A systematic comparison of particle filter and EnKF in assimilating time-averaged observations
1 2
3 Huaran Liu1,2, Zhengyu Liu2, and Feiyu Lu2
4
5
1
6
China
7
2
College of Atmospheric Sciences, Nanjing University of Information Science & Technology,
Department of Atmospheric and Oceanic Sciences, University of Wisconsin-Madison, USA.
8 9 10 11 12
Corresponding author: Huaran Liu (
[email protected])
13
Key Points:
14
● Pseudo proxy experiments using particle filter with simple importance resampling and
15
ensemble Kalman filter to evaluate the performance for paleoclimate reconstruction
16
● The strengths and weaknesses of these two filters for paleoclimate data assimilation
17
● In sparse observation network, large localization radius is crucial to EnKF
18
19
Abstract
20 21
The pParticle filter (PF) and the ensemble Kalman filter (EnKF) are two promising and popularly
22
adopted types of ensemble-based data assimilation methods for paleoclimate reconstruction.
23
However, no systematic comparison between them has been attempted. We compare these two
24
uncertainty based methods in pseudo proxy experiments where synthetic seasonal mean sea
25
surface temperature (SST) observations are assimilated. Their skills were evaluated with regards
26
to local performance, large scale performance, and their ability to capture the modes of variability.
27
It is found that the EAKF (Ensemble Adjstment Kalman Filter, a variant of EnKF) performs better
28
than the particle filterPF with only one third of the ensemble size, despite particle filter’sPF’s
29
theoretical superiority in allowing for non-Gaussian statistics and nonlinear dynamics. The success
30
of the EAKF is attributed to the existence of large-scale co-variability and hence the adoption of a
31
large localization scale in the experiment, which allows broader areas to be impacted by a sparse
32
observation networkthe fact that: 1) Gaussian assumption is appropriate for this application; 2)The
33
EAKF is less sensitive to sampling errors than the PF due to the different methodological natures:
34
16 members are enough to capture accurate covariance for the EAKF, but 48 (even 96) members
35
still underrepresent the state space of such high-dimensional system for the PF. Our study
36
highlights the importance of using a large localization radius to constrain large-scale patterns for
37
the application of the EnKF to paleoclimate reconstruction given such sparse proxy network,
38
andThis study suggests that additional techniques, such as localization or clustered particle filter,
39
are needed to improve the particle filter for paleoclimate reconstruction, in addition to the simple
40
importance resampling which is currently adopted by most research.
41
42
1 Introduction
43
Past climate can be simulated in the state-of-the-art climate models (Otto-Blieser et al.2006;
44
Braconnot et al. 2012) or inferred from proxy-based reconstructions (Mann et al., 2008, 2009;
45
Marcott et al. 2013). Both approaches provide independent yet complementary information about
46
past climate and have been used independently for the reconstruction and understanding of the
47
past climate. Paleoclimate data assimilation (PDA) offers a new approach to further improve the
48
reconstruction of past climate by combining the observational information from the proxy records
49
and the dynamic constraints from climate models. PDA will constrain the model with proxy
50
observations and provide a complete field of the past climate state containing information that
51
cannot be directly derived from the proxy records, equivalent to reanalysis datasets that fill in the
52
spatial and temporal gaps in a sparse observation network (Kalnay, 2003). Furthermore, with
53
careful calibration of the uncertainty in the model and proxy data, PDA can also quantify the
54
uncertainty of the final analysis product.
55
In contrast to data assimilation (DA) techniques in meteorology, PDA is still in its infancy
56
and no universal methodology has been established. The difficulties posed by spatially and
57
temporally sparse, noisy and mostly indirect proxy records make it challenging to apply most
58
modern DA techniques directly to PDA. Various DA methods have been attempted since the
59
concept was introduced to the paleo field by Von Storch (2000). Early methods include pattern
60
nudging (von Storch et al., 2000) and forcing singular vectors (Barkmeijer et al., 2003). These
61
methods share the same idea by adding an artificial forcing term to the prognostic equations to
62
nudge the model towards a desired large-scale pattern derived from available observations. The
63
drawbacks of these methods are that they require a relatively dense observation network to capture
64
the large-scale pattern and that they cannot address the uncertainty of the analysis. Recently,
65
ensemble filter methods have been widely used for paleoclimate studies, in which a finite ensemble
66
of model simulations are used to represent the model’s statistical behavior and Bayesian estimation
67
theory is used to derive the posterior estimates (Leeuwen 2009; Evensen, 1994, 2003). These
68
ensemble methods fall into two categories: the particle filter and the ensemble-based Kalman filter.
69
The application of the particle filter to paleoclimate has experienced two three stages. In
70
the first stage, a simplified (off-line) particle filter was used and a single member that best matches
71
the proxy record is selected among an ensemble of pre-existing model simulations as the final
72
analysis (Goosse et al., 2006a, 2006b), hence no forwarding model is needed. In the second stage,
73
this method was modified towards an on-line degenerate particle filter; (Goosse et al. 2009, 2010a;
74
Crespin et al., 2009; Widmann et al., 2010), in which one member was selected as the best-fit of
75
the observations at the target period and is then used as the initial conditions for the subsequent
76
simulation. In the second third stage, the on-line degenerate particle filter evolved into the standard
77
particle filter with simple importance resampling (PF-SIR) is being used (Dubinkina et al. 2011;
78
Goosse et al., 2010, Goosse et al., 2012a, b; Annan and Hargreaves 2012; Mathiot et al., 2013;
79
Mairesse et al., 2013). In the PF-SIR, the analysis is derived as a weighted sum across all ensemble
80
members. The weights are calculated based on the closeness of each member to the available
81
observations. A resampling step is performed on the ensembles before proceeding to the next cycle
82
where the members with trivial weights are removed and the ones with non-trivial weights are
83
duplicated to maintain the same ensemble size. One advantage of the particle filter, in theory, is
84
its capability to provide the optimal analysis whether the error distribution is Gaussian or not
85
(Leeuwen 2009). However, the particle filter in general suffers from the so-called “curse of
86
dimensionality”, or particle impoverishment: A phenomenon where the non-trivial weights tend
87
to collapse to only a few particles while majority of the particles degenerate to trivial weights.
88
Resampling after each analysis step is used to alleviate such degeneracy, but PF-SIR still requires
89
a considerable amount of particles because the ensemble size required for a successful filtering
90
increases exponentially with the problem size (Snyder et al., 2008). Considering the large degree
91
of freedom of geophysical applications and the limited computation resources, the “curse of
92
dimensionality” has been a severe problem. However, some recent studies seem to be successful
93
in applying the PF-SIR for PDA using affordable ensemble sizes (N~100) (Dubinkina et al. 2011;
94
Goosse et al. 2011a, b; Mathiot et al., 2013, Mairesse et al., 2013). These studies did not encounter
95
filter degeneracy because they reduced the degree of freedom by performing time average, spatial
96
filtering, such as the long-term climatological temperature averaged in a single hemisphere.
97
Particle filter has also been used to investigate the causes and mechanisms of some climate
98
variability in the past (Goosse et al. 2011b; Mathiot et al., 2013).
99
The EnKF (Evensen, 1994, 2003) is a modern DA method that can be modified to
100
accommodate the time-averaged observations in paleo context (Dirren and Hakim 2005; Huntley
101
and Hakim 2010; Steiger et al. 2014; Hakim et al. 2016; Steiger and Hakim 2016). In the EnKF,
102
the final analysis equation is a linear combination of observation information and model estimate
103
with the weight determined according to the uncertainties (Evensen, 1994, 2003; Kalnay, 2003).
104
In the case of Gaussian error, the EnKF can achieve the optimal analysis. This optimal analysis in
105
theory maximizes the likelihood of achiving such an posterior estimates and minimizes the error
106
between itself and the prior estimates and the observations. In general, when the error is non-
107
Gaussian, however, there is no guarantee that the EnKF can achieve the optimal analysis as the
108
PF-SIR can. Nevertheless, the EnKF can provide robust and consistent analysis with relatively
109
small ensemble size as has been shown in numerous studies (e.g. Evensen, 1994, 2003; Zhang et
110
al. 2007)
111
Our goal is to perform a systematic comparison of the PF-SIR and the EnKF for
112
paleoclimate field reconstruction. We adopted on-line DA approach for both PF-SIR and EnKF,
113
which provides better temporal consistency (Matsikaris et al. 2015). The two methods will be
114
performed in pseudo proxy experiments (PPEs) which provide a synthetic, controlled test bed
115
(Smerdon 2012). We found that the EnKF with 16 ensemble members outperforms the PF-SIR
116
using 48 members in both the local reconstruction and hemispheric reconstruction. The conclusion
117
is robust even when the ensemble size of the PF-SIR is doubled to 96. This paper is organized as
118
follows. In section 2, we introduce the model and proxy network, evaluation metrics, and the
119
details of the assimilation methods. Section 3 compares the assimilation of the two filters and
120
discuss the advantages and disadvantages of each filter. The conclusion and some general
121
discussions will be given in section 4.
