Interacting Two-Fluid Viscous Dark Energy Models In Non-Flat Universe
Hassan Amirhashchi1, Anirudh Pradhan2, H. Zainuddin3

1Young Researchers Club, Mahshahr Branch, Islamic Azad University, Mahshahr,
Iran

E-mail<EMAIL_ADDRESS>
2Department of Mathematics, Hindu Post-graduate College, Zamania-232 331,
Ghazipur, India

E-mail<EMAIL_ADDRESS>
2,3Laboratory of Computational Sciences and Mathematical Physics, Institute
for Mathematical Research, University Putra Malaysia, 43400 UPM, Serdang,
Selangor D.E., Malaysia

E-mail<EMAIL_ADDRESS>

###### Abstract

We study the evolution of the dark energy parameter within the scope of a
spatially non-flat and isotropic Friedmann-Robertson-Walker (FRW) model filled
with barotropic fluid and bulk viscous stresses. We have obtained cosmological
solutions which exhibit without a big rip singularity. It is concluded that in
both non-interacting and interacting cases non-flat open universe crosses the
phantom region. We find that during the evolution of the universe, the
equation of state (EoS) for dark energy $\omega_{D}$ changes from
$\omega^{eff}_{D}<-1$ to $\omega^{eff}_{D}>-1$, which is consistent with
recent observations.

Keywords: FRW universe, Dark energy, Viscous Fluid
PACS number: 98.80.Es, 98.80-k, 95.36.+x

##  1 INTRODUCTION

Observations of distant Supernovae (SNe Ia) (Perlmutter et al. 1997, 1998,
1999; Riess et al. 1998, 2000; Garnavich et al. 1998a,b; Schmidt et al. 1998;
Tonry et al. 2003; Clocchiatti et al. 2006), fluctuation of cosmic microwave
background radiation (CMBR) (de Bernardis et al. 1998; Hanany et al. 2000),
large scale structure (LSS) (Spergel et al. 2003; Tegmark et al. 2004), sloan
digital sky survey (SDSS) (Seljak et al. 2005; Adelman-McCarthy et al. 2006),
Wilkinson microwave anisotropy probe (WMAP) (Bennett. et al 2003) and Chandra
x-ray observatory (Allen et al. 2004) by means of ground and altitudinal
experiments have established that our Universe is undergoing a late-time
accelerating expansion, and we live in a priviledged spatially flat Universe
composed of approximately $4\%$ baryonic matter, $22\%$ dark matter and $74\%$
dark energy. The simplest candidate for dark energy is the cosmological
constant. Recently, a great number of theme have been proposed to explain the
current accelerating Universe, partly such as scalar field model, exotic
equation of state (EoS), modified gravity, and the inhomogeneous cosmology
model. There are several dark energy models which can be distinguished by, for
instance, their EoS ($\omega=\frac{p_{de}}{\rho_{de}}$) during the evolution
of the universe.

The introduction of viscosity into cosmology has been investigated from
different view points (Gr$\o$n 1990; Padmanabhan $\&$ Chitre 1987; Barrow
1986; Zimdahl 1996; Farzin et al. 2012). Misner (1966, 1967;) noted that the
“measurement of the isotropy of the cosmic background radiation represents the
most accurate observational datum in cosmology”. An explanation of this
isotropy was provided by showing that in large class of homogeneous but
anisotropic universe, the anisotropy dies away rapidly. It was found that the
most important mechanism in reducing the anisotropy is neutrino viscosity at
temperatures just above $10^{10}K$ (when the Universe was about 1 s old: cf.
Zel’dovich and Novikov (Zel’dovich $\&$ Novikov 1971)). The astrophysical
observations also indicate some evidences that cosmic media is not a perfect
fluid (Jaffe et al. 2005), and the viscosity effect could be concerned in the
evolution of the universe (Brevik $\&$ Gorbunova, 2005; Brevik et al. 2005;
Cataldo et al. 2005). On the other hand, in the standard cosmological model,
if the EoS parameter $\omega$ is less than $-1$, so-called phantom, the
universe shows the future finite time singularity called the Big Rip (Caldwell
et al. 2003; Nojiri et al. 2005) or Cosmic Doomsday. Several mechanisms are
proposed to prevent the future big rip, like by considering quantum effects
terms in the action (Nojiri $\&$ Odintsov 2004; Elizalde et al. 2004), or by
including viscosity effects for the Universe evolution (Meng et al. 2007). A
well known result of the FRW cosmological solutions, corresponding to
universes filled with perfect fluid and bulk viscous stresses, is the
possibility of violating dominant energy condition (Barrow 1987, 1988;
Folomeev $\&$ Gurovich 2008; Ren $\&$ Meng 2006; Brevikc $\&$ Gorbunovac 2005;
Nojiri $\&$ Odintsov 2005). Setare (Setare 2007a,b,c) and Setare and Saridakis
(Setare $\&$ Saridakis 2000) have studied the interacting models of dark
energy in different context. Interacting new agegraphic viscous dark energy
with varying $G$ has been studied by Sheykhi and Setare (Sheykhi $\&$ Setare
2010).

Recently, Amirhashchi et al. (2011a,b); Pradhan et al. (2011); Saha et al.
(2012) have studied the two-fluid scenario for dark energy in FRW universe in
different context. Very recently Singh and Chaubey (2012) have studied
interacting dark energy in Bianchi type I space-time. Some experimental data
implied that our universe is not a perfectly flat universe and recent papers
(Spergel et al. 2003; Bennett et al. 2003; Ichikawa et al. 2006) favoured a
universe with spatial curvature. Setare et al. (2009) have studied the tachyon
cosmology in non-interacting and interacting cases in non-flat FRW universe.
Due to these considerations and motivations, in this Letter, we study the
evolution of the dark energy parameter within the framework of a FRW open
cosmological model filled with two fluids (i.e., barotropic fluid and bulk
viscous stresses). In doing so we consider both interacting and non-
interacting cases.

##  2 THE METRIC AND FIELD EQUATIONS

We consider the spherically symmetric Friedmann-Robertson-Walker (FRW) metric
as

$ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right],$ (1)

where $a(t)$ is the scale factor and the curvature constant $k$ is $-1,0,+1$
respectively for open, flat and close models of the universe.

The Einstein’s field equations (with $8\pi G=1$ and $c=1$) read as

$R^{j}_{i}-\frac{1}{2}R\delta^{j}_{i}=-T^{j}_{i},$ (2)

where the symbols have their usual meaning and $T^{j}_{i}$ is the two-fluid
energy-momentum tensor due to bulk viscous dark and barotropic fluids written
in the form.

$T^{j}_{i}=(\rho+\bar{p})u^{j}_{i}+\bar{p}g^{j}_{i},$ (3)

where

$\bar{p}=p-\xi u^{i}_{;i}$ (4)

and

$u^{i}u_{i}=-1,$ (5)

where $\rho$ is the energy density; $p$, the pressure; $\xi$, the bulk-viscous
coefficient; and $u^{i}$, the four-velocity vector of the distribution. Here
after the semi-colon denotes covariant differentiation.

The expansion factor $\theta$ is defined by
$\theta=u^{i}_{;i}=3\frac{\dot{a}}{a}$. Hence Eq. (4) leads to

$\bar{p}=p-3\xi H,$ (6)

where $H$ is Hubble’s constant defined by

$H=\frac{\dot{a}}{a}.$ (7)

Now with the aid of Equations (3)-(5) and metric (1), the surviving field
equations (2) take the explicit forms

$\rho=3\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{k}{a^{2}}\right),$ (8)

and

$\bar{p}=-\left(\frac{\dot{a}^{2}}{a^{2}}+2\frac{\ddot{a}}{a}+\frac{k}{a^{2}}\right).$
(9)

Also in space-time (1) the Bianchi identity for the bulk-viscous fluid
distribution $G^{;j}_{ij}=0$ leads to $T^{;j}_{ij}=0$ which yields

$\rho u^{i}+(\rho+\bar{p})u^{i}_{;i}$ (10)

which leads to

$\dot{\rho}+3H(\rho+\bar{p})=0.$ (11)

Using Eq. (7) in Eqs. (8) and (9) we get

$\rho=\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right),$ (12)

and

$\bar{p}=-\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right),$ (13)

where $\bar{p}=p_{m}+\bar{p}_{D}$ and $\rho=\rho_{m}+\rho_{D}$. Here $p_{m}$
and $\rho_{m}$ are pressure and energy density of barotropic fluid and $p_{D}$
and $\rho_{D}$ are pressure and energy density of dark fluid respectively.

The equation of state (EoS) for the barotropic fluid $\omega_{m}$ and dark
field $\omega_{D}$ are given by

$\omega_{m}=\frac{p_{m}}{\rho_{m}},$ (14)

and

$\omega_{D}=\frac{\bar{p}_{D}}{\rho_{D}},$ (15)

respectively.
From Eqs. (11)-(13) we obtain

$\frac{\dot{\rho}}{3H}=\frac{2k}{a^{2}}e^{-2Ht}.$ (16)

Now we assume

$\rho=\alpha\theta^{2}~{}\mbox{or}~{}\rho=9\alpha H^{2},$ (17)

where $\alpha$ is an arbitrary constant. Eq. (17) ensure us that our universe
approaches homogeneity (Collins 1977). This condition has also been used by
Banerjee et al. (1986) for deriving a viscous-fluid cosmological model with
Bianchi type II space time.
Putting Eq. (17) in Eq. (16) and after integrating we get

$e^{-2Ht}=-\frac{3\alpha A^{2}}{2kt^{2}},$ (18)

which yields

$H=\frac{1}{2t}\ln\left(-\frac{2kt^{2}}{3\alpha A^{2}}\right),$ (19)

where $A$ is an arbitrary constant. From Eq. (19), we observe that the
condition given by (17) restrict our study to the case when $k=-1$ (i.e. only
for open universe). In the following sections we deal with two cases, (i) non-
interacting two-fluid model and (ii) interacting two-fluid model.

##  3 NON-INTERACTING TWO-FLUID MODEL

In this section we assume that two-fluid do not interact with each other.
Therefor, the general form of conservation equation (11) leads us to write the
conservation equation for the dark and barotropic fluid separately as,

$\dot{\rho}_{m}+3\frac{\dot{a}}{a}\left(\rho_{m}+p_{m}\right)=0,$ (20)

and

$\dot{\rho}_{D}+3\frac{\dot{a}}{a}\left(\rho_{D}+\bar{p}_{D}\right)=0.$ (21)

Integration Eq. (20) and using (7) leads to

$\rho_{m}=\rho_{0}a^{-3(1+\omega_{m})}~{}\mbox{or}~{}\rho_{m}=\rho_{0}Be^{-3H(1+\omega_{m})t},$
(22)

where $\rho_{0}$ is an integrating constant and $B=A^{-3(1+\omega_{m})}$. By
using Eq. (22) in Eqs. (12) and (13), we first obtain the $\rho_{D}$ and
$p_{D}$ in term of Hubble’s constant $H$ as

$\rho_{D}=\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\rho_{0}Be^{-3H(1+\omega_{m})t},$
(23)

and

$\bar{p}_{D}=\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\omega_{m}\rho_{0}Be^{-3H(1+\omega_{m})t}.$
(24)

Figure 1: The plot of $\rho_{D}$ vs $t$ for $\alpha=0.1,A=100,\omega_{m}=0.5$
in both non-interacting and interacting two-fluid model Figure 2: The plot of
EoS parameter $\omega^{eff}_{D}$ vs $t$ for
$\rho_{0}=10,\omega_{m}=0.5,\alpha=0.01,B=1$ in non-interacting two-fluid
model

respectively. By using Eqs. (23) and (24) in Eq. (15), we can find the EoS of
dark energy in term of time as

$\omega_{D}=-\frac{\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right)+\omega_{m}\rho_{0}Be^{-3H(1+\omega_{m})t}}{\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\rho_{0}Be^{-3H(1+\omega_{m})t}}.$
(25)

Therefore the effective EoS parameter for viscous DE can be written as

$\omega^{eff}_{D}=\omega_{D}-\frac{3\xi
H}{\rho_{D}}=-\frac{\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right)+3\xi
H+\omega_{m}\rho_{0}Be^{-3H(1+\omega_{m})t}}{\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\rho_{0}Be^{-3H(1+\omega_{m})t}}.$
(26)

The expressions for the matter-energy density $\Omega_{m}$ and dark-energy
density $\Omega_{D}$ are given by

$\Omega_{m}=\frac{\rho_{m}}{3H^{2}}=\frac{4t^{2}\rho_{0}Be^{-\frac{3}{2}\ln(\frac{2t^{2}}{3\alpha
A^{2}})(1+\omega_{m})}}{3\ln^{2}(\frac{2t^{2}}{3\alpha A^{2}})},$ (27)

and

$\Omega_{D}=\frac{\rho_{D}}{3H^{2}}=-\frac{6\alpha}{\ln^{2}(\frac{2t^{2}}{3\alpha
A^{2}})}+1-\frac{4t^{2}\rho_{0}Be^{-\frac{3}{2}\ln(\frac{2t^{2}}{3\alpha
A^{2}})(1+\omega_{m})}}{3\ln^{2}(\frac{2t^{2}}{3\alpha A^{2}})},$ (28)

Figure 3: The plot of density parameter ($\Omega$) vs $t$ for
$A=1,\alpha=0.01$ in non-interacting two-fluid model

respectively. Adding Eqs. (27) and (28), we obtain

$\Omega=\Omega_{m}+\Omega_{D}=-\frac{6\alpha}{\ln^{2}(\frac{2t^{2}}{3\alpha
A^{2}})}+1.$ (29)

From the right hand side of Eq. (29), it is clear that for open universe,
$\Omega<1$ but at late time we see that $\Omega\to 1$ i.e. the flat universe
scenario. This result is also compatible with the observational results. Since
our model predicts a flat universe for large times and the present-day
universe is very close to flat, so being flat, the derived model is thus
compatible with the observational results.

