are typically larger than the a priori, it implies the initial errors
in P and R are underestimated. Conversely, if they are
smaller the initial errors are likely overestimated.
Figure a shows that sea surface height errors are
consistent, while sea surface temperature, shown in Fig. b, suggests
the a priori errors are overestimated. The a priori observation errors for
NAVO SST observations are defined with a minimum error of 0.4 K, but the a
posteriori errors are more typically around 0.25 K. The a priori background
errors also appear overestimated.
Sea surface salinity observation errors (Fig. c) are
slightly underestimated but generally consistent, as are the background
errors. The HFR observation errors (Fig. d) also appear to
be underestimated, with most a priori errors close to the minimum value of
7 cm s-1. The a posteriori errors suggest that a typical value of around
12–15 cm s-1 would be more appropriate. The a priori background
errors are reasonably consistent with the a posteriori; if anything, they are
slightly overestimated.
This error consistency analysis supports the conclusions in
Sect. that the SST observation errors are overestimated and HFR
values are underestimated. It is worth noting that these diagnostics are only
estimates used to characterize the errors and are not the true posterior
error.
Time series of root mean squared anomalies (RMSAs) between remotely
sensed observations and two model realizations: the state estimate (orange)
and the forecast (blue). (a) Sea surface height; (b) sea
surface temperature; (c) sea surface salinity; and
(d) HFRs.
Comparison with observations
Because I4D-Var relies on the model physics to represent observations through
time, it should provide better forecasts. Time-invariant methods (3D-Var,
optimal interpolation) that perturb the state at single times may better
reduce the time-fixed cost function, but can add nonphysical structures that
generate noisy forecasts.
In this section, we examine the state-estimate solution by comparing the
model to observations. For reference, the observations are also compared
against the forecast starting from the same time as each state-estimate
cycle. The initial and boundary as well as atmospheric and tidal forcings are
initially the same for both runs; however, the initial and boundary
conditions and atmospheric forcing are altered as part of the state-estimate
solution.
For comparing fields we use the root mean squared anomaly (RMSA) and the
anomaly correlation coefficient (ACC), defined as
RMSA(x,y)=1N∑i=1N(xi-x‾)-(yi-y‾)2andACC(x,y)=∑i=1N(xi-x‾)(yi-y‾)∑i=1N(xi-x‾)2∑i=1N(yi-y‾)2,
where N is the number of observations and x represents the model values at the
same location and time as the observations y. The RMSA provides a measure
of the residual between the model and observations, while the ACC determines
the strength of the relationship between the two. We can calculate values for
a single spatial point throughout time or for all spatial points at a single
time; however, we require at least 20 available observation values
to get a representative statistic. The gridded satellite products
are ideally suited to this analysis, while the depth profiles from in situ
measurements are binned into 50 m depth layers to ensure a minimum number
of values. Here it must be noted that our validation is limited to data that
have been employed for the assimilation. The I4D-Var scheme uses the
linearized model dynamics to produce the covariance between the model and the
observations. This allows the model to optimally represent the observations
in time and space rather than replicate them. As such, the desire is to use
every available observation to constrain this representation. Unlike
time-invariant statistical methods, we do not withhold any observations
because they are sampling the dynamical subspaces of a system of unknown
dimension. Since the observations covary in space and time, some particular
observations may not have a significant impact on the cost function and their
representation may suffer. We seek to identify these results.
Time series of anomaly correlation coefficients (ACC) between
remotely sensed observations and two model realizations; the state estimate
(orange) and the forecast (blue). (a) Sea surface height;
(b) sea surface temperature; (c) sea surface salinity; and
(d) HFRs.
Spatial maps of RMSA for SST observation sources for the
forecast (a, c) and the state estimate (b, d).
(a, b) OSTIA data (2007–2008); (c, d) NAVO data
(2008–2017). The typical error of representativeness is around
0.4 K.
Remotely sensed observations
Figure shows the RMSA between the observations and the
models for each source of remotely observed data. The state-estimate solution
reduces the RMSA compared with the forecast by 1.58 cm (17 %),
0.07 K (24 %), 0.01 ppt (3 %), and 8.39 cm s-1
(37 %) for sea surface height, sea surface temperature, sea surface
salinity, and HFR, respectively. In Fig. a the RMSA of the
state-estimate solution is close to the typical observational error of
7 cm, while in Fig. b we see the RMSA is comfortably less
than the 0.4 K representative error. Sea surface salinity is only
marginally improved by the state-estimate solution, but is slightly over the
prescribed observational error of 0.2 ppt. The RMSA of the currents
associated with HFR observations, shown in Fig. d, is
improved greatly by the state estimation; however, the mean value of 14 cm
is around double the typical error prescribed a priori of 7 cm. As shown
in the previous sections, this error was underestimated.