122 123
2 Model, Methods and Experimental Design
124 125
2.1 Model description
126
The Fast Ocean Atmosphere Model (FOAM) is used in this study. FOAM is a fully coupled global
127
atmosphere and ocean model (Jacob 1997). The atmosphere component is based on NCAR CCM2
128
and CCM3 models with R15 resolution (40 latitudes× 48 longitudes) and 18 vertical layers. The
129
ocean component is based on Modular Ocean Model (MOM) created by Geophysical Fluid
130
Dynamics Laboratory (GFDL), It has a regular 128x128 polar grid and 24 vertical levels. The
131
quality of the simulated climate compares well with higher resolution models. FOAM has been
132
used for various paleoclimate studies especially in Holocene (Liu et al. 2000; Liu et al. 2004; Liu
133
et al. 2006; Liu et al. 2007). It has also been used to test DA techniques and algorithms successfully
134
(Liu et al. 2014a; Liu et al. 2014b; Lu et al. 2015 part 2; Lu et al. 2016).
135 136
2.2 EAKF and its implementation.
137
Ensemble Adjustment Kalman Filter (EAKF) is used in FOAM. The EAKF is a variant of
138
EnKF and is a deterministic filter that does not require the perturbation of observations as EnKF
139
does (Anderson 2001, 2003). A brief introduction of the EnKF is provided below, which also
140
applies to the EAKF. For full comprehensive description, please refer to Anderson 2001, 2003 and
141
Kalnaey 2003.
142
The EnKF solves for the best least-square estimate of the system state given model
143
simulations and observations. Model simulations serve as a prior estimate (background) of the true
144
state, which is then updated with the observational information to solve for the posterior estimate
145
(analysis). The final analysis represents the most likely state of the climate system given model
146
estimates and observations. This process expressed in Bayesian framework is:
147
P(𝑿 | 𝒚) =
𝑷(𝒚|𝑿)𝑷(𝑿) 𝑷(𝒚)
(1)
148
Where 𝑷(𝑿) represents the prior distribution of the state X, P(𝒚 | 𝑿) represents the likelihood of
149
obtaining observation 𝒚 given model estimate 𝑿. 𝑷(𝒚) is a normalization term. P(𝑿 | 𝒚) is the
150
posterior distribution after considering observation information. The observations and the prior
151
estimate of the state are assumed to be unbiased with a Gaussian distribution. The model estimate
152
is characterized by its mean ̅̅̅ 𝑥 𝑏 and covariance matrix 𝑃𝑏 , where 𝑥 𝑏 is the prior state vector with
153
the dimension of the state size. The model estimate of the observation 𝑦 is then extracted from the
154
̅̅̅𝑏 ), where ℋ is the observation operator which projects the model space background ̅̅̅ 𝑥 𝑏 as ℋ(𝑥
155
into the observation space. The final analysis ̅̅̅ 𝑥 𝑎 is derived as a weighted sum of the observation
156
𝒚 and the model background ̅̅̅ 𝑥 𝑏 , and the weights are inversely proportional to their relative
157
uncertainty/error. The classic Kalman Filter update equations are:
158
̅̅̅𝑏 )) ̅̅̅ 𝑥 𝑎 = ̅̅̅ 𝑥 𝑏 + 𝐾(𝑦 − ℋ(𝑥
(2)
159
𝐾 = 𝑃𝑏 ℋ 𝑇 (ℋ𝑃𝑏 ℋ 𝑇 + 𝑅)−1
(3)
160
̅̅̅𝑏 )) of a grid point from a From the analysis equation, it can be see that the correction 𝐾(𝑦 − ℋ(𝑥
161
non-local observation is determined by the covariance between these two locations 𝑃𝑏 ℋ 𝑇 in the
162
gain matrix equation (3). If these two highly co-vary, the value of K is larger and hence it receives
163
a heavier correction, and vice versa. Therefore the EnKF and its variants are a covariance-based
164
method: as long as there is covariance, there is correction. In paleoclimatology, all proxy
165
observations contain time-averaged information, such as annual mean and seasonal mean
166
temperature, instead of instantaneous values like those of modern instrumental observations. The
167
algorithm is modified based on Hakim and Tardif (2005) to assimilate the time-averaged
168
observations. For example, if the observation 𝑦 in equation (2) represents seasonal mean
169
temperature, the corresponding model background ̅̅̅ 𝑥 𝑏 is also the seasonally averaged temperature.
170
Note that the covariance matrix 𝑃𝑏 is calculated from the time-averaged ensemble, therefore it
171
captures the low-frequency variability since the high-frequency variability has been smoothed out
172
during the time average.
173
To remove the long-distance spurious correlations caused by a limited ensemble size, a
174
covariance localization is implemented using the fifth order function of Gaspari and Cohn (1999).
175
Localization is an important technique used in the EnKF, which determines the influence radius
176
of the observation at a given location (Houtekamer and Mitchell 2001; Hamill and Whitaker 2001).
177
In a dense observation network, the localization radius is usually chosen to be small since most
178
model grids have a local observation constraint (Liu et al. 2014; Lu et al. 2014b, 2016). In a
179
spatially sparse observation network such as in the paleo context, a broader localization scale is
180
neccessary to extend the observation impact to larger regions. The localization scale in our EAKF
181
experiment is roughly 16 grid points in the ocean, which gives a distance of 5000 km on the equator,
182
and decreases poleward with the cosine of latitude. We will use “5000 km” to denote the EAKF
183
experiment with a localization scale of 16 grid points. Covariance inflation is also applied using
184
the relaxation method by Zhang et al. 2004 and the relaxation factor of 0.5. Further sensitivity tests
185
show that the performance is not sensitive to this parameter.. For the analysis cycle, we use an
186
“on-line” approach (with cycling).i.e. the analysis of the current step is used as the initial condition
187
of the next cycle. This EAKF method has been used to develop the first ensemble-based coupled
188
data assimilation (CDA) system in a fully coupled general circulation model (CGCM) (Zhang et
189
al.,2007). It has been implemented in FOAM to complete the first ensemble-based parameter
190
estimation experiment in a CGCM (Liu et al., 2014a, b).
191
2.3 Particle filter and its implementations
192
A standard particle filter with simple importance resampling (Gordon et al. 1993; Doucet et al.
193
2001; Leeuwen 2009), denoted as PF-SIR, is adopted in this work. The theoretical frameworkof
194
PF-SIR is also Bayes theorem as in equation (1). Here the particles refer to ensemble members,
195
and they provide a Monte Carlo approximation of the model distributions. For example,
196
given N particles, the prior distribution 𝑷(𝑿) can be constructed as a sum of 𝑁 delta functions
197
centered at each particle:
198
𝑃(𝑥) =
1 𝑁
∑𝑁 𝑖=1 𝛿(𝑥 − 𝑥𝑖 )
(4)
199
Given the prior distribution 𝑷(𝑿) and the likelihood 𝑷(𝒚|𝑿), the posterior density P (𝑿 | 𝒚) can
200
be approximated as:
201
𝑃(𝑥|𝑦) = ∑𝑁 𝑖=1 𝑤𝑖 𝛿(𝑥 − 𝑥𝑖 )
202
𝑤𝑖 =
with
𝑃(𝑦|𝑥𝑖 ) ∑𝑁 𝑗=1 𝑃(𝑦|𝑥𝑖 )
(5)
(6)
203
where 𝒘𝒊 is called the weight of the particle and it is a normalized likelihood. The likelihood
204
𝑃(𝑦|𝑥𝑖 ) is often taken to be Gaussian:
205
𝑃(𝑦|𝑥𝑖 ) = exp 𝑒 −
(ℋ(𝑥𝑖 )−𝑦)2 2𝐶
(7)
206
Where C is the error matrix and ℋ is the observation projector. Note that the likelihood of each
207
particle is a product of Gaussian weights at every observation location. Hence if the model estimate
208
differs significantly from the observation at only a few locations, the final weight will be reduced
209
to zero even if the majority of locations show a good match between the model and observation.