Fig. $1$ depicts the energy density of DE ($\rho_{D}$) versus $t$. From this
figure, we observe that ($\rho_{D}$), in both non-interacting and interacting
cases, is a decreasing function of time and approaches a small positive value
at late time and never go to infinity. Thus, in both cases the universe is
free from big rip.

The behavior of EoS for DE in term of cosmic time $t$ is shown in Fig. $2$. It
is observed that for open universe, the $\omega^{eff}_{D}$ is an decreasing
function of time, the rapidity of its decrease at the early stage depends on
the larger value of bulk viscous coefficient. The EoS parameter of the DE
begins in non-dark ($\omega_{D}>-\frac{1}{3}$) region at early stage and cross
the phantom divide or cosmological constant ($\omega_{D}=-1$) region and then
pass over into phantom ($\omega_{D}<-1$) region. The property of DE is a
violation of the null energy condition (NEC) since the DE crosses the Phantom
Divide Line (PDL), in particular depending on the direction (Rodrigues 2008;
Kumar $\&$ Yadav 2011; Pradhan $\&$ Amirhashchi 2011). In theory, despite the
observational constraints, extensions of general relativity are the prime
candidate class of theories consistent with PDL crossing (Nesseris $\&$
Perivolaropoulos 2007). On the other hand, while the current cosmological data
from SN Ia (Supernova Legacy Survey, Gold Sample of Hubble Space Telescope)
(Riess et al. 2004; Astier et al. 2006). CMB (WMAP, BOOMERANG) (Komatsu et al.
2009; MacTavish et al. 2006) and large scale structure (SDSS) (Eisenstein et
al. 2005) data rule out that $\omega_{D}\ll-1$, they mildly favour dynamically
evolving DE crossing the PDL (see Rodrigues 2008; Kumar $\&$ Yadav 2011;
Pradhan $\&$ Amirhashchi 2011; Nesseris $\&$ Perivolaropoulos 2007; Zhao et
al. 2007; Coperland et al. 2006) for theoretical and observational status of
crossing the PDL). Thus our DE model is in good agreement with well
established theoretical result as well as the recent observations. From Fig.
$2$, it is observed that in absence of viscosity (i.e. for $\xi=0$), the
universe does not cross the PDL but approaches to cosmological constant
($\omega_{D}=-1$) scenario. Thus, it clearly indicates the impact of viscosity
on the evolution of the universe.

The variation of density parameter ($\Omega$) with cosmic time $t$ for open
universe has been shown in Fig. $3$. From the figure, it can be seen that in
an open universe, $\Omega$ is an increasing function of time and at late time,
it approaches to the flat universe’s scenario.

##  4 INTERACTING TWO-FLUID MODEL

In this section we consider the interaction between dark viscous and
barotropic fluids. For this purpose we can write the continuity equations for
barotropic and dark viscous fluids as

$\dot{\rho}_{m}+3\frac{\dot{a}}{a}(\rho_{m}+p_{m})=Q,$ (30)

and

$\dot{\rho}_{D}+3\frac{\dot{a}}{a}(\rho_{D}+\bar{p}_{D})=-Q,$ (31)

where the quantity $Q$ expresses the interaction between the dark components.
Since we are interested in an energy transfer from the dark energy to dark
matter, we consider $Q>0$ which ensures that the second law of thermodynamics
is fulfilled (Pavon $\&$ Wang 2009). Here we emphasize that the continuity
Eqs. (11) and (30) imply that the interaction term ($Q$) should be
proportional to a quantity with units of inverse of time i.e
$Q\propto\frac{1}{t}$. Therefor, a first and natural candidate can be the
Hubble factor $H$ multiplied with the energy density. Following Amendola et
al. (2007) and Gou et al. (2007), we consider

$Q=3H\sigma\rho_{m},$ (32)

where $\sigma$ is a coupling constant. Using Eq. (32) in Eq. (30) and after
integrating, we obtain

$\rho_{m}=\rho_{0}a^{-3(1+\omega_{m}-\sigma)}~{}\mbox{or}~{}\rho_{m}=\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t}.$
(33)

By using Eq. (33) in Eqs. (12) and (13), we again obtain the $\rho_{D}$ and
$p_{D}$ in term of Hubble’s constant $H$ as

$\rho_{D}=\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t},$
(34)

and

$\bar{p}_{D}=\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right)-(\omega_{m}-\sigma)\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t},$
(35)

respectively. By using Eqs. (34) and (35) in Eq. (15), we can find the EoS of
dark energy in term of time as

$\omega_{D}=-\frac{\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right)+(\omega_{m}-\sigma)\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t}}{\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t}}.$
(36)

Again we can write the effective EoS parameter of viscous DE as

$\omega^{eff}_{D}=-\frac{\left(\frac{k}{A^{2}}e^{-2Ht}+3H^{2}\right)-3\xi
H+(\omega_{m}-\sigma)\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t}}{\left(\frac{3k}{A^{2}}e^{-2Ht}+3H^{2}\right)-\rho_{0}Be^{-3H(1+\omega_{m}-\sigma)t}}.$
(37)

Figure 4: The plot of EoS parameter $\omega^{eff}_{D}$ vs $t$ for
$\rho_{0}=10,\omega_{m}=0.5,\alpha=0.01,B=1,\sigma=0.3$ in interacting two-
fluid model

The expressions for the matter-energy density $\Omega_{m}$ and dark-energy
density $\Omega_{D}$ are given by

$\Omega_{m}=\frac{\rho_{m}}{3H^{2}}=\frac{4t^{2}\rho_{0}Be^{-\frac{3}{2}\ln(\frac{2t^{2}}{3\alpha
A^{2}})(1+\omega_{m}-\sigma)}}{3\ln^{2}(\frac{2t^{2}}{3\alpha A^{2}})},$ (38)

and

$\Omega_{D}=\frac{\rho_{D}}{3H^{2}}=-\frac{6\alpha}{\ln^{2}(\frac{2t^{2}}{3\alpha
A^{2}})}+1-\frac{4t^{2}\rho_{0}Be^{-\frac{3}{2}\ln(\frac{2t^{2}}{3\alpha
A^{2}})(1+\omega_{m}-\sigma)}}{3\ln^{2}(\frac{2t^{2}}{3\alpha A^{2}})},$ (39)

respectively. Adding Eqs. (38) and (39), we obtain

$\Omega=\Omega_{m}+\Omega_{D}=-\frac{6\alpha}{\ln^{2}(\frac{2t^{2}}{3\alpha
A^{2}})}+1,$ (40)

which is the same expression as in previous case of non-interacting two-fluid.
Fig. $4$ shows a plot of EoS parameter ($\omega^{eff}_{D}$) versus $t$. The
characteristic of $\omega^{eff}_{D}$ in this case is the same as in the
previous case.

##  5 CONCLUSION

In this Letter, we have studied the evolution of dark energy parameter within
the frame work of an open FRW space-time filled with barotropic and bulk
viscous dark fluid. In both non-interacting and interacting cases, we have
observed that for all values of bulk viscous coefficient, the universe has
transition from non-dark region ($\omega^{eff}_{D}>-\frac{1}{3}$) to phantom
region ($\omega^{eff}_{D}<-1$). In summary, we have investigated the
possibility of constructing a two-fluid dark energy models which have the
equation of state ($\omega^{eff}_{D}$) crossing - 1 by using the two-fluid
(barotropic and bulk viscous dark fluid) naturally. Therefore, the two-fluid
scenario discussed in the present paper is a viable candidate for dark energy.
It is also worth mentioned here that in both interacting and non-interacting
cases, our models are free from big rip.

## ACKNOWLEDGMENT

This work has been supported by the FRGS Grant by the Ministry of Higher
Education, Malaysia under the Project Number 02-10-10-969 FR. H. Amirhashchi &
A. Pradhan also thank the Laboratory of Computational Sciences and
Mathematical Physics, Institute for Mathematical Research, Universiti Putra
Malaysia for providing facility where this work was done.