The ACC is also improved by the state estimate for all variables, as shown in
Fig. . For sea surface height, temperature, and salinity the
improvement is small due to a significant agreement in the forecast with
gains of 0.03, 0.02, and 0.01, respectively. The improvement in HFR is
much more significant, with an average correlation improvement from 0.35 to
0.68. As shown in Fig. d the free-running forecast model
can diverge from the observations enough to become negatively correlated over
a cycle, while the state-estimate solution is consistently positively
correlated.
Spatial maps of HFR statistics for south O`ahu for the
forecast (a, c) and the state estimate (b, d).
(a, b) RMSA; (c, d) ACC.
Figure shows the spatial RMSA between the forecast and
analyses model solutions and the observations for both sources of sea surface
temperature observations: OSTIA and NAVO. In both cases there is a clear
reduction in the RMSA, with the largest source of error in the areas leeward
of the islands, most notably the island of Hawai`i. This is due to higher
heat flux variability from a reduction in cloud cover
. Even in this peak area, the state-estimate
solution is around the observational error of representativeness of 0.4 K,
meaning the model is performing well with regards to SST.
Both RMSA and ACC between the experiments and HFR observations are shown in
Fig. for the island of O`ahu. The RMSA of the free-running
forecast is reasonably uniform across the region covered by the HFR, around
20–25 cm s-1, with some varying values around the extent of the radar
coverage. The inclusion of HFR observations in the state-estimate solution
leads to significantly reduced values of 12–15 cm s-1, a reduction of
almost half. The ACC is also significantly improved from a weak correlation
to a consistently strong positive one.
As discussed in , there are several reasons the model can
differ from surface current observations: the discretization of the model,
imperfect stratification, differing barotropic-to-baroclinic tide conversion
at Kaena Ridge, or mixing parameters that do not capture the real baroclinic
mixing. This may lead to a different location of the currents in the model
from those observed by the HFR; however, the model does a good job of reducing
these errors . The HFRs located on the island of Hawai`i
have a smaller coverage region, but the level of improvement from the
forecast to the state-estimate solution is consistent with the O`ahu results
shown here.
RMSA (solid) and ACC (dashed) profiles of subsurface
temperature (a) and salinity (b) for Argo floats,
Seagliders, and HOT CTDs for the forecast (blue) and the state estimate
(orange). Data were binned into 50 m intervals.
Subsurface observations
The in situ observation sources are Argo floats, Seagliders, and HOT CTDs, which also
show an improvement in the state estimate over the forecast. The subsurface
temperature RMSA values are reduced by an average of 0.03 K and salinity
by 0.01 ppt. The average RMSA is within the representative errors for both
variables at 0.8 K and 0.15 ppt, respectively. However, there are several
occasions when the RMSA value for a cycle exceeds that limit when there are
very few in situ observations available.
Figure shows the RMSA and ACC profiles for temperature and
salinity for each source of subsurface observation. For all
three sources, the greatest RMSA between the models and observations is along
the thermocline where minor differences in thermocline depth lead to
temperature differences. The state estimate improves the RMSA in this region
by 10–15 %. The thermocline location is also the source of the lowest
correlation between the observations and the model, which is improved by the
state estimate by ∼ 5 %. There is a high RMSA for Seagliders at the
base of their profiles (close to 1000 m). In this instance the
state estimate does not result in an improvement of the forecast. Many of the
glider missions operated in the shallow waters off the south coast of O`ahu
where processes are at much finer scale than can be resolved at 4 km
resolution. As such, the observational representation errors were higher.
For subsurface salinity (Fig. b), the improvements made by the
state-estimate solution occur almost exclusively above 500 m for Argo
floats and HOT CTDs. As with temperature the largest improvement is at the
top of the thermocline. There are some low ACC values lower down in the
profile between both models and the observations, but both the forecast and
state estimate perform equally at this depth. Seagliders produce the biggest
improvement in subsurface salinity model performance, with the state-estimate
solution up to 20 % better than the forecast for both RMSA and ACC. The
peak improvement is at the top of the thermocline, but there are improvements
throughout the profile.
Mean skill metric for remotely sensed observations as a function of
forecast length. Solid lines: skill (see Eqs.
and ); dashed lines: standard deviation of skill.