210
To mitigate the effects of the trivial weights at some locations on the final weight of the particle,
211
we add inflation to the weight calculated at every observation location before taken the product of
212
them. With the posterior distribution and weights, the final analysis will be:
213
214
̅̅̅̅̅̅ 𝑓(𝑥) = ∫ 𝑓(𝑥)𝑃(𝑥|𝑦)𝑑𝑥 ≈ ∑𝑁 𝑗=1 𝑤𝑖 𝑓(𝑥𝑖 )
(8)
215
Equations (4) through (7) construct the procedure of particle filter of sequential importance
216
sampling. The proposal density in this case is chosen to be the prior probability density and the
217
weights of the prior particles are equal (Doucet et al. 2001). This equal prior weight assumption is
218
achieved by resampling the particles at every filtering time step. Therefore, after importance
219
weights are calculated in equations (5) through (6), the particles are resampled according to their
220
importance weights. The particles with weights less than 𝑁 are dropped and particles with large
221
weights are duplicated. Though there are several resampling methods, we apply a systematic
222
resampling scheme (Hol et al. 2006) in this paper. After resampling, a small perturbation is added
223
to the identical particles. The identitical particles share the same ocean state, but differ in
224
atmospheric fields. These different atmospheric fields act as the perturbations. These new model
225
states will be used as initial conditions for the next cycle. In our study, about half of the 48 particles
226
end up having trivial weights and get removed at every analysis step, they are then replaced by the
227
duplications of the remaing particles.
1
228
Again, note that we are assimilating observations that contain time-averaged information,
229
hence the background is temporally averaged before calculating the importance weights. This
230
time-averaging also help to reduce the number of degree of freedom (Dubinkina et al. 2011). The
231
assimilation procedure is essentially the same as in the previous studies (Dubinkina et al. 2011;
232
Goosse et al. 2012a, b) except that no spatial filtering is applied. Both our PF-SIR and the PF-SIRs
233
in the previous studies referenced above do not have the localization implemented (The more
234
recently developed techniques in Poterjoy 2016 and Lee and Majda 2016).
235
2.4 Pseudo proxy network and experiments
236
We use pseudo seasonal mean sea surface temperature (SST) observations to conduct the
237
PPEs. The pseudo proxy locations are chosen from Marcott et al. 2013, which provides 47
238
observations globally (Figure 1a) with a higher density in the northern hemisphere. The pseudo
239
proxy time series are generated by adding Gaussian noise with a standard deviation of 0.5 °𝐶 to a
240
specific model control run representing the actual state or “truth”. The Gaussian noise mimics the
241
errors associated with the proxy observation. The performance of each DA method is measured
242
using the root-mean-square error (RMSE) and correlation coefficient between the analysis and the
243
truth.
1
𝑇 2 𝑅𝑀𝑆𝐸𝑖 =√𝑛𝑡 ∑𝑡[𝑋𝑖,𝑡 − 𝑋𝑖,𝑡 ]
244
(9)
245
𝑇 where 𝑖 represents the location, 𝑡 represents the time, 𝑋𝑖,𝑡 is the truth and 𝑋𝑖,𝑡 is the analysis. In all
246
experiments, we use 16 members for EAKF and 48 for PF-SIR. We did not use the same ensemble
247
size because EAKF shows decent performance even using a relatively small number of ensemble
248
members (Lu et al. 2014b, 2016). Our results will show that even with only 16 members, EAKF
249
still significantly outperforms PF-SIR with 48 members.
250
A 200-year-long experiment is carried out for both EAKF and PF-SIR assimilating
251
seasonal SST pseudo observations. Two EAKF sensitivity experiments were also carried out using
252
smaller localization scale, 2000 km and 500km, to illustrate the importance of localization radius
253
to the performance of the EAKF. Furthermore, to test the sensitivity of ensemble size, we also
254
carried out a PF-SIR experiment of 96 members and a EAKF-5000km experiment of 32 members.
255
These two ensemble size sensitivity tests are only 50-year long because of the expensive
256
computational costs.
257
3 Results
258
The performance of the two filters was mainly assessed by computing the correlation on
259
both local scale and hemispheric scale between the analysis SST and the truth. We also investigate
260
the ability of each filter in capturingto capture the modes of variability by performing empirical
261
orthogonal function (EOF) analysis on both the simulation and the truth.
262
3.1 Local reconstruction skills
263
Figure 1 displays the analysis quality of SST characterized by RMSE and correlation
264
coefficient for the PF-SIR and the EAKF of 5000 km localization scale. The RMSE is calculated
265
from the differences between seasonally-averaged ensemble mean analysis and the “truth” at each
266
grid point and normalized by its corresponding natural variability from Figure 1a. The local
267
reconstruction skill of the EAKF is significantly better than that of the PF-SIR at both the proxy
268
location and the nearby regions. In the North Atlantic where the observations are dense, the
269
correlation with truth is above 0.6 basin-wide for the EAKF (Figure 1e) but below 0.6 for the PF-
270
SIR (Figure 1c); and the RMSE ranges from 0.4 to 0.6 for the EAKF (Figure 1d) but above 0.8 for
271
the PF-SIR (Figure 1b). In regions that have observation clusters such as the south of Iceland and
272
the east of Newfoundland, the correlation coefficient for the EAKF reaches 0.85. In the tropical
273
eastern Pacific where there is only one observation in the far east, the EAKF still maintains a
274
correlation of around 0.4 and RMSE around 0.9 (Figure 1e and d), but the PF-SIR’s reconstruction
275
skill is fairly weak (Figure 1b and c).
276
The spatial reconstruction skill of the EAKF greatly depends on the localization scale.
277
Recall that localization determines how far can the observation at a given location influence its
278
nearby locations. In the bottom two rows of Figure 1, we show the results of the EAKF with
279
decreasing localization scales of 2000 km (Figure 1f, g) and 500 km (Figure 1h, i). When the
280
localization scale is reduced, the area of higher correlation and lower RMSE also shrink towards
281
the observation location. When the localization scale is reduced to 500 km, well-constrained
Formatted: Left, Indent: First line: 0.5", Widow/Orphan control
282
regions shrink to small areas tightly surrounding the proxy locations, as colored patches in
283
Figure 1d and 1e diminish to colored dots in Figure 1h and 1i.
284
In PDA, larger localization scale can extend the observational impact of the sparse proxy
285
network, especially in regions where large-scale variability exists and only a few proxies are
286
available. For example, there are basin-wide covarying patterns, such as Pacific Decadal
287
Oscillation (PDO) in the north Pacific and El Nino-Southern Oscillation (ENSO) in the tropical
288
Pacific. One observation along the west coast of North America (Figure 1a) is able to partially
289
constrain the evolution of PDO in the entire basin. This is also demonstrated in section 3.3 in terms
290
of modes of variability. When localization radius is small, the reconstruction skill is lost away
291
from the observation sites.
292 293
3.2 Hemispheric averaged SST time series
294 295
Figure 2 shows the hemispheric averaged SST time series, derived as area weighted mean
296
of the analysis. For the northern hemisphere (NH) where most of the observations are located and
297
where the ocean area is not overwhelmingly dominant (~60% coverage) (Figure 2d), the EAKF
298
with 5000 km localization scale (red line) and PF-SIR (blue line) have a correlation with the true
299
state (black line) of 0.85 and 0.67, respectively. They both capture the decadal scale variability
300
very well, except that during the year 30 to 80 the PF-SIR starts to suffer from particle
301
impoverishment where it cannot select good members that match the observations out of its 48
302
members. This mismatch greatly diminishes the final correlation coefficient calculated between
303
PF-SIR and the truth. To take such situations into account, we calculate the correlation coefficient
304
between the truth curve and the other three curves in Figure 2 a, d, and g at different time segments
305
(Figure 3), such as 20 years, 40 years and 100 years. These segments are sampled from the 200-
306
year simulation. Thus each dot in Figure 3 represents the correlation coefficient of a sample. We
307
can see that the EAKF- 5000km has higher correlation coefficient than the PF-SIR at 20 years, 40
308
years, 100 years and 200 years sample length (Figure 3a). The distribution of the dots is more
309
compact for the EAKF-5000km than the PF-SIR, which suggests that the performance of the
310
EAKF-5000km is more stable than the PF-SIR. Similar results can also be found in Figure 3b-c.