## References

  * [1] Adelman-McCarthy, J. K., et al. 2006, Astrophysical Journal Supplement, 162, 38
  * [2] Allen, S. W., et al. 2004, Monthly Notices of the Royal Astronomical Society,353, 457
  * [3] Amendola, L., et al. 2007, Phys. Rev. D, 75, 083506
  * [4] Amirhashchi, H., Pradhan, A $\&$ Saha, B. 2011a, Chinese Physics Letters,28, 039801
  * [5] Amirhashchi, H., Pradhan, A., $\&$ Zainuddin, H. 2011b, International Journal of Theoretical Physics,50, 3529
  * [6] Astier, P., et al. 2006, Astronomy $\&$ Astrophysics, 447, 31
  * [7] Banerjee, A., Duttachoudhury, S. B., $\&$ Sanyal, A. K. 1986, General relativity $\&$ Gravitation, 18, 461
  * [8] Barrow, J. D. 1986, Physics Letters B, 180, 335
  * [9] Barrow, J. D. 1987, Physics Letters B, 180, 335
  * [10] Barrow, J. D. 1988, Nuclear Physics B, 310, 743
  * [11] Bennett, C. L., et al. 2003, Astrophysical Journal Supplement, 148, 1
  * [12] Brevikc, I., $\&$ Gorbunovac, O. 2005, General relativity $\&$ Gravitation, 37, 2039
  * [13] Brevik, I., Gorbunova. O., $\&$ Shaido, Y. A. 2005, International Journal of Theoretical Physics D, 14, 1899
  * [14] Caldwell, R. R., Kamiionkowski, M., $\&$ Weinberg, N. N. 2003, Physical Review Letters, 91, 07301
  * [15] Cataldo, M., Cruz, N., $\&$ Lepe, S. 2005, Physics Letters B, 619, 5
  * [16] Clocchiatti, A., et al. 2006, Astrophysical Journal, 642, 1
  * [17] Collins, C. B. 1977, Journal of Mathematical Physics, 18, 2116
  * [18] Coperland, E. J., et al. 2006, International Journal of Modern Physics D, 15, 1753
  * [19] De Bernardis, P., et al. 1998, Nature, 391, 5
  * [20] Eisenstein, E., et al. 2005, Astrophysical Journal, 633, 560
  * [21] Elizalde, E., Nojiri, S., $\&$ Odintsov, S. D. 2004, Phys. Rev. D, 70, 0343539
  * [22] Farzin, A., et al. 2012, RAA, 12, 26
  * [23] Folomeev, V., $\&$ Gurovich, V. 2008, Physics Letters B, 661, 75
  * [24] Garnavich, P., et al. 1998, Astrophysical Journal, 493, L53
  * [25] Garnavich, P. M., et al. 1998, Astrophysical Journal, 509, 74
  * [26] Gr$\o$n, $\O$. 1990, Astrophysics $\&$ Space Science, 173, 191
  * [27] Guo, Z. K., Ohta, N., $\&$ Tsujikawa, S. 2007, Phys. Rev. D, 76, 023508
  * [28] Hanany, S., et al. 2000, Astrophysical Journal, 493, L53
  * [29] Ichikawa, K., et al. 2006, Journal of Cosmology and Astroparticle Physics,0612, 005
  * [30] Jaffe, T. R., et al. 2005, Astrophysical Journal, 629, L1-L4
  * [31] Komatsu, E., et al. 2009, Astrophysical Journal Supplement Series, 180, 330
  * [32] Kumar, S., $\&$ Yadav, A. K. 2011, Modern Physics Letters A, 26, 647
  * [33] MacTavish, C. J., et al. 2006, Astrophysical Journal, 647, 799
  * [34] Meng, X. H., Ren, J., $\&$ Hu, M. 2007, Communications in Theoretical Physics, 27, 379
  * [35] Misner, C. W. 1966, Astrophysical Journal, 151, 431
  * [36] Misner, C. W, 1967, Physical Review Letters, 19, 533
  * [37] Nesseris, S., $\&$ Perivolaropoulos, L. 2007, Journal of Cosmology and Astroparticle Physics, 0701, 018
  * [38] Nojiri, S., $\&$ Odintsov, S. D. 2005, Phys. Rev. D, 72, 023003
  * [39] Nojiri, S., $\&$ Odintsov, S. D. 2004, Physics Letters B, 595, 1
  * [40] Nojiri, S., Odintsov, S. D., $\&$ Tsujikawa, S. 2005, Phys. Rev. D, 71, 063004
  * [41] Padmanabhan, T., Chitre, S. M. 1987, Phys. Lett. A, 120, 433
  * [42] Pavon, D. $\&$ Wang, B. 2009, General Relativity and Gravitation, 41, 1
  * [43] Perlmutter, S., et al. 1999, Astrophysical Journal, 517, 5
  * [44] Perlmutter, S., et al. 1998, Nature, 391 51
  * [45] Perlmutter, S., et al. 1997, Astrophysical Journal, 483, 565
  * [46] Pradhan, A., Amirhashchi, H. 2011, Modern Physics Letters A, 30, 2261
  * [47] Pradhan, A., Amirhashchi, H., $\&$ Saha, B. 2011, Astrophysics and Space Science, 333 343
  * [48] Ren, J., $\&$ Meng, X. H. 2006, Physics Letters B, 633, 1
  * [49] Riess, A. G., et al. 2004, Astrophysical Journal, 607, 665
  * [50] Riess, A. G., et al. 2000, Publications of the Astronomical Society of the Pacific, 112, 1284
  * [51] Riess, A. G., et al. 1998, Astronomical Journal, 116, 1009
  * [52] Rodrigues, D. C., 2008, Phys. Rev. D, 77, 023534
  * [53] Saha, B., Amirhashchi, H., $\&$ Pradhan, A. 2012, Astrophysics and Space Science, DOI:10.1007/s10509-012-1155-x
  * [54] Schmidt, B. P., et al. 1998. Astrophysical Journal, 507, 46
  * [55] Seljak, U., et al. 2005, Phys. Rev. D, 71, 103515
  * [56] Setare, M. R. 2007, Physics Letters B, 644, 99
  * [57] Setare, M .R. 2007, European Physical Journal C, 50, 991
  * [58] Setare, M. R. 2007, Physics Letters B, 654, 1
  * [59] Setare, M. R., $\&$ Saridakis, E. N. 2008, Physics Letters B, 668, 177
  * [60] Setare, M. R., Sadeghi, J., $\&$ Amani, R. R. 2009, Physics Letters B, 673, 241
  * [61] Sheykhi, A., $\&$ Setare, M. R. 2010, International Journal of theoretical Physics, 49, 2777
  * [62] Spergel, D. N., et . 2003, Astrophysical Journal Supplement Series, 148, 175
  * [63] Tegmark, M., et al. 2004, Phys. Rev. D, 69 103501
  * [64] Tonry, J. L., et al. 2003, Astrophysical Journal, 594 1
  * [65] Singh, T., $\&$ Chaubey, R. 2012, RAA,12, 473
  * [66] Zel’dovich, Ya. B., $\&$ Novikov, I. D. 1971, Relativistic Astrophysics, Vol. 2, The University Chicago Press, Chicago, p. 519\.
  * [67] Zhao, G. B., et al. 2007, Physics Letters B, 648, 8
  * [68] Zimdahl, W. 1996, Phys. Rev. D, 53, 53483


11institutetext:  College of Computer and Information Science, Chongqing
Normal University
22institutetext: Chongqing Engineering Research Center of Educational Big Data
Intelligent Perception and Application
33institutetext: College of Electronic and Information Engineering, Southwest
University
Corresponding author<EMAIL_ADDRESS>
# RFAConv: Innovating Spatial Attention and Standard Convolutional Operation

Xin Zhang 11 Chen Liu 11 Tingting Song 11** Degang Yang 1122** Yichen Ye 33 Ke
Li 11 Yingze Song 11

###### Abstract

Spatial attention has been widely used to improve the performance of
convolutional neural networks. However, it has certain limitations. In this
paper, we propose a new perspective on the effectiveness of spatial attention,
which is that the spatial attention mechanism essentially solves the problem
of convolutional kernel parameter sharing. However, the information contained
in the attention map generated by spatial attention is not sufficient for
large-size convolutional kernels. Therefore, we propose a novel attention
mechanism called Receptive-Field Attention (RFA). Existing spatial attention,
such as Convolutional Block Attention Module (CBAM) and Coordinated Attention
(CA) focus only on spatial features, which does not fully address the problem
of convolutional kernel parameter sharing. In contrast, RFA not only focuses
on the receptive-field spatial feature but also provides effective attention
weights for large-size convolutional kernels. The Receptive-Field Attention
convolutional operation (RFAConv), developed by RFA, represents a new approach
to replace the standard convolution operation. It offers nearly negligible
increment of computational cost and parameters, while significantly improving
network performance. We conducted a series of experiments on ImageNet-1k,
COCO, and VOC datasets to demonstrate the superiority of our approach. Of
particular importance, we believe that it is time to shift focus from spatial
features to receptive-field spatial features for current spatial attention
mechanisms. In this way, we can further improve network performance and
achieve even better results. The code and pre-trained models for the relevant
tasks can be found at https://github.com/Liuchen1997/RFAConv.

##  1 Introduction

Convolutional neural networks [1, 2] have dramatically reduced the
computational overhead and complexity of models by using the convolutional
operation with shared parameters. Driven by classical networks, such as LeNet
[3], AlexNet [4], and VGG [5], convolutional neural networks have now
established a complete system and formed advanced convolutional neural network
models [6, 7, 8, 9, 10]. After carefully studying the convolutional operation,
we gained inspiration. For classification, object detection, and semantic
segmentation tasks, on one hand, the shape, size, color, and distribution of
objects in different locations in the image are variable. However, during the
convolutional operation, the convolutional kernel uses the same parameters in
each receptive field to extract information, which does not consider the
differential information from different locations. So the performance of the
network is limited, as demonstrated by recent works [11, 12, 13]. On the other
hand, the convolutional process does not take into account the significance of
each feature, which further reduces the efficiency of the extraction features
and ultimately restricts the performance of the model. In addition, the
attention mechanism [14, 15, 16] enables the model to concentrate on
significant features, which can enhance the benefits of feature extraction and
the ability of convolutional neural networks to capture detailed feature
information.

By examining the intrinsic limitations of convolutional operations and the
properties of attention mechanisms, we assert that while the current spatial
attention mechanism has fundamentally addressed the issue of parameter sharing
in convolutional operations, but it remains restricted to the recognition of
spatial features. The current spatial attention mechanism does not fully
address the parameter sharing problem for larger convolutional kernels.
Furthermore, they are unable to emphasize the significance of each feature in
the receptive field, such as the existing Convolutional Block Attention Module
(CBAM) [17] and Coordinate Attention (CA) [18]. Consequently, we propose a
novel receptive-field attention (RFA) that comprehensively addresses the issue
of parameter sharing for convolutional kernels and takes into account the
significance of each feature in the receptive field. The RFA-designed
convolutional operation (RFAConv) is a groundbreaking method that can replace
standard convolutional operations in current neural networks. With only a few
additional parameters and computational overhead, RFAConv enhances network
performance. Numerous experiments conduct on ImagNet-1k [19], COCO [20], and
VOC [21] have demonstrated the efficacy of RFAConv. As an attention-based
convolutional operation, RFAConv outperforms CAMConv, CBAMConv, CAConv
(constructed by CAM [17], CBAM, and CA, respectively), as well as the standard
convolutional operation. Moreover, to address the issue of slow to extract
receptive field features for current methods, we propose a lightweight
operation. During the construction of RFAConv, we also design an upgraded
version of CBAM and CA and conduct relevant experiments. We assert that
spatial attention mechanisms should focus on receptive-field spatial features
to further advance their development and enhance the advantages of
convolutional neural networks.

##  2 Related Works

###  2.1 Convolutional neural network architecture

The convolutional operation, which serves as a basic operation in
convolutional neural networks, has led the development of many advanced
network mode, such as vehicle detection [22], UAV images [23], medicine [24],
etc. He et al. [25] suggested that as the depth of the network increases, the
model becomes harder to train and may experience a degradation phenomenon. To
address this issue, they proposed to use residual connections to revolutionize
the design of network. Huang et al. [26] improved feature information by
reusing features to address the issue of network gradient disappearance. After
conducting a thorough study on convolutional operations, Dai et al. [27]
claimed that the convolutional operation with a fixed sampling position can
restrict the performance of the network to a certain extent. So, they proposed
Deformable Conv, which alters the sampling positions of convolutional kernels
by learning offsets. Building upon this approach, Deformable Conv V2 [28], and
Deformable Conv V3 [29] were subsequently developed to further enhance the
performance of convolutional networks. Zhang et al. [30] observed that group
convolution could decrease the number of parameters and computational overhead
of the model, however, insufficient interaction between information within the
group can adversely affect the final network performance. Although the
1$\times$1 convolution can interact with information, it will bring more
parameters and computational overhead, so they proposed the parameter-free
Channel Shuffle operation to interact with the information between groups. Ma
et al. [31] discovered that models with few parameters do not always result in
faster inference times, and similarly, small computational effort does not
guarantee quick performance. After careful study, they proposed the ShuffleNet
V2. The YOLO [32] object detection network divided the input image into a grid
to predict the location and class of objects. As research has progressed,
eight versions of object detectors based on YOLO have been proposed, such as
YOLOv5 [33], YOLOv7 [34], YOLOv8 [35], etc. While the previously-mentioned
convolutional neural network architectures have achieved significant success,
they do not directly address the problem of parameter sharing during the
feature extraction process. Our work focuses on utilizing the attention
mechanism to tackle the problem of convolutional parameter sharing from a
fresh perspective.

###  2.2 Attention Mechanism

Attention mechanism, as a technique to improve the performance of network
models, allows models to focus on key features. The theory of attention
mechanism has now established a complete and mature system in the field of
deep learning. Hu et al. [36] proposed a Squeeze-and-Excitation (SE) block to
obtain the weights corresponding to each channel. This is achieved by
compressing features to aggregate global channel information. Wang et al. [37]
asserted that the correspondence between individual channels and weights is
indirect when the SE interacts with information. Therefore, they designed the
Efficient Channel Attention (ECA) by replacing the Fully Connected (FC) layer
in the SE with a one-dimensional convolution of adaptive kernel size. Woo et
al. [17] proposed the Convolutional Block Attention Module (CBAM), which
combines channel attention and spatial attention. As a plug-and-play module,
it can be embedded into convolutional neural networks to enhance network
performance. Although SE and CBAM have allowed the network to achieve good
performance, Hou et al. [18] found that the compressing feature in SE and CBAM
lost too much information. Therefore, they proposed the lightweight coordinate
attention (CA) to solve this problem. Fu et al. [38] designed a spatial
attention module and a channel attention module to extend Fully Convolutional
Networks (FCN) for modeling semantic interdependencies in the spatial and
channel dimensions, respectively. Zhang et al. [39] generated feature maps at
different scales on channels to build a more efficient channel attention
mechanism. This paper introduces a new approach to address the issue of
parameter sharing in standard convolutional operations. Our proposal involves
combining attention mechanisms to create convolutional operations. Although
existing attention mechanisms have demonstrated good performance, they do not
specifically target the spatial features of receptive fields. To tackle this
limitation, we developed RFAConv with non-shared parameters to improve
performance of the network.

##  3 Methods

###  3.1 Reviewing Standard Convolutional Operation

Figure 1: It simply represents a 3$\times$3 convolution operation. The
features are obtained by multiplying the convolution kernel with a receptive-
field slider of the same size and then summing.