(a) Sea surface height; (b) sea surface temperature;
(c) sea surface salinity; (d) HFRs; and
(e) subsurface temperature.
Forecast skill
In this section we quantify the model skill by using a skill score evaluated
as the improvement against a reference field . For the
reference, we take the model value at the spatial location of each
observation at the time of initialization for each 8-day cycle and assume
persistence of this value throughout the 8-day cycle (persistence
assumption). The skill score (SS) for the state-estimate analysis and
forecast is then defined using the ratios of RMSAs with respect to the
observations:
SSa=1-RMSA(xa,y)RMSA(x0,y),SSf=1-RMSA(xf,y)RMSA(x0,y),
where the superscripts a, f, and 0 refer to the analysis, free-running
forecast, and persistence, respectively, and y indicates the observations.
Under this measure, a SS of 1 represents a perfect fit between the model
and observations, while a value of zero indicates where the model and
persistence values perform exactly the same. If the model is better than
persistence, then the skill score will lie in the range 0<SS<1
and the degree of improvement over persistence is determined by how close to
1 the score is. Conversely, a negative SS means the model is further from the
observations than persistence.
For this verification we wish to examine the effect of forecast length on the
skill. Starting with the same initial conditions as each state-estimate cycle
we produce an 8-day forecast, the length of two state-estimate cycles.
The RMSA is calculated every 3 h for each 8-day forecast, the
corresponding state-estimate cycles, and the persistence field from the start
of the forecast.
EOF1 and PC1 of initial condition increments for temperature,
east–west velocity, and north–south velocity (all averaged 0–100 m) and of
forcing perturbations applied to surface heat flux. The EOFs were calculated
using the routines described in .
Figure shows the mean SS over all cycles for remotely sensed
observations. For SSH, SST, and HFR, the skill for both the state-estimation
and free-running forecast is positive throughout, indicating that both models
are successful over persistence in representing those variables. SSS,
however,
is close to zero and slightly negative, meaning the models provide no better
information than persistence. SST values are consistently the highest, with a
reduction in skill versus persistence for both models once per day. This is
expected as initial conditions are used for persistence values and the
diurnal cycle will move ocean temperatures close to this persistence value
once per day. The state-estimate skill for HFR has a consistent value of
0.5 regardless of the forecast day, while the skill of the free-running
forecast decreases within the first 12 h and is closer to 0.2 for the rest
of the forecast period. This decrease in skill is driven by the fact that the
radials are dominated by the semidiurnal baroclinic and barotropic tides.
Analysis of increments
During each I4D-Var 4-day window, the initial model field and
time-varying boundary and surface forcings are adjusted to minimize the
residuals. The initial condition increments form a single record for each
cycle, while the boundary and surface forcings are perturbed every time they
are applied to the model. The perturbations applied to the boundary exhibit
only a minor influence on the model (not shown) due to the mean advection
speed (≈20 cm s-1) and sponge layer dampening near the
boundaries. We focus our analysis on the increments of the initial conditions
and the surface forcing.
Because we are analyzing the increments (rather than the state) to the
initial conditions and forcing fields, the mean increment should be zero
(unless there is a bias in the model), and we are looking to examine the
variability. Over the entire reanalysis period, the mean biases between the
model and observations for the different types are temperature
(-0.0048 K), salinity (0.0049 ppt), SSH (-7 mm), and HFR
(0.06 cm s-1). A consistent pattern or principal component may suggest
a repeated correction to account for missing or misrepresented physics in
the model.
Over the 10-year reanalysis, there are 917 analysis cycles with 16
surface forcing adjustments (four per day) per cycle. We calculated the
empirical orthogonal functions (EOFs) of the increments
applied to the forcing and the initial conditions to analyze the dominant
spatial patterns of the adjustments.
For each cycle, the initial perturbation of the primary model prognostic
variables are examined: sea surface height, temperature, salinity, east–west
velocity, and north–south velocity. With the exception of sea surface height,
each variable is averaged over the upper 100 m to cover the mixed layer
depth in the domain . The increments for salinity and sea
surface height as a percentage of the initial conditions are insignificant
(<1 %), while temperature increments (2–10 %) and the two velocity
fields (10–20 %) are significant enough to analyze.