311
In addition, despite that PF-SIR captures the general variability in the northern hemisphere, there
312
is a small phase shift between the PF-SIR and the truth. This again reflects the nature of the PF-
313
SIR in that the analysis is reconstructed from existing ensemble members that are closest to the
314
observation and the analysis will deviate from the truth if all the members are far away from the
315
observations. In contrast, the EAKF corrects each member using the observation increment so that
316
every member in the ensemble will be “dragged” towards the true state. Hence the final analysis
317
has better synchronization with the truth time series compared to the PF-SIR. For the southern
318
hemisphere (SH), both methods failed to track the true state due to very limited observations and
319
the large area of ocean (~80% coverage) (Figure 2g). On the global scale (Figure 2a), the
320
correlation with truth for the global averaged SST is 0.61 for the EAKF and 0.24 for the PF-SIR.
321
The low correlation coefficient of the PF-SIR is further reduced by the poor performance from the
322
SH. Therefore, even though the PF-SIR is deficient at local reconstruction as shown in Figure 1b
323
and c, it seems to be able to capture some hemispheric-scale variability if the ensemble size is
324
sufficiently large. However, even with 3 times the computational expense, the PF-SIR with 48
325
members still underperforms the EAKF with 16 members in the reconstruction of hemispheric-
326
scale SST.
327
The uncertainty of the final analysis is also shown in Fig.2, as represented by its ensemble
328
members. The second column is for the EAKF with 5000 km localization radius (the pink curves
329
as the 16 ensemble members)PF-SIR (light blue curves as the 48 ensemble members) and the third
330
column is for the PF-SIR (light blue curves as the 48 ensemble members) EAKF with 5000 km
331
localization radius (the pink curves as the 16 ensemble members). In general, the PF-SIR tends to
332
underestimate the uncertainty of the model forecasts, an indication of the problem of filter
333
degeneracy. All the 48 ensemble members cluster tightly at the hemispheric and global scales and
334
the spread is not large enough to encompass the truth time series. This is because the posterior
335
ensemble are heavily bounded by the prior ensemble: it is impossible to generate the posterior
336
ensemble outside the span of the prior ensemble even when the observation is outside the span.
337
For the EAKF, the truth series always lies within the uncertainty of the 16 ensemble members. In
338
the NH (Figure 2f), abundant observations lead to stronger constraint of the analysis and hence
339
smaller uncertainty, while in the SH (Figure 2i), weak observational constraint causes large spread
340
among the 16 ensemble members. This suggests that the EAKF provides a better representation of
341
the uncertainty with only 16 members than the PF-SIR with 48 members.
342 343
3.3 Modes of climate variability
344 345
In this section, we investigate the ability of the two methods in capturing the modes of
346
climate variability. Empirical Orthogonal Function (EOF) analysis is performed on the four PDA
347
analyses (PF-SIR and three EAKF experiments with different localization radius) as well as the
348
true state on different regions. The EOF patterns and principle components (PCs) of the first mode
349
of variability are shown in Figure 4 and Figure 6-8. In the observation-dense North Atlantic (Right
350
column in Figure 4), both the PF-SIR and the EAKF-5000 km capture the 1st EOF pattern that is
351
almost identical to the truth in both pattern and strength, and the pattern correlation is close to 1
352
for both experiments (Table 1). The corresponding PCs of these two methods show similar decadal
353
variability to the truth. The temporal correlation coefficients between the PC time series of each
354
filter and that of the truth is as high as 0.78 for EAKF-5000km and 0.67 for PF-SIR (Table 1).
355
However, the explained variance of the first mode is much higher in the PF-SIR (25.9%) and
356
EAKF-5000km (24.7%) than that in truth (18.8%), which suggests that both filters have trouble
357
constraining the precise variability given so limited observational information. As the localization
358
radius in the EAKF is decreased, the spatial pattern gradually becomes weaker (Figure 4f, h and j)
359
and the temporal correlation coefficient also decreases (Table 1). Consistent with the grid point
360
analysis and hemispheric analysis, a large localization scale is necessary even in relatively dense
361
observation area.
362
In the North Pacific, PF-SIR (Figure 4c) captured clearly a PDO pattern similar to the truth
363
(Figure 4a).The correlation coefficient between the corresponding PC time series and the truth in
364
the northern hemisphere is 0.23 for PF-SIR and 0.57 for EAKF- 5000 km (Table 1). Note that there
365
are very few observations in the North Pacific: one around the Oregon coast and a few around east
366
China Sea (Figure 1a). The vast majority of the ocean basin remains unobserved. Nevertheless,
367
both PF-SIR and EAKF- 5000 km can capture the PDO spatial pattern and constrain the temporal
368
variability. However, the temporal variability of the PF-SIR is much less accurate than that of the
369
EAKF- 5000 km. The ability of the EAKF to constrain the entire basin using very few observations
370
is due to two factors: the existence of a large-scale co-varying pattern, namely PDO, and the
371
existence of observations in the critical region, in this case, around the Oregon coast. The control
372
of this basin-wide co-variability through the critical observation location is reflected by the one-
Formatted: Indent: First line: 0.5"
373
point correlation map in Figure 5a. The observation in the crucial region therefore provides
374
information about the phase of the large-scale co-variability. With this information, the PF-SIR
375
can select members that have the correct phase and the EAKF can nudge the impact area towards
376
the targeted phases. In this case, EAKF requires a large localization radius such that the entire co-
377
varied area can be corrected. As can be seen from Figure 4e, g and i, the spatial pattern gradually
378
fades away as the localization scale is decreased, as does the temporal correlation with truth (Table
379
1).
380
As is the case in the North Pacific, the tropical Pacific also has large-scale co-variability
381
with the observation location (Figure 5b) and only one observation located in the far eastern
382
portion of the basin (Figure 1a). Like north Pacific, both PF-SIR and EAKF 5000 km captured the
383
pattern of ENSO with pattern correlation coefficient close to 1, although the magnitude is weaker
384
than truth. The EAKF-5000km shows better temporal variability, as the correlation in the first PC
385
is 0.43 compared to 0.25 for the PF-SIR. The weaker spatial pattern but better temporal variability
386
in the tropics and North Pacific for the EAKF compared to the PF-SIR is due to the fundamental
387
differences of the two filters. The analysis of EAKF is ensemble mean output, thus when
388
observational constraint is weak, the ensemble mean tends to average out the internal variability
389
and result in a weaker spatial pattern. Also, recall that in the EAKF the physical field in each
390
member is modified by the observation innovation. Hence given the phase information of the large-
391
scale pattern from the key observation, the EAKF will directly correct all members and drag them
392
towards the correct phase. The PF-SIR does not have this feature because the analysis of the PF-
393
SIR is a weighted sum across all members (The physical field in each member is not modified),
394
and the weight attached to each member is based on observations from all locations. Therefore,
395
the local constraint of a given observation will be mitigated by observations from other locations.
396
Thus, the temporal constraint of these key observations on their neighboring regions is weaker in
397
the PF-SIR than that in the EAKF. As localization radius is decreased in the two additional
398
experiments of the EAKF, the first EOF pattern gets weaker in the tropics and the PC correlation
399
with truth also decreases. When localization radius is decreased to 500 km, the EAKF barely has
400
any constraint of the large-scale variability in this area (Figure 8 and Table 1).
401
In the SH where there are very few observations, both methods failed to reconstruct the
402
leading modes of similar patterns to the truth (Figure 7). The spatial field of the PF-SIR bears some
403
similarities to the truth in the south Pacific, but the temporal variability of this pattern has no
404
correlation with that of truth at all. The three EAKF experiments capture a pattern that co-varies
405
with the truth with a correlation coefficient around 0.3 (Table 1), but these patterns do not
406
resemble the truth that much. The EAKF- 5000 km maintains some performance near observation
407
locations such as around South Atlantic and the west coast of Chile and New Zealand (green
408
patches in Figure 1e). But these limited observations are far from enough to constrain the entire
409
southern hemisphere considering that there is no large-scale co-variability in the southern
410
hemisphere as in the northern hemisphere (Figure 5c and d). Hence it is not surprising that both
411
methods perform poorly in the southern hemisphere (Table 1).