The standard convolutional operation serves as the fundamental building block
for constructing convolutional neural networks. It utilizes sliding windows
with shared parameters to extract feature information and overcome the
inherent issues of neural networks constructed with fully connected layers,
such as a large number of parameters and high computational overhead. Let X
$\in$ $R^{C\times H\times W}$ denote the input feature maps, where C, H, and W
represent the number of channels, height, and width of the feature map,
respectively. To clearly demonstrate the feature extraction process through
convolutional kernels, we use the example of C = 1. The convolutional
operation to extract feature information from each receptive field slider can
be expressed as follows:

$\displaystyle F_{1}=X_{11}\times K_{1}+X_{12}\times K_{2}+X_{13}\times
K_{3}+...+X_{1S}\times K_{S}$ (1) $\displaystyle F_{2}=X_{21}\times
K_{1}+X_{22}\times K_{2}+X_{23}\times K_{3}+...+X_{2S}\times K_{S}$
$\displaystyle...$ $\displaystyle F_{N}=X_{N1}\times K_{1}+X_{N2}\times
K_{2}+X_{N3}\times K_{3}+...+X_{NS}\times K_{S}$

Here, $F_{i}$ represents the value obtained by each convolutional slider after
computation, $X_{i}$ represents the pixel value at the corresponding position
within each slider, K represents the convolutional kernel, S denotes the
number of parameters in the convolutional kernel, and N represents the total
number of the receptive-field slider. As Shown in Figrue 1, K is a 3$\times$3
convolutional kernel and S is 9. Moreover, there are a total of 16 receptive-
field sliders in the Figrue 1. For simplicity of representation, we just draw
three receptive-field sliders. It can be seen that features at the same
position within each slider share the same parameter $K_{i}$. Therefore, the
standard convolutional operation fails to capture the difference in
information brought by different positions, which ultimately limits the
performance of the convolutional neural network to a certain extent.

###  3.2 Reviewing Spatial Attention

Currently, the spatial attention mechanism uses the attention map obtained
through learning to highlight the importance of each feature. Similar to the
previous section, C=1 is taken as an example. The spatial attention mechanism
highlighting key features can be expressed simply as follows:

$\displaystyle F_{1}=X_{1}\times A_{1}$ (2) $\displaystyle F_{2}=X_{2}\times
A_{2}$ $\displaystyle...$ $\displaystyle F_{N}=X_{N}\times A_{N}$

Here, $F_{i}$ represents the value obtained after the weighting operation.
$X_{i}$ and $A_{i}$ represent the values of the input feature map and the
learned attention map at different positions, respectively, N is the product
of the height and width of the input feature map, which represents the total
number of pixel values. In general, the whole process can be simply
represented in Fig. 2.

Figure 2: The original feature map highlights the key features by learned
attention map. This process of highlighting is the Re-weight ($\times$)
operation.

###  3.3 Spatial Attention and Standard Convolutional Operation

As it is widely recognized, incorporating attention mechanisms into
convolutional neural networks can enhance their performance. After a careful
study of the standard convolution operation and the existing spatial attention
mechanism, we argue that the spatial attention mechanism effectively overcomes
the inherent limitation of convolutional neural networks, which is parameter
sharing. Currently, the most commonly used kernel sizes in convolutional
neural networks are 1 $\times$ 1 and 3 $\times$ 3\. The convolutional
operation for extracting features after introducing the spatial attention
mechanism is either a 1 $\times$ 1 or a 3 $\times$ 3 convolutional operation.
To intuitively show the process, the spatial attention mechanism is inserted
into the front of the 1 $\times$ 1 convolutional operation. Through the
attention map to weigh operation the input feature map (Re-weight ”$\times$”),
finally, the slider feature information of the receptive-field is extracted
through 1 $\times$ 1 convolution operation. The overall process can be simply
represented as follows:

$\displaystyle F_{1}=X_{1}\times A_{1}\times K$ (3) $\displaystyle
F_{2}=X_{2}\times A_{2}\times K$ $\displaystyle...$ $\displaystyle
F_{N}=X_{N}\times A_{N}\times K$

Here, the convolution kernel K represents only one parameter value. If we take
the value of $A_{i}\times K$ as a new convolution kernel parameter, the
interesting thing is that the problem of parameter sharing in the extraction
of features by 1$\times$1 convolution operations is solved. As shown in Fig.
3, it visualizes the combination of a 1$\times$1 convolutional kernel and a
spatial attention mechanism. This combination essentially solves the problem
of parameter sharing.

Figure 3: The convolutional kernel parameter $K_{i}$ obtained by multiplying
the attentional weight $A_{i}$ with the convolutional kernel parameter K is
different in each receptive-field slider, i.e., $Kernel1\neq Kernel2\neq
Kernel3\neq...\neq KernelN$. $\displaystyle F_{1}=X_{11}\times A_{11}\times
K_{1}+X_{12}\times A_{12}\times K_{2}+X_{13}\times A_{13}\times
K_{3}+...+X_{19}\times A_{19}\times K_{9}$ (4) $\displaystyle
F_{2}=X_{21}\times A_{21}\times K_{1}+X_{22}\times A_{22}\times
K_{2}+X_{23}\times A_{23}\times K_{3}+...+X_{29}\times A_{29}\times K_{9}$
$\displaystyle...$ $\displaystyle F_{N}=X_{N1}\times A_{N1}\times
K_{1}+X_{N2}\times A_{N2}\times K_{2}+X_{N3}\times A_{N3}\times
K_{3}+...+X_{N9}\times A_{N9}\times K_{9}$ Figure 4: It is obvious that there
is an overlap of features in each receptive-field slider, which leads to the
problem of sharing of attentional weights across sliders.

Specifically, the output features are obtained by multiplying the receptive-
field slider, the attention weights and the 1$\times$1 convolutional kernel
parameters. In contrast to Fig. 1, if the product of the attention weights and
the convolution kernel parameters is treated as a new convolution parameter,
then the convolution parameter in each receptive field are different and not
shared. However, when the spatial attention mechanism is inserted in front of
the 3$\times$3 convolutional operation, it will be limited. As mentioned
above, if we take the value of $A_{i}\times K$ as a new convolution kernel
parameter, the Equation (4) completely solves the problem of parameter sharing
for large-scale convolutional kernels. As shown in the Fig. 4, it represents
the process in which the input features are weighted by spatial attention and
then operated by a 3$\times$3 convolution. If the product of the attention
weight corresponding to the receptive-field slider and the convolution kernel
is taken as the convolution parameter, then the problem of convolution
parameter sharing is solved. However, the most important point is that the
convolutional kernel will share some of the features when it extracts features
in each receptive field slider. In other words, there will be an overlap
within each receptive-field slider, and when spatial attention is weighted,
the size of the attention map is the same as the size of the input features,
so the attention weights are shared in the sense field slider.

After careful analysis will find that
$A_{12}=A_{21},A_{13}=A_{22},A_{15}=A_{24},...$. In this case, the weights of
the spatial attention map are shared across each sliding window. As a result,
the spatial attention mechanism cannot effectively address the problem of
parameter sharing for large-scale convolutional kernels, such as the
3$\times$3 convolution, because it does not consider the spatial features of
the entire receptive field. Consequently, the effectiveness of the spatial
attention mechanism is limited.

###  3.4 Innovating Spatial Attention and Standard Convolutional Operation

RFA is proposed to address the limitations of the existing spatial attention
mechanism and provides an innovative solution to spatial processing. Drawing
inspiration from RFA, a series of spatial attention mechanisms have been
developed that can further enhance the performance of convolutional neural
networks. RFA can be considered as a lightweight plug-and-play module, and the
convolutional operation (RFAConv) designed by RFA can replace the standard
convolution to improve the performance of convolutional neural networks.The
spatial attention mechanism can solve the problem of convolutional parameter
sharing by focusing on the receptive-field spatial feature, which is the
direction for upgrading the existing spatial attention. The performance can be
improved by combining the upgraded spatial features with the convolutional
operation. So, we predict that the combination of spatial attention mechanisms
and standard convolution operations will continue to evolve and lead to new
breakthroughs in the future.

Receptive-Field Spatial Feature: In order to better understand the concept of
receptive-field spatial feature, We will provide the relevant definition. The
receptive-field spatial feature is specifically designed for convolutional
kernels and is dynamically generated based on the kernel size. As shown in
Fig. 5, the 3$\times$3 convolutional kernel is used as an example.

Figure 5: The receptive-field spatial features are obtained by transforming
the spatial features.

In Figure 5, the ”Spatial Feature” refers to the original feature map. The
”Receptive-Field Spatial Feature” is the feature map transformed by spatial
features, which is composed of non-overlapping sliding windows. Each
3$\times$3 size window in the receptive-field spatial feature represents a
receptive-field slider.

Figure 6: In the figure, it shows in detail an example of extracting
3$\times$3 receptive-field spatial features by the Unfold method. For the
convenience of display, we set the step size to 3.

Rceptive-Filed Attention Convolution (RFAConv): Regarding the receptive-field
spatial feature, we propose receptive-field attention (RFA). This approach not
only emphasizes the significance of different features within the receptive
field slider, but also prioritizes the receptive-field spatial feature.
Through this method, the problem of convolution kernel parameter sharing is
completely solved. The receptive-field spatial features are dynamically
generated depending on the size of the convolution kernel, therefore, The RFA
is a fixed combination of convolution and cannot be separated from the help of
convolution operations, which simultaneously rely on RFA to improve
performance, so we propose receptive-field attention convolution (RFAConv).
The overall structure of RFAConv with a 3$\times$3 size convolutional kernel
is shown in Fig. 7.

Figure 7: The detailed structure of RFAConv, which dynamically determines the
importance of each feature in the receptive-field and solves the problem of
parameters sharing.

As one of the popular frameworks in the field of deep learning, Pytorch
provides the Unfold method to extract receptive-field spatial features. The
detailed structure is shown in Fig. 6, which extracts 3$\times$3 receptive-
field spatial features. Let input X $\in$ $R^{C\times H\times W}$, after the
Unfold method, its dimension becomes 9C$\times$H$\times$W. Where C, H, and W
represent the number of channels, height and width of input. Although Unfold
is able to extract receptive-field spatial features by parameterless way, it
is slow. Therefore, in RFAConv, we utilize a fast method to extract receptive-
field spatial features, i.e., Group Conv. As mentioned in the previous
section, each 3$\times$3 size window in the receptive-field spatial feature
represents a receptive-field slider when the 3$\times$3 convolution kernel is
used to extract features. However, after using fast Group Conv to extract the
receptive-field features, original features are mapped into new features. This
method is fast and more efficient than the original Unfold method. As shown in
Table 1, experiments based on the YOLOv5n and VisDrone dataset [40]
demonstrate it. It can be seen that RFAConv based on GroupConv obtains good
performance while training 300 epochs requires less training time than the
Unfold method. Morefore, we need to explain that the Unfold method is
parameterless, while in the Table 1, it can be seen that the number of
parameters required for the GroupConv-based method is the same as for the
Unfold method. Because we use a lightweight approach to interacting
information in receptive-field.

Methods | mAP50($\%$) | mAP($\%$) | FLOPS(G) | Param(M) | Training Time (Hours)
---|---|---|---|---|---
YOLOv5n | 26.43 | 13.66 | 4.3 | 1.78 | 6.81
YOLOv5n + RFAConv (Unfold) | 27.43 | 14.22 | 4.6 | 1.85 | 10.42
YOLOv5n + RFAConv (GroupConv) | 27.58 | 14.36 | 4.7 | 1.85 | 7.37
Table 1: Object detection experiments based on YOLOv5n and VisDrone datasets
to illustrate the advantages of RFAConv built on GroupConv.

Recent research has shown that interacting information can enhance network
performance, as demonstrated in [41, 42, 43]. Similarly, for RFAConv,
interacting receptive-field feature information to learn the attention map can
enhance network performance. However, interacting with each receptive-field
feature can result in additional computational overhead,so to minimize
computational overhead and number of parameters, the AvgPool is utilized to
aggregate the global information of each receptive-field feature. Then, a 1×1
group convolutional operation is used to interact information. Finally, we use
softmax to emphasize the significance of each feature within the receptive-
field feature. In general, the calculation of RFA can be expressed as:

$\displaystyle F=Softmax(g^{1\times 1}(AvgPool(X)))\times ReLU(Norm(g^{k\times
k}(X)))$ (5) $\displaystyle~{}~{}~{}=A_{rf}\times F_{rf}$

Here, $g^{i\times i}$ represents a grouping convolution of size $i\times i$, k
represents the size of the convolution kernel, Norm stands for normalization,
X represents the input feature maps, F is obtained by multiplying the
attention map $A_{rf}$ with the transformed receptive-field spatial feature
$F_{rf}$. Unlike CBAM and CA, RFA is capable of generating attention maps for
each receptive-field feature. The performance of convolutional neural networks
is limited by the standard convolutional operations due to the fact that the
convolution operation relies on shared parameters and is not sensitive to
differences in information brought about by positional variations. However,
RFAConv can completely address this issue by emphasizing the significance of
different features within the receptive field slider and prioritizing the
receptive-field spatial feature.