The assimilation was configured to optimize the surface forcing increments
every 6 h (to avoid overadjustment). The time of day potentially impacts
forcing variables, particularly surface heat flux, so we calculate EOFs on
the increments for each of the four distinct times of day they occur (00:00,
06:00, 12:00, 18:00 UTC). Due to the size of the model grid, the number of
records, and the computational resources available the EOF calculation is
limited to a 4-year period, with approximately 1500 records. Several different
periods were examined with no significant differences in the structure of the
modes or their percentage of variance explained. The time of day does impact the
percentage of variance explained by each mode, most notably for surface heat
flux for which the effect of diurnal solar heating occurs. However, the overall
locations and magnitudes of the peaks and troughs as well as the temporal
evolution of PCs do not exhibit significant differences for each time of day,
so we present one of the modes for each considered variable.
Spatial EOF patterns and principal components (PCs) of wind stress
perturbations for the period prior to the assimilation of HFR measurements
(June 2007–September 2010).
The four key surface forcing terms are surface heat flux, surface salinity
flux, east–west wind stress, and north–south wind stress. Of these,
increments in surface salinity flux are quite small compared to their initial
value, while increments in surface heat flux (10 %–15 % of initial
value) and the wind stresses (15 %–20 % of initial value) are
significant.
For surface heat flux and near-surface temperature, we observe that the EOF1
modes represent 63 % and 20.8 % of the variability, respectively, with
a consistent sign over the region (Fig. ). This mode
essentially accounts for the bias between our ocean model and the WRF
atmospheric model used to force the surface. Unfortunately, WRF was not
integrated loosely coupled to the ROMS using the ROMS SST field; rather, it
was run using persistent estimates of daily SST during its integration. It
must be noted, however, that the monopole structure of the EOF1 does not
represent a constant offset between ROMS and WRF since the actual
perturbation of surface heat flux and increment applied to near-surface
temperature are given by the products of the respective EOF1 and the PC1. As
can be seen in the lower panels of Fig. , the temporal
evolution of the PC1 for both surface heat flux and near-surface temperature
adjustments is dominated by high-frequency, nonphysical variance.
Spatial EOF patterns and principal components (PCs) of wind stress
perturbations for the period including the assimilation of HFR measurements
(January 2011–January 2014).
The EOF1 modes of the near-surface velocity increments explain 26.1 % and
20.8 % of the variance, respectively. Both modes exhibit a strong impact
south of the main Hawaiian Islands. The structure of the wind stress curl in
this region results in the spin-up of cyclonic and anticyclonic eddies to the
north and south of the lee side of each island, respectively
. As a consequence, a zone of strong current shear is
created between the North Equatorial Current and the Hawaiian Lee Counter
Current (see also Fig. ). The EOF1 modes
of the near-surface velocity increments are responsible for adjusting the
state estimate for the significant eddy activity in the lee of Hawai`i.
The EOFs of surface wind stress increments are confined to relatively small
regions of the model domain (Figs.
and ). A significant change occurs after the HFR
observations come online. During the period prior to the availability of the
HFR data (June 2007–September 2010), the wind stress was primarily adjusted
in the lee regions where the winds are forced between islands (e.g., Kaiwi,
`Alenuihāhā Channels, and to a smaller degree over the Kaua`i
Channel; Fig. ). The wind stress curl in these
regions plays an important role as a vorticity source to the ocean
. Hence the adjustment of wind stress in the channels between
the islands is critical for a reliable representation of ocean conditions.
The magnitude and sign of the PCs of the wind stress adjustments for this period
are driven by day-to-day variability (Fig. ,
lower panels). Also, the PCs of the east–west wind stress and north–south
wind stress are largely uncorrelated, aggravating an interpretation of the
adjustments in terms of a larger-scale atmospheric pattern or wind stress
curl.
With the integration of the HFR measurements (October 2010), the dominant
wind stress increments occur across the shallow region close to the south
coast of O`ahu (Fig. ). The first mode for both
east–west and north–south wind stress exhibits a monopole structure
adjusting the wind stress uniformly across the area covered by the HFR and
its vicinity. The second modes have an east–west dipole structure that will
either increase or decrease the wind stress shear around the HFR region.
Similarly to the pre-HFR period, the PCs of the wind stress increments are
dominated by day-to-day variability and do not represent a physical mode.
Conclusions
We have presented a 10-year reanalysis of the PacIOOS Hawaiian Island Ocean
Forecast System and assessed the performance of the state-estimate solution
and free-running forecasts. Using a time-dependent incremental strong-constraint
four-dimensional variational data assimilation (I4D-Var) scheme, we
show that the model represents the observational data well over the time
period. The state-estimate solution reduces the RMSA compared to the forecast
by 3 % (salinity) to 37 % (surface velocities). A limitation of the
model–observation comparison is given by the fact that in the absence of a
sufficient number of independent observations, only assimilated data could
be used for the validation.