412 413
3.4 Reconstruction skills on other variables
414
From the reconstructed SST analysis, we see that the EAKF-5000km has better
415
performance than PF-SIR is both the local reconstruction and hemispherically averaged
416
reconstruction. In this section, we will take look at the spatial reconstruction skills on other
417
variables, namely air surface temperature (AST), 500hPa geopotential height (Z) and precipitation
418
(PRECP) for both filters (Figure 9). Since RMSE ratio map shows similar skills to spatial
419
correlation map, we only show correlation map for these variables. The correlation coefficients are
420
calculated over the entire 200-year simulation after smoothing the time series at each grid point to
421
remove the high-frequency noise in the atmosphere. The AST reconstructions (Figure 9a-b) are
422
similar to the SST reconstruction (Figure 1c and e) over the ocean area, but have a slightly larger
423
impact region away from the observation location (Figure 9a-b). Over land, the reconstruction
424
skills mainly exist in coastal regions where there are observations in the nearby ocean, such as
425
Eastern North America, Greenland, North Africa and East Asia. Again, EAKF outperforms the
426
PF-SIR by showing larger area of high correlation coefficients especially in the Pacific ocean and
427
the south Atlantic ocean. For 500hPa geopotential height, the patterns are similar to AST (Figure
428
9c-d). Both PF-SIR and EAKF have poor reconstruction skills for precipitation. This is because
429
that precipitation is not only influenced by temperature, but also strongly influenced by the
430
moisture distribution and the wind field. Therefore it is not surprising to see such poor
431
reconstruction skills on precipitation from both filters.
432 433
3.5 sensitivity to ensemble size and localization scale
434 435
Considering a sparse observation network in PDA, the large localization radius is crucial
436
for EAKF to extend the observation impact and hence to constrain large scale variability as seen
437
from previous sections. This leads to the question on the optimal localization radius for such PDA
438
application. To find such optimal localization radius, we further increased 5000km to 10,000km
439
in the EAKF. This is almost equivalent to that each observation updates the entire globe. We found
440
that in terms of local reconstruction skill, North Pacific and tropical Pacific have stronger PDO
441
and ENSO pattern compared to the EAKF-5000km, but except these regions, the patterns in other
442
regions are almost identical (Figure not shown). In terms of hemispheric averaged SST time series,
443
the correlation for the EAKF-10000km is even slightly smaller than the EAKF-5000km. This is
444
because in regions that do not have the large-scale co-variability, a large localization radius allows
445
sampling error to come into play, which eventually deteriorates the hemispheric-scale performance.
446
Therefore, although large localization scale is necessary for PDA applications, it is not “the larger,
447
the better”. This suggests that the optimal localization radius for PDA applications is not a globally
448
uniform value, but a spatially-varying parameter. Zhen and Zhang 2014 has proposed a variational
449
approach to adaptively determines the optimum radius of influence for ensemble covariance
450
localization. However, the spatially-varied localization parameter is beyond the scope of this paper,
451
but is worthy of future investigation.
452
Ensemble size is an important factor for ensemble filters, especially for the PF-SIR which
453
requires an ensemble size that increases exponentially with the dimension of the system (Snyder
454
et al. 2008), and previous studies adopting on-line PF-SIR uses 96 ensemble members (Dubinkina
455
et al. 2011; Goosse et al. 2012). We double the ensemble size of the EAKF-5000km and PF-SIR
456
and run the experiment for 50 years to see if this will greatly improve the performance of the two
457
filters and potentially change the relative performance. For the EAKF-5000km, doubling the
458
ensemble size yield slight improvement in local skill (Figure 10b-c) compared to previous case
459
using only 16 members (Figure 1d-e). One noticeable improvement is in the tropical Pacific. For
460
the northern hemispheric SST time series, the correlation with truth has increased from 0.79 to
461
0.91. The global mean temperature correlation has increased from 0.54 to 0.72 (Figure 2). For the
462
PF-SIR, there is noticeable improvement in the local reconstruction scale (Figure 10d-e) such as
463
in the North Pacific and South Pacific. But for hemispheric averaged SST, there is no significant
464
improvement. EAKF-5000km still outperforms the PF-SIR when ensemble size is doubled for
465
both filters. In fact, EAKF-5000km with only 16 members still significantly outperforms the PF-
466
SIR with 96 members in both local skill and hemispheric SST time series reconstruction.
467 468
4 Summary and discussion
469 470
Particle filter with simple importance resampling and ensemble Kalman filter are emerging
471
as two popular ensemble-based data assimilation methods for paleoclimate studies. Here, using a
472
CGCM (FOAM), we systematically compare the performance of the two filters in assimilating
473
seasonally averaged observations of SST in PPEs and provide some insights into the choice for
474
PDA methodology. It is found that EnKF has overall better performance than the PF-SIR using
475
only one third the number of ensemble members. On local scales, EnKF shows good reconstruction
476
skill in the northern hemisphere where observations are relatively abundant, as well as in regions
477
that have few observations but has large-scale variability of coherent pattern, such as the tropical
478
Pacific and the North Pacific. For the PF-SIR, the correlation with truth is much smaller compared
479
to the EnKF and only the north Atlantic region has non-trivia correlation coefficient. In the
480
hemispheric reconstruction, both methods can capture the decadal variability in the data-rich
481
northern hemisphere and lose skill in the data-sparse southern hemisphere. In general, the EnKF
482
has higher correlation with the actual time series than the PF-SIR. PF-SIR is also subject to particle
483
impoverishment where all members end up far from the observations and hence few particles can
484
be selected out of its 48 ensemble members. When such degeneracy occurs, the reconstruction will
485
deviate from the truth substantially as shown in the year 30 - 80 in the 200-year-long experiment
486
(Figure 2). Doubling the ensemble size of the PF-SIR improve the local reconstruction, but the
487
overall reconstruction skill is still smaller than the EnKF with only 16 members. Hence we
488
conclude that the EnKF is more consistent and stable with much cheaper computation cost
489
compared to the PF-SIR.
490
The success of the EAKF over the PF-SIR in this PPE is attributed to the fact that a)
491
Gaussian assumption is appropriate for this application. b) The EAKF is less sensitive to sampling
492
errors than PF-SIR due to different natures of the two filters. 16 members is enough to capture the
493
covariance used in equation (3), but 48 (even 96) members still underrepresent the state space of
494
such high-dimensional system. This underrepresentation is manifested as the bounding of the
495
posterior particles of prior ensemble, since it is impossible to generate posterior samples outside
496
the span of the prior ensemble even when the observation is far outside of the prior ensemble.
497
Therefore a sufficient ensemble size is critical to the PF-SIR, and this “sufficient” ensemble size
498
is much greater than that required by the EAKF. c)Note that one important practice of applying
499
EAKF to PDA is the choice of a large localization radius since there There exists is large-scale co-
500
variability on slow time scales., which This practice benefits the covariance-based filter
501
substantially. As the localization radius of the EnKF is decreased in the two addition experiments,
502
the performances on both the local scale and hemispheric scale are diminished. This localization
503
radius effect is most obvious in regions that have large-scale co-variability such as tropical Pacific
504
and the North Pacific. One observation in these regions is able to constrain the entire basin given
505
a larger observation impact radius. Three EAKF experiments with different localization scales has
506
clearly illustrated this point, but further study can be conducted to apply different localization
507
radius at different grid points: regions that have large-scale co-variability are assigned larger
508
localization radius, while regions that do not have such feature or have relatively dense
509
observations are assigned smaller localization scale. This localized localization radius network
510
will exploit the sparse observation information and at the same time avoid the long-distance
511
spurious correlation, which will be very suitable to PDA applications.
512
These results raise the question of which filter should be preferred in the future for
513
paleoclimate studies. In our experiments, we show that the EnKF with only 16 members
514
outperforms the PF-SIR with 48 (even 96) members in almost every way. In theory particle filter
515
allows non-Gaussianity, which is an advantage over the EnKF. The challenge for particle filter is,
516
however, the “curse of dimensionality”. To better apply particle filter for paleoclimate studies,
517
more techniques such as localized and clustered particle filter (Poterjoy 2016; Lee and Majda 2016)
518
should be considered instead of using just the particle filter with simple importance resampling.