The feature map obtained through RFA is the receptive-field spatial feature,
which does not overlap after ”Adjust Shape”. Therefore, the learned attention
map aggregate the feature information of each receptive field slider. In other
words, the attention map is no longer shared within each receptive-field
slider. This completely compensates for the shortcomings of the existing CA
and CBAM attention mechanisms. RFA provides significant benefits to the
standard convolution kernel. However, after adjusting shape, the features are
k times in height and width, requiring a k $\times$ k convolution operation
with a stride = k to extract feature information. The convolution operation
RFAConv designed by RFA brings good gains for convolution, and it innovates
the standard convolution.

Figure 8: Detailed structure of RFCAConv and RFCBAMConv, which focus on
receptive-field spatial features. Comparing the original CBAM, We use SE
attention to replace CAM in RFCBAM. Because, this can reduce computational
overhead.

Furthermore, We assert that existing spatial attention mechanisms, such as
CBAM [17] and CA [18], should prioritize receptive field spatial features to
improve network performance. As is well-known, the network model based on the
self-attention mechanism [44, 45, 46] has achieved great success, because it
solves the problem of convolution parameter sharing and models long-distance
information. However, the self-attention mechanism also introduces significant
computational overhead and complexity to the model. We argue that directing
the attention of existing spatial attention mechanisms to the receptive-field
spatial feature can solve the problems of parameter sharing and modeling of
long-range information in a way similar to self-attention. This approach
requires significantly fewer parameters and computational resources than self-
attention. The answers are as follows:

(1) The combination of the spatial attention mechanism, which focuses on the
receptive-field spatial feature, with convolution eliminates the problem of
convolution parameter sharing. (2) The current spatial attention mechanism
already considers long-distance information and can obtain global information
through global average pooling or global maximum pooling, which explicitly
takes into account long-range information.

Therefore, we design new CBAM and CA models called RFCBAM and RFCA, which
focus on the receptive-field spatial feature. Similar to RFA, the final
convolution operation of $k\times k$ with stride = k is used to extract the
feature information. The specific structure of these two new convolution
methods, as shown in Fig. 8, we call these two new convolution operations
RFCBAMConv and RFCAConv. Comparing the original CBAM, We use SE attention to
replace CAM in RFCBAM. Because this can reduce computational overhead.
Moreover in RFCBAM, channel and spatial attention are not performed in
separate steps. Instead, they are weighted simultaneously, allowing the
attention map obtained for each channel to be different.

Layer Name |  Output Size |  Resnet18 |  Resnet34
---|---|---|---
Conv1 |  112 $\times$ 112 |  |
Layer1 |  56 $\times$ 56 |  $\begin{bmatrix}NewConv&3\times 3\\\ Conv&3\times 3\end{bmatrix}\times 2$ |  $\begin{bmatrix}NewConv&3\times 3\\\ Conv&3\times 3\end{bmatrix}\times 3$
Layer2 |  28 $\times$ 28 |  $\begin{bmatrix}NewConv&3\times 3\\\ Conv&3\times 3\end{bmatrix}\times 2$ |  $\begin{bmatrix}NewConv&3\times 3\\\ Conv&3\times 3\end{bmatrix}\times 4$
Layer3 |  14 $\times$ 14 |  $\begin{bmatrix}NewConv&3\times 3\\\ Conv&3\times 3\end{bmatrix}\times 2$ |  $\begin{bmatrix}NewConv&3\times 3\\\ Conv&3\times 3\end{bmatrix}\times 6$
Layer4 |  7 $\times$ 7 |  $\begin{bmatrix}NewConv&3\times 3\\\ Conv&3\times 3\end{bmatrix}\times 2$ |  $\begin{bmatrix}NewConv&3\times 3\\\ Conv&3\times 3\end{bmatrix}\times 3$
|  1 $\times$ 1 |  AvgPool 1000-d
Table 2: The Resnet18 and Resnet34 are construct by the new convolution
operation.

##  4 Experiments and Discussions

To verify the effectiveness of our method, we perform classification, object
detection and semantic segmentation experiments. The equipment for all
experiments are based on RTX3090. In the classification experiments, we use
four RTX3090 to train the model in parallel.

###  4.1 Classification experiments on ImageNet-1k

We perform experiments to validate our method on the ImageNet-1k, which
contains a number of 1281167 training sets and 50000 validation sets. Similar
to RFAConv, we construct CBAMConv and CAConv by combining the CBAM and CA,
respectively, with an additional 3 $\times$ 3 convolution layer at the end of
the CBAM and CA modules. We also compare the CAMConv, which is constructed
using the channel attention mechanism CAM [17]. we evaluate in ResNet18,
ResNet34. Specifically, RFAConv, CBAMConv, CAConv, and CAMConv are used to
replace (r) the first convolutional layer of BasicBlock in ResNet18 and
ResNet34, respectively. In general, the new convolution is structured as shown
in Table 2.

Models | FLOPS(G) | Param(M) | Top1(%) | Top5(%)
---|---|---|---|---
Resnet18 | 1.82 | 11.69 | 69.59 | 89.05
\+ CAMConv(r) | 1.83 | 11.75 | 70.76 | 89.74
\+ CBAMConv(r) | 1.83 | 11.75 | 69.38 | 89.12
\+ CAConv(r) | 1.83 | 11.74 | 70.58 | 89.59
\+ RFAConv(r) | 1.91 | 11.85 | 71.23 | 90.29
Resnet34 | 3.68 | 21.80 | 73.33 | 91.37
\+ CAMConv(r) | 3.68 | 21.93 | 74.03 | 91.69
\+ CBAMConv(r) | 3.68 | 21.93 | 72.95 | 91.26
\+ CAConv(r) | 3.68 | 21.91 | 73.76 | 91.68
\+ RFAConv(r) | 3.84 | 22.16 | 74.25 | 92.03
Table 3: Classification results on ImageNet-1K using the Resnet18 and
Resnet34. The different convolutional operation constructed by the attention
mechanism is compared.

The NewConv in Table 2 represents the convolution mode constructed by the
attention mechanism. For CAConv, CBAMConv, and CAMConv, it can be considered
that CA, CBAM, and CAM are added before the first layer of convolution in
BasicBlock. In the image classification experiments, we train 100 epochs for
each model with batch-size of 128. The learning rate start from 0.1 and
decrease every 30 epochs, 0.1 times each time. In experiments, we follow most
of the previous work and report the accuracy for TOP1 and TOP5, respectively.
Table 3 shows the results produced by different networks on the ImageNet-1K
validation set. It is clear that replacing the 3$\times$3 convolutional
operation with RFAConv significantly improves the recognition results.
Compared to the baseline models RestNet18 and ResNet34 , the network
constructed by RFAConv achieves the best recognition results at the cost of
only a small increase in parameters and computational overhead. Such as
ResNet18 constructed based on RFAConv only adds 0.16 M parameters and 0.09 G
computational overhead over the original model, and increases the accuracy by
1.64% and 1.24% on TOP1 and TOP5, respectively.

Moreover, as we mentioned earlier, spatial attention can be enhanced again by
focusing on receptive-field spatial features. Therefore, we design RFCBAMConv
and RFCAConv, which are improvements of CBAM and CA. In order to verify their
advantages, we conduct experiments based on ResNet18 and report the relevant
data in Table 4. It is obvious that RFCBAMConv and RFCAConv achieve better
recognition accuracy, compared with CBAMConv and CAConv in Table 3. Most
importantly, they also significantly improve performance at the cost of only a
small increase in parameters and computational overhead. This is a strong
demonstration that spatial attention can be improved by placing attention into
the receptive-field spatial features. It fully demonstrates that spatial
attention can be improved by placing attention into the receptive-field
spatial features.

Models | FLOPS(G) | Param(M) | Top1(%) | Top5(%)
---|---|---|---|---
Resnet18 | 1.82 | 11.69 | 69.59 | 89.05
\+ RFCBAMConv(r) | 1.90 | 11.88 | 72.15 | 90.71
\+ RFCAConv(r) | 1.92 | 11.89 | 72.01 | 90.64
Table 4: RFCBAMConv and RFCAConv improve the performance of CBAMConv and
CAConv. The table shows that the classification accuracy is significantly
improved on ImageNet-1k.

All classification experiments clearly demonstrate the significant advantages
of our methods, because RFAConv, RFCBAMConv, and RFCAConv completely address
the problem of convolution kernel parameter sharing. Moreover, it is worth
noting that RFCBAMConv and RFCAConv outperform RFAConv, because they not only
solve the problem of convolution kernel parameter sharing but also consider
long-distance information through global pooling.

Moreover, to provide a more intuitive analysis, as with the most work, we use
the Grad-CAM [47] algorithm for visualization. Grad-CAM highlights the regions
of interest of different networks for a particular class of objects. To some
extent, it is possible to see how the networks utilize the features. We
randomly selected some images in the validation set of ImageNet-1K and
visualized the results of the networks constructed with different attention
convolutions based on ResNet18 separately. As shown in the Fig. 9, compared
with other attention convolutions, our RFAConv can help the network to better
recognize and highlight the key regions of objects.

Figure 9: Each network is built on ResNet18 based on attention convolution,
and the construction process is shown in Table 2. We use Grad-CAM as our
visualization tool to visualize networks without the last layer of
classifiers. Compared to other attention convolution methods, our RFAConv can
help the network to better recognize and highlight the key regions of objects

We also use Grad-CAM to visualize ResNet18 constructed by CBAMConv,
RFCBAMConv, CAConv, and RFCAConv. RFCBAM is obtained by improving the
attention of CBAM by putting it into the recepitve-field spatial feature.
Similarly, RFCA is obtained by CA after the same method. As shown in Fig. 10,
the improved RFCA and RFCBAM can help the network to better recognize and
highlight the key regions of objects after combining with the convolution
operation.

Figure 10: We put the attention of CBAM and CA into the receptive-field
spatial features and improve them to obtain RFCBAM and RFCA. Then RFCBAMConv
and RFCAConv are constructed by the same method as RFA. As in Fig. 10, we
visualize the different networks separately. Obviously, compared to CBAMConv
and CAConv, the improved obtained RFCBAMConv and CAConv can help the network
to better recognize and highlight the key regions of objects.

###  4.2 Object detection experiments on COCO2017

We conduct object detection experiments on COCO2017 to re-evaluate our
methods. COCO2017 contains 118287 training sets and 5000 verification sets. We
select YOLOv5n, YOLOv7-tiny, and YOLOv8n models to perform a series of
experiments. All parameters except for epoch and batch-size are set to the
default values. We train each model for 300 epochs with a batch-size of 32. To
be similar in classification, we replace some convolution operation in the
baseline model with novel convolution operations constructed using attention
mechanisms. Specifically, we replace all 3$\times$3 convolution operations in
the yaml files for YOLOv5 and YOLOv8 using attention convolution. And for
YOLOv7, we replace the first 3$\times$3 convolution operation in all ELAN [34]
in the backbone.