The largest reduction of the cost function of the state-estimate solution
occurs when minimizing the residuals to HFR data, with SST also accounting
for a significant improvement. On average, the assimilation achieves the
near-optimal solution; however, the variability is heavily influenced by the
HFR observations. The analysis suggests that the observational errors
associated with HFR are too low and results could be improved by redefining
these errors. This is supported by the increase in variability and upward
trend of optimality towards the end of the time period during which HFR observations
are most numerous.
The increments made by the reanalysis have revealed that sea surface height
and salinity initial conditions are not significantly adjusted by the I4D-Var
procedure, whereas temperature and velocity account for a significant change
from the forecast field. For the atmospheric forcing, surface salinity is
insignificant, but the adjustments made to surface heat flux and wind stresses
alter the forcings by up to 20 %. This corresponds to cost function
statistics that point to HFR and temperature as the two dominant observation
sources.
The dominant EOF mode for adjustments of surface heat flux and near-surface
temperature exhibits a monopole structure, indicating a slight bias correction
between the ocean and atmospheric model. The leading modes of wind stress
increments are concentrated in the region south of O`ahu. The wind stress
heavily influences the surface currents and adjustments are mostly made as a
consequence to HFR data. Additional analysis reveals that wind stress
adjustments in the channels between the islands dominated the increments in
the period prior to the radar-based measurements of surface currents.
The reanalysis has provided the testing for improvements to the PacIOOS
operational forecast system. The data are being used to update the back
catalog available to the public at
http://www.pacioos.hawaii.edu (last access: 22 December 2018)
and will influence the future results from daily forecasts. Analysis
of the I4D-Var increments has provided a greater understanding of the
variability in the region and will provide the basis for a move towards
ensemble forecasting in the region.
The specific ROMS Fortran source for this package is under
the MIT license and is available from
10.5281/zenodo.1493617. Model initial conditions and
boundary forcing come from the HYbrid Coordinate Ocean Model
(http://hycom.org, last access: 22 December 2018).
Atmospheric surface forcing and HF radar observations are
distributed through the PacIOOS data portal at
http://pacioos.hawaii.edu (last access: 22 December 2018). Satellite measurements come from
two sources; sea surface temperature and salinity are provided by the
Physical Oceanography Distributed Active Archive Centre at
http://podaac.jpl.nasa.gov (last access: 22 December 2018), and surface height
anomalies are provided by the Copernicus Marine Environment Monitoring
Service at http://marine.copernicus.eu (last access: 22 December 2018). In situ
measurements used are available from three sources: Argo measurements through
the Global Ocean Data Assimilation Experiment at
http://usgodae.org (last access: 22 December 2018), Seagliders through the School of
Ocean and Earth Science and Technology at the University of Hawai`i at
Mānoa at http://hahana.soest.hawaii.edu/seagliders (last access: 22 December 2018), and
CTDs through the Hawai`i Ocean Time-Series project at
http://hahana.soest.hawaii.edu/hot (last access: 22 December 2018). Reanalysis output is
produced as 3-hourly snapshots of the 3-D field temperature, salinity, and
velocities interpolated onto a z grid from the native s grid, as well as the
2-D sea surface height field for the full time period. These data are
archived through the PacIOOS data server at
http://oos.soest.hawaii.edu/thredds/idd/ocn_mod_hiig.html?dataset=roms_hiig_reanalysis (last access: 22 December 2018).
DP and BSP designed and conducted the reanalysis simulations.
All three authors contributed to the analysis and interpretation of the model results and to writing the paper.
The authors declare that they have no conflict of
interest.
Acknowledgements
The authors would like to thank the GODAE for hosting the Argo observations
and the HOT project for CTD and Seaglider data. The authors would also like
to thank Yi-Leng Chen of the University of Hawai`i Department of Meteorology
for the atmospheric model data MM5 and WRF. The authors are grateful to two
anonymous reviewers and the editor for helping improve this paper. This work
was supported by PacIOOS (http://pacioos.org, last access: 22 December 2018), which is a part of the US Integrated Ocean
Observing System (IOOS®), funded in part by
National Oceanic and Atmospheric Administration (NOAA) award
no. NA16NOS0120024. This is SOEST publication no. 10525. Edited by: Steven Phipps Reviewed by: two
anonymous referees
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