519 520
Acknowledgement
521 522
We gratefully appreciate the help of Zachary R. Hansen in editing the manuscript. This work is
523
supported by a grant from NUIST, Chinese NSF41630527 and NSF P2C2 program. The data used
524
in this paper is stored in UCAR HPSS system and is available upon request from
525
[email protected].
526
527
References
528
Anderson, J. L., (2001), An Ensemble Adjustment Kalman Filter for Data Assimilation. Mon.
529
Weather Rev., 129, 2884–2903, doi:10.1175/15200493(2001)1292.0.CO;
530
2.
531 532
533 534
Anderson, J. L., (2003), A local least squares framework for ensemble filtering. Mon. Weather Rev., 131, 634–642, doi:10.1175/1520-0493(2003)1312.0.CO;2.
Annan, J. D., and J. C. Hargreaves, (2012), Identification of climatic state with limited proxy data. Clim. Past, 8, 1141–1151, doi:10.5194/cp-8-1141-2012.
535
Braconnot, P., S. P. Harrison, M. Kageyama, P. J. Bartlein, V. Masson-Delmotte, A. Abe-Ouchi,
536
B. Otto-Bliesner, and Y. Zhao, (2012), Evaluation of climate models using palaeoclimatic
537
data. Nat. Clim. Chang., 2, 417–424, doi:10.1038/nclimate1456.
538 539
Barkmeijer, J., Iversen, T. and Palmer, T. N. (2003), Forcing singular vectors and other sensitive model structures. Q.J.R. Meteorol. Soc., 129: 2401–2423. doi:10.1256/qj.02.126
540
Bhend, J., J. Franke, D. Folini, M. Wild, and S. Brönnimann, (2012), An ensemble-based approach
541
to climate reconstructions, Clim. Past, 8(3), 963–976, doi:10.5194/cp-8-963-2012.
542
Crespin, E., H. Goosse, T. Fichefet, and M. E. Mann, (2009), The 15th century Arctic warming
543
in coupled model simulations with data assimilation. Clim. Past, 5, 389–401,
544
doi:10.5194/cp-5-389-2009.
545 546 547
Doucet, A., N. de Freitas, and N. Gordon, (2001), Sequential MonteCarlo Methods in Practice. Springer-Verlag, 581 pp.
548 549
Dirren, S., and G. Hakim, (2005), Toward the assimilation of timeaveraged observations. Geophys. Res. Lett., 32, L04804, doi:10.1029/2004GL021444.
550
Dubinkina S, Goosse H, Sallaz-Damaz Y, Crespin E, Crucifix M, (2011) Testing a particle filter
551
to reconstruct climate changes over the past centuries. Int J Bifurc Chaos 21(12):3611–3618
552
Evensen, G., (1994), Sequential data assimilation with a nonlinear quasi-geostrophic model using
553
Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99, 10143,
554
doi:10.1029/94JC00572.
555 556
557 558
Evensen, G., (2003), The Ensemble Kalman Filter : theoretical formulation and practical implementation. 343–367, doi:10.1007/s10236-003-0036-9.
Gordon N J, Salmond D J, Smith A F M. (1993), Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE P F Radar Sign Proc,140: 107–113
559
Goosse, H., H. Renssen, A. Timmermann, R. Bradley, and M. Mann (2006a), Using paleoclimate
560
proxy-data to select optimal realisations in an ensemble of simulations of the climate of the
561
past millennium, Clim. Dyn., 27, 165–184, doi:10.1007/s00382-006-0128-6.
562
Goosse, H, O. Arzel, J. Luterbacher, M. E. Mann, H. Renssen (2006b), The origin of the European
563
"Medieval Warm Period". Climate of the Past Discussions, European Geosciences Union
564
(EGU), 2 (3), pp.285-314.
565
Goosse, H, E. Crespin, A. de Montety, M.E. Mann, H. Ressen and A. Timmermann (2010a),
566
Reconstructing surface temperature changes over the past 600 years using climate model
567
simulations
with
data
568
doi:10.1029/2009JD012737.
assimilation,
J.
Geophys.
Res.,
115,
D09108,
569
Goosse, H., Lefebvre, W., de Montety, A., Crespin, E., Orsi, A., (2009). Consistent past half-
570
century trends in the atmosphere, the sea ice and the ocean at high southern latitudes. Climate
571
Dynamics 33, 999e1016.
572
Goosse, H., J. Guiot, M. E. Mann, S. Dubinkina, and Y. SallazDamaz, (20121a): The medieval
573
climate anomaly in Europe: Comparison of the summer and annual mean signals in two
574
reconstructions and in simulations with data assimilation. Global Planet. Change,
575
doi:10.1016/j.gloplacha.2011.07.002
576
Goosse, H., J.Guiot, M.E. Mann, S.Dubinkina, and Y. Sallaz-Damaz (20121b), The Medieval
577
Climate Anomaly in Europe in simulations with data assimilation. Quat. Int., 279–280, 173–
578
174, doi:10.1016/j.quaint.2012.08.246.
579 580
581 582
Hakim, G. J., J. Emile-geay, E. J. Steig, D. Noone, and M. David, (2016), The Last Millennium Climate Reanalysis Project : Framework and First Results. 98195, 1–56.
Huntley, H. S., and G. J. Hakim (2010), Assimilation of time-averaged observations in a quasigeostrophic atmospheric jet model, Clim. Dyn., 35(6), 995–1009.
583
Houtekamer, P. L., and H. L. Mitchell, (2001), A sequential ensemble {Kalman} filter for
584
atmospheric data assimilation. Mon. Weather Rev., 129, 123–137, doi:10.1175/1520-
585
0493(2001)1292.0.CO;2.
586
Hamill, T. M., J. S. Whitaker, and C. Snyder, (2001): Distance-Dependent Filtering of Background
587
Error Covariance Estimates in an Ensemble Kalman Filter. Mon. Weather Rev., 129, 2776–
588
2790, doi:10.1175/1520-0493(2001)1292.0.CO;2.
589
Hol, J. D., T. B.Schon, and F.Gustafsson (2006), On resampling algorithms for particle filters,
590
paper presented at Nonlinear Statistical Signal Processing IEEE Workshop, Linkoping,
591
Sweden.
592
593
594 595
596 597
598 599 600
Jacob, R., (1997): Low Frequency Variability in a Simulated Atmosphere Ocean System.
University of Wisconsin-Madison, 159 pp.
Kalnay E. (2003), Atmospheric Modeling, Data Assimilation and Predictability. 4th ed. Cambridge: Cambridge University Press
Liu, Z., J. E. Kutzbach, and L. Wu (2000), Modeling climatic shifts of el Nino variability in the Holocence. Geophys. Res. Lett., 27, 2265–2268, doi:10.1029/2000GL011452.
Liu, Z., Harrison, S.P., Kutzbach, J., Otto-Bliesner, B., (2004). Global monsoons in the MidHolocene and oceanic feedback. Clim. Dyn. 22, 157–182. Liu, Z., Wang, Y., Gallimore, R., Notaro, M., Prentice, I.C., (2006). On the cause of abrupt
601
vegetation collapse in North Africa during the Holocene: climate variability vs. vegetation
602
feedback. Geophys. Res. Lett. 33, L22709. doi:10.1029/2006GL028062.
603
Liu, Z., Wang, Y., Gallimore, R., Gasse, F., Johnson, T., deMenocal, P., Adkins, J., Notaro, M.,
604
Prentice, I.C., Kutzbach, J., Jacob, R., Behling, P., Wang, L., Ong, E., (2007), Simulating
605
the transient evolution and abrupt change of Northern Africa atmosphere–ocean–terrestrial
606
607 608
609
ecosystem in the Holocene. Quaternary Science Reviews 26, 1818–1837.
Leeuwn, P., (2009), Particle Filtering in Geophysical Systems. Monthly Weather Reviews, 4089– 4114, doi:10.1175/2009MWR2835.1.
Liu, Y., Z. Liu, S. Zhang, X. Rong, R. Jacob, S. Wu, and F. Lu, (2014a), Ensemble-based
610
parameter estimation in a coupled GCM using the adaptive spatial average method. J.
611
Climate, 27, 4002–4014, doi:10.1175/JCLI-D-13-00091.1.