Models | FLOPS(G) | Param(M) |  $AP_{50}$(%) |  $AP_{75}$(%) | AP(%) |  $AP_{S}$(%) | $AP_{M}(\%)$ | $AP_{L}(\%)$ | Time(ms)
---|---|---|---|---|---|---|---|---|---
YOLOv5n | 4.5 | 1.8 | 45.6 | 28.9 | 27.5 | 13.5 | 31.5 | 35.9 | 4.4
\+ CAMConv(r) | 4.5 | 1.8 | 45.6 | 28.3 | 27.4 | 13.8 | 31.4 | 35.8 | 5.2
\+ CBAMConv(r) | 4.5 | 1.8 | 45.5 | 28.6 | 27.6 | 13.6 | 31.2 | 36.6 | 5.4
\+ CAConv(r) | 4.5 | 1.8 | 46.2 | 29.2 | 28.1 | 14.3 | 32 | 36.6 | 4.8
\+ RFAConv(r) | 4.7 | 1.9 | 47.3 | 30.6 | 29 | 14.8 | 33.4 | 37.4 | 5.3
YOLOv7-tiny | 13.7 | 6.2 | 53.8 | 38.3 | 35.9 | 19.9 | 39.4 | 48.8 | 6.8
\+ RFAConv(r) | 14.1 | 6.3 | 55.1 | 40.1 | 37.1 | 20.9 | 41.1 | 50 | 8.4
YOLOv8n | 8.7 | 3.1 | 51.9 | 39.7 | 36.4 | 18.4 | 40.1 | 52 | 4.2
\+ CAMConv(r) | 8.8 | 3.1 | 51.6 | 39 | 36.2 | 18 | 39.9 | 51.2 | 4.5
\+ CBAMConv(r) | 8.8 | 3.1 | 51.5 | 39.6 | 36.3 | 18.3 | 40.1 | 51.5 | 4.6
\+ CAConv(r) | 8.8 | 3.1 | 52.1 | 39.9 | 36.7 | 17.8 | 40.3 | 51.6 | 4.3
\+ RFAConv(r) | 9.0 | 3.2 | 53.4 | 41.1 | 37.7 | 18.9 | 41.8 | 52.7 | 4.5
\+ RFCAConv(r) | 9.1 | 3.2 | 53.9 | 41.7 | 38.2 | 19.7 | 42.3 | 53.5 | 4.7
Table 5: Object detection $AP_{50}$, $AP_{75}$, $AP$, $AP_{S}$, $AP_{M}$, and
$AP_{L}$ on the COCO2017 validation sets. We adopt the YOLOv5n, YOLOv7-tiny,
and YOLOv8n detection framework and replace the original convolution with the
novel convolutioanl operation constructed by attention mechanism.

Following the previous work, we report $AP_{50}$, $AP_{75}$, $AP$, $AP_{S}$,
$AP_{M}$ and $AP_{L}$ separately. Moreover, in order to better display the
performance of different networks, we chose the training process of YOLOv5n
for visualization. We visualize how the AP50 changes with the number of
iterations. The experimental results are shown in Table 5 and Fig. 11. When
RFAConv is used to replace some convolution, the network achieved
significantly improved detection results with only a small increase in the
number of parameters and computational overhead. Compared to other attention,
RFA still brings considerable benefits to the detection network. In some
experiments, we once again verify the effectiveness of RFCA, which exhibits
better performance of convolutional operation compared to the original CA.
Time represents the total time spent processing an image during validation. It
can be clearly seen that the model constructed with the novel convolutioanl
operation has an increase in time for processing a image. Therefore, if real-
time is pursued, the number of replacement convolutions should not be too
many.

Figure 11: The $AP_{50}$ change during training for the different YOLOv5n constructed by attention convolution. Models | FLOPS(G) | Param(M) | mAP(%) | Time(ms)
---|---|---|---|---
YOLOv5n | 4.2 | 1.7 | 41.5 | 2.7
\+ CAMConv(r) | 4.2 | 1.7 | 41.4 | 2.9
\+ CBAMConv(r) | 4.3 | 1.7 | 41.9 | 3
\+ CAConv(r) | 4.3 | 1.7 | 42.4 | 3
\+ RFAConv(r) | 4.5 | 1.8 | 43.3 | 3
YOLOv5s | 15.9 | 7.1 | 48.9 | 3
\+ CAMConv(r) | 16 | 7.1 | 48.5 | 3.5
\+ CBAMConv(r) | 16 | 7.1 | 49 | 3.7
\+ CAConv(r) | 16.1 | 7.1 | 49.6 | 3.1
\+ RFAConv(r) | 16.4 | 7.2 | 50 | 5.1
\+ RFCBAMConv(r) | 16.4 | 7.2 | 50.1 | 3.9
\+ RFCAConv(r) | 16.6 | 7.2 | 51 | 4.4
YOLOv7-tiny | 13.2 | 6.1 | 50.2 | 5
\+ CAMConv(r) | 13.2 | 6.1 | 50.3 | 5.4
\+ CBAMConv(r) | 13.2 | 6.1 | 50.1 | 5.4
\+ CAConv(r) | 13.2 | 6.1 | 50.5 | 5.4
\+ RFAConv(r) | 13.6 | 6.1 | 50.6 | 7.5
YOLOv8n | 8.1 | 3 | 53.5 | 3
\+ CAMConv(r) | 8.1 | 3 | 52.8 | 3.1
\+ CBAMConv(r) | 8.2 | 3 | 53.3 | 3.1
\+ CAConv(r) | 8.2 | 3 | 53.8 | 2.9
\+ RFAConv(r) | 8.4 | 3.1 | 54 | 3.2
Table 6: Object detection mAP50 and mAP on the VOC7+12 validation set.

###  4.3 Object detection experiments on VOC7+12

In order to validate our method again, we select the VOC7+12 dataset for
experiments. VOC7+12 is a mixture of VOC2007 and VOC2012, with a total of
16551 training sets and 4952 verification sets. Similar to the experiments on
COCO2017, we conduct experiments on advanced detection models such as YOLOv5n,
YOLOv5s, YOLOv7-tiny, and YOLOv8n. All hyperparameter settings and network
structure are the same as in the previous section.

Following most of the work, we also report mAP. As shown in Table 6. As with
the previously obtained conclusions, in all experiments, after we replace some
of the convolution operations in the network using RFAConv, the network gained
significant improvement by adding only a small number of parameters and
computational overhead. Meanwhile, RFA obtains an outstanding performance
compared to other attentions. Moreover, in some experiments, in some
experiments, we similarly experiment with networks constructed by RFCBAMConv
and RFCAConv. The results again validate their advantages. Also compared with
CBAMConv and CAConv, they still obtained better results.

###  4.4 Semantic segmentation experiments on VOC2012

In order to validate the advantages of our method again, we conduct semantic
segmentation experiments on the VOC2012 dataset, selecting DeepLabplusV3 [48]
and the backbone network Resnet18 to conduct related experiments. The pre-
training weights of each backbone network are obtained in the ImageNet-1k
experiment. We report the results for outputs at two different step sizes, 8
and 16, respectively. In the experiment, we found that the semantic
segmentation network constructed by RFAConv achieved better results than the
original model, but compared to CAConv, CAMConv, the performance of RFAConv is
not good. After thinking, we assert that RFAConv lacks consideration for long-
distance information, while semantic segmentation tasks rely on long-distance
information. CAConv, CAMConv and CBAMConv capture long range information by
global averaging pooling to obtain global information. Although CBAMConv
produce poor results for semantic segmentation, the improved RFCBAM obtain the
good performance. This again demonstrates that spatial attention can again
improve network performance through our approach, which simply requires making
spatial attention attend to the receptive-field spatial feature.

Backbone |  Stride |  MIOU(%)
---|---|---
Resnet18 |  8 |  58.9
\+ CAMConv(r) |  8 |  60.9
\+ CBAMConv(r) |  8 |  59.3
\+ CAConv(r) |  8 |  62.1
\+ RFAConv(r) |  8 |  60.8
\+ RFCBAMConv(r) |  8 |  62.1
\+ RFCAConv(r) |  8 |  63.9
Resnet18 |  16 |  64.6
\+ CAMConv(r) |  16 |  65.5
\+ CBAMConv(r) |  16 |  63.6
\+ CAConv(r) |  16 |  66.6
\+ RFAConv(r) |  16 |  65.4
\+ RFCBAMConv(r) |  16 |  67.7
\+ RFCAConv(r) |  16 |  68.0
Table 7: Results of experiments comparing different the novel convolutional
operation based on DeepLabPlusV3.

###  4.5 Discussions

As all the experiments have shown, RFAConv serves as an alternative to
standard convolution, bringing significant improvements to classification,
target detection, and semantic segmentation visual tasks at the cost of a
small increase in parameters and computational effort. As shown in the
classification visualization in Fig. 9 and Fig. 10, networks built based on
RFAConv are able to better focus on important information and features. This
is due to the fact that RFA takes into account the receptive-field spatial
features and is able to highlight the importance of each feature within the
receptive-field, while combining it with convolution to turn it into a non-
parametric shared convolution operation. Also as mentioned earlier, increasing
the focus of CA and CBAM to the receptive-field spatial feature can again
improve performance. Thus RFCBAM,RFCA was designed by us to significantly
improve the network performance. What is clear is that RFAConv is advantageous
but not outstanding in semantic segmentation tasks. We think it is that the
design of RFA does not take long range information into account. As Hou et
al.[18] argued, semantic segmentation relies on long range information. In
contrast to RFA, RFCBAM and RFCA take into account the receptive-field spatial
features and consider long range information through maximum pooling and
average pooling. So they obtain outstanding performance in every visual task.
By carefully analyzing RFAConv, RFCBAMConv and RFCAConv, our task can put some
existing spatial attention into the receptive field to enhance its
performance.

##  5 Conclusion

By analyzing the standard convolution and spatial attention, we conclude that
spatial attention mechanisms address the problem of parameter sharing and have
the advantage of considering long-distance information. However, the
performance of the spatial attention mechanism is limited for the large-size
convolution kernel. To address this issue, we propose a novel attention
mechanism called RFA and devise a novel convolution operation, which further
improves network performance. We also emphasize the importance of directing
attention to the receptive-field spatial feature to enhance network
performance. Through extensive experiments, we demonstrate the effectiveness
and advanced nature of our approach. Going forward, we hope that more and more
spatial attention mechanisms will adopt our proposed approach to further
improve performance, and we also hope that the novel convolution method will
be widely adopted to enhance network performance.

## References

  * [1] M. Lin, Q. Chen, S. Yan, Network in network, arXiv preprint arXiv:1312.4400 (2013).
  * [2] C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, A. Rabinovich, Going deeper with convolutions, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2015, pp. 1–9.
  * [3] Y. LeCun, L. Bottou, Y. Bengio, P. Haffner, Gradient-based learning applied to document recognition, Proceedings of the IEEE 86 (11) (1998) 2278–2324.
  * [4] A. Krizhevsky, I. Sutskever, G. E. Hinton, Imagenet classification with deep convolutional neural networks, Communications of the ACM 60 (6) (2017) 84–90.
  * [5] K. Simonyan, A. Zisserman, Very deep convolutional networks for large-scale image recognition, arXiv preprint arXiv:1409.1556 (2014).
  * [6] F. Chollet, Xception: Deep learning with depthwise separable convolutions, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp. 1251–1258.
  * [7] A. Howard, M. Sandler, G. Chu, L.-C. Chen, B. Chen, M. Tan, W. Wang, Y. Zhu, R. Pang, V. Vasudevan, et al., Searching for mobilenetv3, in: Proceedings of the IEEE/CVF international conference on computer vision, 2019, pp. 1314–1324.
  * [8] K. Han, Y. Wang, Q. Tian, J. Guo, C. Xu, C. Xu, Ghostnet: More features from cheap operations, in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2020, pp. 1580–1589.
  * [9] H. Zhang, C. Wu, Z. Zhang, Y. Zhu, H. Lin, Z. Zhang, Y. Sun, T. He, J. Mueller, R. Manmatha, et al., Resnest: Split-attention networks, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 2736–2746.
  * [10] Z. Liu, H. Mao, C.-Y. Wu, C. Feichtenhofer, T. Darrell, S. Xie, A convnet for the 2020s, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 11976–11986.
  * [11] P. Ramachandran, N. Parmar, A. Vaswani, I. Bello, A. Levskaya, J. Shlens, Stand-alone self-attention in vision models, Advances in neural information processing systems 32 (2019).
  * [12] G. Elsayed, P. Ramachandran, J. Shlens, S. Kornblith, Revisiting spatial invariance with low-rank local connectivity, in: International Conference on Machine Learning, PMLR, 2020, pp. 2868–2879.
  * [13] D. Li, J. Hu, C. Wang, X. Li, Q. She, L. Zhu, T. Zhang, Q. Chen, Involution: Inverting the inherence of convolution for visual recognition, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp. 12321–12330.
  * [14] J. Park, S. Woo, J.-Y. Lee, I. S. Kweon, Bam: Bottleneck attention module, arXiv preprint arXiv:1807.06514 (2018).
  * [15] X. Li, W. Wang, X. Hu, J. Yang, Selective kernel networks, in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2019, pp. 510–519.
  * [16] A. Luo, F. Yang, X. Li, S. Liu, Learning optical flow with kernel patch attention, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 8906–8915.
  * [17] S. Woo, J. Park, J.-Y. Lee, I. S. Kweon, Cbam: Convolutional block attention module, in: Proceedings of the European conference on computer vision (ECCV), 2018, pp. 3–19.
  * [18] Q. Hou, D. Zhou, J. Feng, Coordinate attention for efficient mobile network design, in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2021, pp. 13713–13722.
  * [19] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, L. Fei-Fei, Imagenet: A large-scale hierarchical image database, in: 2009 IEEE conference on computer vision and pattern recognition, Ieee, 2009, pp. 248–255.
  * [20] T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Dollár, C. L. Zitnick, Microsoft coco: Common objects in context, in: Computer Vision–ECCV 2014: 13th European Conference, Zurich, Switzerland, September 6-12, 2014, Proceedings, Part V 13, Springer, 2014, pp. 740–755.
  * [21] M. Everingham, S. A. Eslami, L. Van Gool, C. K. Williams, J. Winn, A. Zisserman, The pascal visual object classes challenge: A retrospective, International journal of computer vision 111 (2015) 98–136.
  * [22] M. Hassaballah, M. A. Kenk, K. Muhammad, S. Minaee, Vehicle detection and tracking in adverse weather using a deep learning framework, IEEE transactions on intelligent transportation systems 22 (7) (2020) 4230–4242.
  * [23] X. Zhu, S. Lyu, X. Wang, Q. Zhao, Tph-yolov5: Improved yolov5 based on transformer prediction head for object detection on drone-captured scenarios, in: Proceedings of the IEEE/CVF international conference on computer vision, 2021, pp. 2778–2788.
  * [24] Z. Ning, S. Zhong, Q. Feng, W. Chen, Y. Zhang, Smu-net: saliency-guided morphology-aware u-net for breast lesion segmentation in ultrasound image, IEEE transactions on medical imaging 41 (2) (2021) 476–490.
  * [25] K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image recognition, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp. 770–778.
  * [26] G. Huang, Z. Liu, L. Van Der Maaten, K. Q. Weinberger, Densely connected convolutional networks, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp. 4700–4708.
  * [27] J. Dai, H. Qi, Y. Xiong, Y. Li, G. Zhang, H. Hu, Y. Wei, Deformable convolutional networks, in: Proceedings of the IEEE international conference on computer vision, 2017, pp. 764–773.
  * [28] X. Zhu, H. Hu, S. Lin, J. Dai, Deformable convnets v2: More deformable, better results, in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2019, pp. 9308–9316.
  * [29] W. Wang, J. Dai, Z. Chen, Z. Huang, Z. Li, X. Zhu, X. Hu, T. Lu, L. Lu, H. Li, et al., Internimage: Exploring large-scale vision foundation models with deformable convolutions, arXiv preprint arXiv:2211.05778 (2022).
  * [30] X. Zhang, X. Zhou, M. Lin, J. Sun, Shufflenet: An extremely efficient convolutional neural network for mobile devices, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2018, pp. 6848–6856.
  * [31] N. Ma, X. Zhang, H.-T. Zheng, J. Sun, Shufflenet v2: Practical guidelines for efficient cnn architecture design, in: Proceedings of the European conference on computer vision (ECCV), 2018, pp. 116–131.
  * [32] J. Redmon, S. Divvala, R. Girshick, A. Farhadi, You only look once: Unified, real-time object detection, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp. 779–788.
  * [33] J. Glenn, Yolov5 release v6.1, https://github.com/ultralytics/yolov5/releases/tag/v6.1 (2022).
  * [34] C.-Y. Wang, A. Bochkovskiy, H.-Y. M. Liao, Yolov7: Trainable bag-of-freebies sets new state-of-the-art for real-time object detectors, arXiv preprint arXiv:2207.02696 (2022).
  * [35] J. Glenn, Ultralytics yolov8, https://github.com/ultralytics/ultralytics (2023).
  * [36] J. Hu, L. Shen, G. Sun, Squeeze-and-excitation networks, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2018, pp. 7132–7141.
  * [37] Q. Wang, B. Wu, P. Zhu, P. Li, W. Zuo, Q. Hu, Eca-net: Efficient channel attention for deep convolutional neural networks, in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2020, pp. 11534–11542.
  * [38] J. Fu, J. Liu, H. Tian, Y. Li, Y. Bao, Z. Fang, H. Lu, Dual attention network for scene segmentation, in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2019, pp. 3146–3154.
  * [39] H. Zhang, K. Zu, J. Lu, Y. Zou, D. Meng, Epsanet: An efficient pyramid squeeze attention block on convolutional neural network, in: Proceedings of the Asian Conference on Computer Vision, 2022, pp. 1161–1177.
  * [40] P. Zhu, L. Wen, D. Du, X. Bian, H. Fan, Q. Hu, H. Ling, Detection and tracking meet drones challenge, IEEE Transactions on Pattern Analysis and Machine Intelligence 44 (11) (2021) 7380–7399.
  * [41] I. Bello, B. Zoph, A. Vaswani, J. Shlens, Q. V. Le, Attention augmented convolutional networks, in: Proceedings of the IEEE/CVF international conference on computer vision, 2019, pp. 3286–3295.
  * [42] A. Srinivas, T.-Y. Lin, N. Parmar, J. Shlens, P. Abbeel, A. Vaswani, Bottleneck transformers for visual recognition, in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2021, pp. 16519–16529.
  * [43] A. Vaswani, P. Ramachandran, A. Srinivas, N. Parmar, B. Hechtman, J. Shlens, Scaling local self-attention for parameter efficient visual backbones, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp. 12894–12904.
  * [44] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, I. Polosukhin, Attention is all you need, Advances in neural information processing systems 30 (2017).
  * [45] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, et al., An image is worth 16x16 words: Transformers for image recognition at scale, arXiv preprint arXiv:2010.11929 (2020).
  * [46] Z. Liu, Y. Lin, Y. Cao, H. Hu, Y. Wei, Z. Zhang, S. Lin, B. Guo, Swin transformer: Hierarchical vision transformer using shifted windows, in: Proceedings of the IEEE/CVF international conference on computer vision, 2021, pp. 10012–10022.
  * [47] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, D. Batra, Grad-cam: Visual explanations from deep networks via gradient-based localization, in: Proceedings of the IEEE international conference on computer vision, 2017, pp. 618–626.
  * [48] L.-C. Chen, Y. Zhu, G. Papandreou, F. Schroff, H. Adam, Encoder-decoder with atrous separable convolution for semantic image segmentation, in: Proceedings of the European conference on computer vision (ECCV), 2018, pp. 801–818.


[figure]style=plain,subcapbesideposition=top [table]style=plaintop

Synoptic circulation patterns during temperature extremes in southeastern
Europe

Fragkoulidis G.${}^{\text{1*}}$, Wirth V.${}^{\text{1}}$

1 Institute for Atmospheric Physics, Johannes Gutenberg University, Mainz,
Germany

* corresponding author e-mail<EMAIL_ADDRESS>
###### Abstract

This study investigates the synoptic circulation patterns associated with
temperature extremes in southeastern (SE) Europe. Using ERA-Interim reanalysis
data we report on the typical patterns that characterize the middle and upper
tropospheric flow a few days before and during hot and cold extremes. The
analysis is done separately for each season, while a further distinction
between short-lived and persistent extremes reveals differences and
similarities in the associated circulation, as inferred by the spatiotemporal
evolution of Rossby wave packets (RWPs). Finally, the performance of ECMWF
deterministic forecasts is evaluated for persistent cold and hot extremes
during winter and summer respectively. Overall, this work suggests that a
correct representation of the RWP evolution is crucial in determining the
magnitude and persistence of temperature extremes in SE Europe.

##  1 Introduction

A better understanding of the physical processes that lead to weather extremes
is essential for various reasons. From a weather forecast perspective, it
proves beneficial in the challenges of identifying the predictability limits
and interpreting forecast biases or “busts” associated with these events
(Rodwell _et al._ 2013). From a climate perspective, investigating further the
relevant processes will help in formulating hypotheses and evaluating model
projections on the weather extremes characteristics of futures decades
(Shepherd 2014). In this regard, recent studies have examined the possibility
of far-upstream precursors to weather extremes by tracking transient Rossby
Wave Packets (RWPs) in the upper-tropospheric flow (Martius _et al._ 2008;
Fragkoulidis _et al._ 2018; Wirth _et al._ 2018).

This study focuses on the role of the middle and upper tropospheric
circulation during abnormally hot and cold spells in southeastern (SE) Europe.
This region is particularly interesting in terms of large-scale dynamics as it
lies to the southeast of the climatological North Atlantic jet exit (where
Rossby waves are typically in their mature nonlinear phase and frequently wave
breaking is observed) and slightly northward of the subtropical jet (Sprenger
_et al._ 2017). Using anomaly composites of reanalysis and forecast data we
report on typical circulation patterns and forecast errors in the daily
evolution of short-lived and persistent temperature extremes of both signs.

##  2 Data and Methodology

###  2.1 Data

Reanalysis data for meridional wind $v$ at 300 hPa, geopotential height $Z$ at
500 hPa, and temperature $T$ at 850 hPa spanning a 38-year period (1979-2016)
are retrieved on a 2° $\times$2° horizontal grid with 6-hourly temporal
resolution from the ERA-Interim reanalysis project (Dee _et al._ 2011). In
addition, we use deterministic forecasts that were produced from the same
ECMWF model (Integrated Forecasting System (IFS) model cycle 31r2) with the
ERA-Interim analyses as initial conditions (Berrisford _et al._ 2009). Namely,
10-day forecasts at 12-hourly steps of $v$ at 300 hPa and $T$ at 850 hPa,
issued daily at 1200 UTC, are retrieved on a 2° $\times$2° horizontal grid.

###  2.2 Methodology

Fig. 1: Region of $T^{\prime}$ averaging in the extremes selection.

For all the retrieved data we calculate the daily mean anomalies
($v^{\prime}$, $Z^{\prime}$, $T^{\prime}$) from a climatological annual cycle
as in Fragkoulidis _et al._ (2018). Since we are primarily interested in
temperature extremes of large spatial extent in SE Europe, the following
methodology to define hot and cold extreme events is used. We first detrend
the field of 850 hPa $T^{\prime}$ by subtracting the 1979–2016 linear trend at
each grid point (the detrended field is only used for the selection of the
temperature extreme events in this study). The detrended 850 hPa $T^{\prime}$
is averaged over the region [34-44°N , 18-28°E ] (Fig. 1) with a cosine
latitude weighting. Days with an area-averaged $T^{\prime}$ greater (lower)
than the 95th (5th ) percentile of the respective season (DJF, MAM, JJA, SON)
are regarded hot (cold) extremes. Episodes of 1-2 extreme days constitute a
_short-lived_ temperature extreme event. Episodes of 3 or more consecutive
extreme days constitute a _persistent_ temperature extreme event. In few cases
where the onset of two subsequent extreme events occurred within 4 days or
less, we did not regard the second one to be distinct and independent from the
first one and we discard it from the event list.

Due to the limited size of the event samples, composite fields in the
following analyses are computed using the mean of the interquartile range of
the sample, which restricts the effect of outliers and renders the statistics
more robust. In addition, the statistical significance of the $v^{\prime}$,
$Z^{\prime}$ and $T^{\prime}$ composites is assessed with a Monte Carlo
technique (e.g. Martius _et al._ 2008). Namely, the values at each grid point
are deemed significant at the $\alpha$=0.10 level, if they belong to either 5%
tail of a distribution created by reconstructing the composites 300 times
using random selections of dates.

##  3 Results

|  Hot extremes |  Cold extremes
---|---|---
Season |  Events |  Duration |  Events |  Duration
DJF |  95 (16) |  1.8 (4.1) |  86 (21) |  2.0 (3.7)
MAM |  82 (26) |  2.1 (3.8) |  62 (30) |  2.8 (4.4)
JJA |  69 (26) |  2.5 (4.5) |  76 (25) |  2.3 (4.0)
SON |  70 (24) |  2.5 (4.4) |  80 (26) |  2.2 (3.6)
Table 1: Number and mean duration (in days) of hot and cold extremes. Values
in the parenthesis correspond to persistent temperature extremes.

The resulting number and mean duration of distinct hot and cold extremes in
all four seasons is shown in Table 1. The winter season is characterized by
the shortest duration in extremes (1.8 days for hot and 2.0 for cold). The
percentage of persistent extremes maximizes in summer for hot extremes (26/69,
with 4.5 days mean duration) and spring for cold extremes (30/62, with 4.4
days mean duration). Finally, with the exception of spring, hot extremes in
all seasons tend to last longer than cold extremes.