612
Liu, Y., Z. Liu, S. Zhang, R. Jacob, F. Lu, X. Rong, and S. Wu, (2014b), Ensemble-based
613
parameter estimation in a coupled general circulation model. J. Climate, 27, 7151–7162,
614
doi:10.1175/ JCLI-D-13-00406.1.
615 616
617
Lee, Y., and A. J. Majda, (2016), State estimation and prediction using clustered particle filters. Proc. Natl. Acad. Sci., 113, 201617398, doi:10.1073/pnas.1617398113.
Lu F, Liu Z, Zhang S, Liu Y, Jacob R, (2015), Strongly coupled data assimilation using leading
618
averaged coupled covariance (LACC). Part II: CGCM experiments*. Mon Weather Rev
619
143:4645–4659. doi:10.1175/MWR-D-15-0088.1
620
Lu, F., Liu, Z., Liu, Y., (2016). Understanding the control of extratropical
621
atmospheric variability on ENSO using a coupled data assimilation approach
622
doi:10.1007/s00382-016-3256-7
623
624
Marcott, S. a., J. D. Shakun, P. U. Clark, and A. C. Mix, (2013), A Reconstruction of Regional.
Science, 339, 1198–1201, doi:10.1126/science.1228026.
625
Mann, M. E., Z. Zhang, M. K. Hughes, R. S. Bradley, S. K. Miller, S. Rutherford, and F. Ni,
626
(2008), Proxy-based reconstructions of hemispheric and global surface temperature variations
627
over the past two millennia. Proc. Natl. Acad. Sci. U. S. A., 105, 13252–13257,
628
doi:10.1073/pnas.0805721105.
629 630
Mann, M. E., and Coauthors, (2009), Global signatures and dynamical origins of the Little Ice Age and Medieval Climate Anomaly. Science, 326, 1256–1260, doi:10.1126/science.117 7303.
631
Mathiot, P., H. Goosse, X. Crosta, B. Stenni, M. Braida, H. Renssen, and C. J. Van Meerbeeck,
632
(2013), Using data assimilation to investigate the causes of Southern BP. Clim. Past, 887–
633
901, doi:10.5194/cp-9-887-2013.
634
Mairesse, A., H. Goosse, P. Mathiot, H. Wanner, and S. Dubinkina, (2013), Investigating the
635
consistency between proxy-based reconstructions and climate models using data assimilation:
636
a mid-Holocene case study. Clim. Past, 5194, doi:10.5194/cp-9-2741-2013.
637 638
Matsikaris, A., M. Widmann, J. Jungclausn, (2015), On-line and off-line data assimilation in palaeoclimatology : a case study. Clim. Past, 11:81–93. doi:10.5194/cp-11-81-2015.
639
Otto-Bliesner, B. L., E. C. Brady, G. Clauzet, R. Tomas, S. Levis, and Z. Kothavala, (2006), Last
640
glacial maximum and Holocene climate in CCSM3. J. Clim., 19, 2526–2544,
641
doi:10.1175/JCLI3748.1.
642 643
Poterjoy, J., 2015: A localized particle filter for high-dimensional nonlinear systems. Mon. Weather Rev., 151020092806005, doi:10.1175/MWR-D-15-0163.1.
Formatted: Font: Italic
644
Snyder, C., T. Bengtsson, P. Bickel, and J. Anderson, (2008), Obstacles to High-Dimensional
645
Particle Filtering. Mon. Weather Rev., 136, 4629–4640, doi:10.1175/2008MWR2529.1.
646
Smerdon, J. E., (2012), Climate models as a test bed for climate reconstruction methods:
647
Pseudoproxy
648
doi:10.1002/wcc.149.
649 650
651 652
653
experiments.
Wiley
Interdiscip.
Rev.
Clim.
Chang.,
3,
63–77,
Steiger, N. J., G. J. Hakim, E. J. Steig, D. S. Battisti, and G. H. Roe (2014), Assimilation of timeaveraged pseudoproxies for climate reconstruction, J. Clim., 27(1), 426–441
Steiger, N., and G. Hakim, (2016), Multi-timescale data assimilation for atmosphere-ocean state estimates. Clim. Past, 12, 1375–1388, doi:10.5194/cp-12-1375-2016.
Tippett, M. K., J. L. Anderson, C. H. Bishop, T. M. Hamill, and J. S. Whitaker, (2003), Ensemble
654
Square
Root
Filters.
Mon.
Weather
655
0493(2003)1312.0.CO;2.
Rev.,
131,
1485–1490,
doi:10.1175/1520-
656
Von Storch H, Cubasch U, Gonzalez-Rouco J F, Jones J M, Voss R, Widmann M, Zorita E. ,
657
(2000), Combining paleoclimatic evidence and GCMs by means of data assimilation through
658
upscaling and nudging (DATUN). In: Proceedings of the 11th Symposium on Global Change
659
Studies. 28–31
660
Whitaker, J. S., and T. M. Hamill, (2002): Ensemble data assimilation without perturbed
661
observations. Mon. Wea. Rev., 130, 1913–1924, doi:10.1175/1520-0493(2002)130,1913:
662
EDAWPO.2.0.CO;2.
663
Widmann, M., H. Goosse, G. van der Schrier, R. Schnur, and J. Barkmeijer, (2010), Using data
664
assimilation to study extratropical Northern Hemisphere climate over the last millennium.
665
Clim. Past, 6, 627–644, doi:10.5194/cp-6-627-2010.
666
Zhang, F. Q., C. Snyder, and J. Z. Sun, 2004: Impacts of initial estimate and observation
667
availability on convective-scale data assimilation with an ensemble Kalman filter. Mon. Wea.
668
Rev., 132, 1238–1253,
669
Zhang, S., M. J. Harrison, a. Rosati, and a. Wittenberg, (2007), System Design and Evaluation of
670
Coupled Ensemble Data Assimilation for Global Oceanic Climate Studies. Mon. Weather
671
Rev., 135, 3541–3564, doi:10.1175/MWR3466.1.
672
Zhen, Y., and F. Zhang, 2014: A probabilistic approach to adaptive covariance localization for
673
serial
674
doi:https://doi.org/10.1175/MWR-D-13-00390.1
675
ensemble
square
root
filters. Mon.
Wea.
Rev., 142, 4499–4518,
676
Hemispheric Scale
Modes of Variability
NH
SH
GL
NH. Pacific
NH. Atlantic
Tropics
SH
PF-SIR
0.67
~0
0.24
0.23 (0.85)
0.67 (0.99)
0.25 (0.98)
~0 (0.95)
EAKF 5000 km
0.85
~0
0.61
0.57 (0.85)
0.78 (0.98)
0.43 (0.95)
0.25 (0.46)
EAKF 2000 km
0.74
~0
0.36
0.42 (0.67)
0.70 (0.99)
0.30 (0.97)
0.27 (0.23)
EAKF 500 km
0.51
~0
~0
~0 (0.95)
0.44 (0.99)
~0 (0.98)
0.30 (0.54)
677 678
Table 1. Correlation coefficients between different experiments and truth based on hemispheric scale averaged SST
679
time series from Figure 2; Correlation coefficients for PCs series between each experiment and the truth for the first
680
mode from Figure 7. Pattern correlation coefficients for the leading mode EOF between each experiment and the truth
681
(in brackets). The correlation coefficients that are not significant at 0.05 confidence level are marked as zero.
682
683 684
Figure 1. Observation locations and spatial reconstruction skill represented by RMSE and correlation coefficient for
685
4 experiments. a) shows the location of 47 proxies, the shading shows the seasonal natural variability/standard
686
deviation of the SST from the truth run. b) and c), d) and e), f) and g), h) and i) correspond to the four experiments:
687
PF-SIR, EAKF with localization 5000km, EAKF with localization 2000km, EAKF with localization 500km. b), d),
688
f), h) are RMSE divided by natural variability from a). c), e), g), i) are the spatial correlation map between each
689
experiment and the truth and the dotted area is not significant at 0.05 significance level.
690
691 692 693
Figure 2. Area averaged SST anomalies for southern hemisphere (g,h,i), northern hemisphere (d,e,f) and global (a,b,c);
694
Black curve is for truth; blue curve is for PF-SIR; red curve is for EAKF with localization scale 5000 km; orange
695
curve is for EAKF with localization scale 2000 km; green curve is for EAKF with localization scale 500 km. The
696
correlation coefficient with truth is shown for each experiment on a), d) and g); In the second column, the pink curves
697
are the 16 ensemble members from EAKF with localization scale 5000km; In the third column, the light blue curves
698
are for 48 ensemble members from PF-SIR.