Composites of 850 hPa $T^{\prime}$, 500 hPa $Z^{\prime}$ and 300 hPa
$v^{\prime}$ during JJA hot extremes (69 distinct events) are shown in Fig.
2a,c. In the $T^{\prime}$ composite over the onset days (Fig. 2a), a dipole
structure is evident with warm anomalies maximized over SE Europe and cold
anomalies over Western Europe, both having a SW-NE orientation. In Fig. 2c,
$v^{\prime}$ and $Z^{\prime}$ composites are shown for selected days before
and after the onset of the events, revealing a distinct pattern of RWP
propagation. More specifically, a trough-ridge sequence moves eastward at slow
phase velocities, while the encompassing RWP as a whole propagates faster
indicating the transfer of energy and generation of new disturbances
downstream. Eventually, the upper-tropospheric flow over SE Europe is
deflected northward forming a ridge with anticyclonic vorticity. As a
consequence, we get a favorable environment for extreme temperatures since
warm and dry air is advected from N. Africa. Previous studies (e.g. Founda and
Giannakopoulos 2009) have shown that this works in conjuction with adiabatic
compressional heating due to enhanced subsidence (which during summer months
is anyway present as a climatic feature of the region) and anomalous radiative
forcing to result in extreme temperatures.

Fig. 2: a) Composite of 850 hPa $T^{\prime}$ (color shading) at the onset of
JJA hot extremes. Stars denote statistically significant values of
$T^{\prime}$ at $\alpha$=0.10. b) Same as a) but for DJF cold extremes. c)
Evolution of 300 hPa $v^{\prime}$ (color shading) and 500 hPa $Z^{\prime}$
(blue/magenta contours indicate negative/positive anomalies at
$\scriptstyle\pm$3, $\scriptstyle\pm$6, $\scriptstyle\pm$9, … gpdam)
composites from Day -6 to Day 2 of JJA hot extremes. Stars (hatching) denote
statistically significant values of $v^{\prime}$ ($Z^{\prime}$) at
$\alpha$=0.10. d) Same as c) but for DJF cold extremes.

Apart from the opposite sign in the anomaly composites during DJF cold
extremes (86 distinct events), we get a rather different circulation pattern
evolution (Fig. 2b,d). The areas affected by cold temperatures have a higher
spatial extent and are surrounded by smaller centres of positive $T^{\prime}$
(Fig. 2b). A distinct blocking anticyclone is a prevalent feature of the North
Atlantic circulation several days before the onset of the events; a typical
flow regime for cold extremes in Europe (Sillmann and Croci-Maspoli 2009). At
Day -2, an elongated trough has formed at the eastern flank of the block,
transporting continental cold air masses from the northeast towards SE Europe.
This pattern persists and two days later we have the onset of the cold
extreme. Afterwards, the North Atlantic block starts to weaken and the RWP
propagates into Asia arching toward the Tropics.

Fig. 3: a) Hovmöller composites of 850 hPa $T^{\prime}$ (color shading) and
300 hPa $v^{\prime}$ (blue/magenta contours indicate negative/positive
anomalies at $\scriptstyle\pm$3, $\scriptstyle\pm$5, $\scriptstyle\pm$7, …
m/s) during DJF persistent (left) and short-term (right) cold extremes. Stars
(hatching) denote statistically significant values of $T^{\prime}$
($v^{\prime}$) at $\alpha$=0.10. b) Same as a) but for hot extremes. c), d)
Same for MAM. e), f) Same for JJA. g), h) Same for SON.

We now consider short-lived and persistent temperature extremes separately, in
order to explore differences in the typical circulation associated with them.
To this end, we show composites of 850 hPa $T^{\prime}$ and 300 hPa
$v^{\prime}$ compressed in Hovmöller diagrams for all seasons and both types
of extremes (Fig. 3). $T^{\prime}$ has been averaged over the 34-44°N
latitudinal band, while $v^{\prime}$ has been averaged over the wider 34-64°N
latitudinal band, in order to capture the main features of the large-scale
midlatitude flow. In composites of persistent extremes our sample size
decreases considerably (Table 1) at the expense of statistical significance,
but indications of qualitative differences from short-lived extremes can still
arise. Some key observations on the various extreme types include:

  1. (i)

Patterns of $T^{\prime}$ for the persistent extremes appear more elongated (in
the time direction, as expected), wider (spanning more longitudes) and more
stagnant (forming an almost right angle with the longitudes axis) than the
ones for short-lived extremes.

  2. (ii)

In many cases (e.g. MAM short-lived cold extremes, DJF persistent hot
extremes, SON short-lived hot extremes), the succession of positive
(southerly) and negative (northerly) $v^{\prime}$ values indicates a far-
upstream RWP signal several days before the extreme onset. In the cases where
an RWP is not traceable several days in advance, either an organized group of
Rossby wave activity was not present (e.g. block cases in DJF cold extremes)
or the RWPs evolve in varying pathways leading to partial destructive
interference and a reduced wave signal from the sample.

  3. (iii)

As is well known, the Hovmöller diagram can provide an estimation of the phase
and the group velocity associated with an RWP (Fragkoulidis _et al._ 2018). In
all cases, the group velocity is greater than the phase velocity of individual
troughs-ridges. This so-called downstream development phenomenon is
particularly pronounced for e.g. DJF short-lived hot extremes and JJA short-
lived cold extremes and less so for e.g. JJA short-lived hot extremes and MAM
short-lived extremes.

  4. (iv)

Persistent extremes are associated with quasi-stationary $v^{\prime}$
patterns, whereas higher phase velocities are observed in short-lived
extremes. This manifests the decisive role of upper-tropospheric circulation
in the duration of temperature extremes (Kyselý 2008).

  5. (v)

The zonal gradient of $v^{\prime}$ above SE Europe is greater for cold than
hot extremes (also evident in Fig. 2). Associated with that, hot extremes tend
to occur close to the centre of the overlying ridge, while cold extremes occur
closer to the area of negative $v^{\prime}$ values (northerlies) of the trough
aloft.

Finally, it’s worth verifying the performance of deterministic forecasts of
these temperature extremes. The availability of a 38-year dataset with
forecasts produced with the same model (section 2) allows us to have a
consistent comparison with the corresponding reanalysis of selected events.
Our verification involves forecasts issued 3 and 5 days prior to DJF cold and
JJA hot persistent extremes (21 and 26 events respectively). Fig. 4 shows
Hovmöller composites of $T^{\prime}$ and $v^{\prime}$ (as in Fig. 3) for these
forecasts. These are complemented by the forecast-minus-reanalysis Hovmöller
composites, that serve to reveal systematic errors in the prediction of
persistent extremes.

Fig. 4: a) Left: Hovmöller composites of 850 hPa $T^{\prime}$ (color shading)
and 300 hPa $v^{\prime}$ forecasts (blue/magenta contours indicate
negative/positive anomalies at $\scriptstyle\pm$3, $\scriptstyle\pm$5,
$\scriptstyle\pm$7, … m/s), issued 3 days prior to JJA persistent hot
extremes. Stars (hatching) denote statistically significant values of
$T^{\prime}$ ($v^{\prime}$) at $\alpha$=0.10. Right: Deviation of the
forecasted 850 hPa $T^{\prime}$ (color shading) and 300 hPa $v^{\prime}$
(solid/dashed contours indicate positive/negative anomalies at
$\scriptstyle\pm$2, $\scriptstyle\pm$3, $\scriptstyle\pm$4, … m/s) from the
respective reanalysis fields (Fig. 3f left). Statistically significant
deviations are assumed where both the forecast and reanalysis fields are
significant at $\alpha$=0.10. b) Same as a) but for DJF persistent cold
extremes. c),d) Same as a),b) but for forecasts issued 5 days prior to the
extremes.

Forecasts issued 3 and 5 days prior to JJA persistent hot extremes show an
underestimation of $T^{\prime}$ above our region of interest (18-28°E ) and an
overestimation to the east and west of that. This pattern can arise due to
positional (some extremes are predicted to occur to the east and others to the
west of the eventually affected region) and/or magnitude errors. Forecasts
issued 3 days prior to DJF persistent cold extremes reveal an overestimation
in both magnitude and duration. In 5-day forecasts, an eastward shift of the
cold anomalies is predicted. In contrast to hot extremes, now an error is also
evident in the $v^{\prime}$ field with an eastward shift of the trough that
can in principle explain the $T^{\prime}$ error pattern.

##  4 Conclusions

In this study, we investigated the synoptic circulation associated with the
most pronounced temperature extremes that affected SE Europe in the last 38
years. Although the amount of such high-impact events is limited and robust
results are not feasible, qualitative differences and similarities between the
synoptic circulation imprints of hot and cold extremes across all seasons were
inferred. In addition, distinguishing between short-lived and persistent
extremes revealed differences in the spatiotemporal evolution of the upper-
tropospheric flow. Far-upstream precursors were evident for several types of
temperature extremes, indicating a recurrent in-phase RWP propagation, while
in other cases the less clear signal in the days preceding the extremes
suggests a larger variability with no preferential RWP pathway.

Future studies on medium-range forecasts of temperature extremes should
investigate further the role of upper-tropospheric dynamical features and how
model limitations lead to systematic errors in the Rossby wave structure (Gray
_et al._ 2014). Moreover, an improved understanding on the interplay between
synoptic circulation patterns, radiative transfer and boundary layer processes
will help in interpreting the observed increase in frequency and severity of
persistent temperature extremes (Kyselý 2008).

## Acknowledgements

We thank the ECMWF for providing access to the data used in this study. The
research leading to these results has been done within the Transregional
Collaborative Research Center SFB/TRR 165 “Waves to Weather” funded by the
German Research Foundation (DFG).

## References

  * Berrisford _et al._ (2009) Berrisford P, _et al._ 2009\. The ERA-Interim Archive - Version 2.0. _ERA report series_ 1(1): 1–16.
  * Dee _et al._ (2011) Dee DP, _et al._ 2011\. The ERA-Interim reanalysis: Configuration and performance of the data assimilation system. _Q. J. R. Meteorol. Soc._ 137(656): 553–597, doi:10.1002/qj.828.
  * Founda and Giannakopoulos (2009) Founda D, Giannakopoulos C. 2009. The exceptionally hot summer of 2007 in Athens, Greece - A typical summer in the future climate? _Glob. Planet. Change_ 67(3-4): 227–236, doi:10.1016/j.gloplacha.2009.03.013.
  * Fragkoulidis _et al._ (2018) Fragkoulidis G, _et al._ 2018\. Linking Northern Hemisphere temperature extremes to Rossby wave packets. _Q. J. R. Meteorol. Soc._ 144(711): 553–566, doi:10.1002/qj.3228.
  * Gray _et al._ (2014) Gray SL, _et al._ 2014\. Systematic model forecast error in Rossby wave structure. _Geophys. Res. Lett._ 41(8): 2979–2987, doi:10.1002/2014GL059282.
  * Kyselý (2008) Kyselý J. 2008. Influence of the persistence of circulation patterns on warm and cold temperature anomalies in Europe: Analysis over the 20th century. _Glob. Planet. Change_ 62(1-2): 147–163, doi:10.1016/j.gloplacha.2008.01.003.
  * Martius _et al._ (2008) Martius O, _et al._ 2008\. Far-upstream precursors of heavy precipitation events on the Alpine south-side. _Q. J. R. Meteorol. Soc._ 134(631 PART B): 417–428, doi:10.1002/qj.229.
  * Rodwell _et al._ (2013) Rodwell MJ, _et al._ 2013\. Characteristics of occasional poor medium-range weather forecasts for Europe. _Bull. Am. Meteorol. Soc._ 94(9): 1393–1405, doi:10.1175/BAMS-D-12-00099.1.
  * Shepherd (2014) Shepherd TG. 2014. Atmospheric circulation as a source of uncertainty in climate change projections. _Nat. Geosci._ 7(10): 703–708, doi:10.1038/ngeo2253.
  * Sillmann and Croci-Maspoli (2009) Sillmann J, Croci-Maspoli M. 2009. Present and future atmospheric blocking and its impact on European mean and extreme climate. _Geophys. Res. Lett._ 36(10), doi:10.1029/2009GL038259.
  * Sprenger _et al._ (2017) Sprenger M, _et al._ 2017\. Global climatologies of Eulerian and Lagrangian flow features based on ERA-Interim. _Bull. Am. Meteorol. Soc._ 98(8): 1739–1748, doi:10.1175/BAMS-D-15-00299.1.
  * Wirth _et al._ (2018) Wirth V, _et al._ 2018\. Rossby Wave Packets on the Midlatitude Waveguide — A Review. _Mon. Weather Rev._ doi:10.1175/MWR-D-16-0483.1.