699 700
Figure 3. Correlation coefficients between hemispheric averaged SST time series of EAKF- 5000 (red dot), EAKF-
701
2000 (magenta dot), EAKF-500 (green dot), PF-SIR (blue dot) and that of the truth at global scale (a), northern
702
hemisphere (b) and southern hemisphere (c). Each dot represents a 20-year, 40-year, or 100-year sample from the 200-
703
year-long simulation. The red and blue diamonds are the correlation coefficient of the 50-year-long experiments for
704
EAKF with 32 members and PF-SIR with 96 members.
705 706
Figure 4. First EOF field of North Pacific hemisphere (left column), North Atlantic (right column) ; The first row (a,
707
b ) is for the truth state, the second row (c, d) is for PF-SIR, the third row (e, f) is for EAKF with localization 5000km,
708
the forth row (g, h) is for EAKF with localization 2000km, and the last row (i, j) is for EAKF with localization 500km.
709
All figures are drawn in the same color scale and the unit is ℃. On top of each figure, the percentage of explained
710
variance is shown.
711
712 713
Figure 5. One-point correlation map calculated from seasonal mean model control at selected observation locations.
714
The black dot represents the observation location. The dotted area is not significant at 0.05 significance level.
715 716 717
Figure 6. First EOF field of the tropics ; a), b), c), d), and e) are for the actual state, PF-SIR, EAKF with localization
718
5000km, EAKF with localization 2000km, and EAKF with localization 500km respectively. All figures are drawn in
719
the same color scale and the unit is ℃.
720 721
Figure 7. Same as Figure 4. But for the southern hemisphere.
North Pacific
PF-SIR 0.23; EAKF: 0.57, 0.42, 0.04 2 0 -2 EAKF500
EAKF2000
truth
PF-SIR
EAKF5000
-4 0
50
0
50
North Atlantic
3
100
150
200
100
150
200
100
150
200
100
150
200
PF-SIR 0.67; EAKF: 0.78, 0.70, 0.44
2 1 0 -1 -2 -3 4
PF-SIR 0.25; EAKF: 0.43, 0.30, -0.11
Tropics
2 0 -2 -4 0
50
3
PF-SIR 0.02; EAKF: 0.26, 0.27, 0.30
2
SH
1 0 -1 -2 -3 0
50
Years
722 723
Figure 8. The normalized PCs for the first EOFs from Figure 3 - 5 for north Pacific, north Atlantic, tropics and
724
southern hemisphere. The black curve is for truth; the blue curve is for PF-SIR, the red curve is for EAKF with
725
localization scale 5000 km, the magenta curve is for EAKF with localization scale 2000 km, the green curve is for
726
EAKF with localization scale 500 km.
727 728
Figure 9. Spatial reconstruction skills for PF-SIR (left column) and EAKF-5000km (right column) on air surface
729
temperature (AST, a and b), 500hPa geopotential height (Z, c and d), and precipitation (PRECP, e and f).
730 731
Figure 10. Same as Figure 1 but for EAKF-5000km with 32 ensemble members (b and c), and PF-SIR with 96
732
ensemble members (d and e).
733 734
Figure 11. Same as Figure 2 a,d,g, but for EAKF-5000km with 32 ensemble members (red dashed line), and PF-SIR
735
with 96 ensemble members (blue dashed line).
736
737
Captions:
Hemispheric Scale
Modes of Variability
NH
SH
GL
NH. Pacific
NH. Atlantic
Tropics
SH
PF-SIR
0.67
~0
0.24
0.23 (0.85)
0.67 (0.99)
0.25 (0.98)
~0 (0.95)
EAKF 5000 km
0.85
~0
0.61
0.57 (0.85)
0.78 (0.98)
0.43 (0.95)
0.25 (0.46)
EAKF 2000 km
0.74
~0
0.36
0.42 (0.67)
0.70 (0.99)
0.30 (0.97)
0.27 (0.23)
EAKF 500 km
0.51
~0
~0
~0 (0.95)
0.44 (0.99)
~0 (0.98)
0.30 (0.54)
738 739
Table 1. Correlation coefficients between different experiments and truth based on hemispheric scale averaged SST
740
time series from Figure 2; Correlation coefficients for PCs series between each experiment and the truth for the first
741
mode from Figure 7. Pattern correlation coefficients for the leading mode EOF between each experiment and the truth
742
(in brackets). The correlation coefficients that are not significant at 0.05 confidence level are marked as zero.
743 744
Figure 1. Observation locations and spatial reconstruction skill represented by RMSE and correlation coefficient for
745
4 experiments. a) shows the location of 47 proxies, the shading shows the seasonal natural variability/standard
746
deviation of the SST from the truth run. b) and c), d) and e), f) and g), h) and i) correspond to the four experiments:
747
PF-SIR, EAKF with localization 5000km, EAKF with localization 2000km, EAKF with localization 500km. b), d),
748
f), h) are RMSE divided by natural variability from a). c), e), g), i) are the spatial correlation map between each
749
experiment and the truth and the dotted area is not significant at 0.05 significance level.
750 751
Figure 2. Area averaged SST anomalies for southern hemisphere (g,h,i), northern hemisphere (d,e,f) and global (a,b,c);
752
Black curve is for truth; blue curve is for PF-SIR; red curve is for EAKF with localization scale 5000 km; orange
753
curve is for EAKF with localization scale 2000 km; green curve is for EAKF with localization scale 500 km. The
754
correlation coefficient with truth is shown for each experiment on a), d) and g); In the second column, the light blue
755
curves are for 48 ensemble members from PF-SIR; In the third column, the pink curves are the 16 ensemble members
756
from EAKF with localization scale 5000km.
757 758
Figure 3. Correlation coefficients between hemispheric averaged SST time series of EAKF- 5000 (red dot), EAKF-
759
2000 (magenta dot), EAKF-500 (green dot), PF-SIR (blue dot) and that of the truth at global scale (a), northern
760
hemisphere (b) and southern hemisphere (c). Each dot represents a 20-year, 40-year, or 100-year sample from the 200-
761
year-long simulation. The red and blue diamonds are the correlation coefficient of the 50-year-long experiments for
762
EAKF with 32 members and PF-SIR with 96 members.
763 764
Figure 4. First EOF field of North Pacific hemisphere (left column), North Atlantic (right column) ; The first row (a,
765
b ) is for the truth state, the second row (c, d) is for PF-SIR, the third row (e, f) is for EAKF with localization 5000km,
766
the forth row (g, h) is for EAKF with localization 2000km, and the last row (i, j) is for EAKF with localization 500km.
767
All figures are drawn in the same color scale and the unit is ℃. On top of each figure, the percentage of explained
768
variance is shown.
769
Figure 5. One-point correlation map calculated from seasonal mean model control at selected observation locations.
770
The black dot represents the observation location. The dotted area is not significant at 0.05 significance level.
771 772
Figure 6. First EOF field of the tropics ; a), b), c), d), and e) are for the actual state, PF-SIR, EAKF with localization
773
5000km, EAKF with localization 2000km, and EAKF with localization 500km respectively. All figures are drawn in
774
the same color scale and the unit is ℃.
775 776
Figure 7. Same as Figure 4. But for the southern hemisphere.
777 778
Figure 8. The normalized PCs for the first EOFs from Figure 3 - 5 for north Pacific, north Atlantic, tropics and
779
southern hemisphere. The black curve is for truth; the blue curve is for PF-SIR, the red curve is for EAKF with
780
localization scale 5000 km, the magenta curve is for EAKF with localization scale 2000 km, the green curve is for
781
EAKF with localization scale 500 km.
782
783
Figure 9. Spatial reconstruction skills for PF-SIR (left column) and EAKF-5000km (right column) on air surface
784
temperature (AST, a and b), 500hPa geopotential height (Z, c and d), and precipitation (PRECP, e and f).
785 786
Figure 10. Same as Figure 1 but for EAKF-5000km with 32 ensemble members (b and c), and PF-SIR with 96
787
ensemble members (d and e).
788 789
Figure 11. Same as Figure 2 a,d,g, but for EAKF-5000km with 32 ensemble members (red dashed line), and PF-SIR
790
with 96 ensemble members (blue dashed line).
791 792
