# SyncNet: Using Causal Convolutions and Correlating Objective for Time Delay
Estimation in Audio Signals

###### Abstract

This paper addresses the task of performing robust and reliable time-delay
estimation in audio-signals in noisy and reverberating environments. In
contrast to the popular signal processing based methods, this paper proposes
machine learning based method, i.e., a semi-causal convolutional neural
network consisting of a set of causal and anti-causal layers with a novel
correlation-based objective function. The causality in the network ensures
non-leakage of representations from future time-intervals and the proposed
loss function makes the network generate sequences with high correlation at
the actual time delay. The proposed approach is also intrinsically
interpretable as it does not lose time information. Even a shallow convolution
network is able to capture local patterns in sequences, while also correlating
them globally. SyncNet outperforms other classical approaches in estimating
mutual time delays for different types of audio signals including pulse,
speech and musical beats.

Index Terms: Time Delay Estimation, Semi-Causal Convolutional Neural Networks,
Sound Source Localization

##  1 Introduction

Synchronization of signals from different sources is a common problem and has
numerous applications, such as communication [1], radar systems [2], source
localization [3, 4], latency estimation [5], inter-aural time delay estimation
and music synchronization [6]. A precise measurement of time delay in two
signals can help synchronize the signals. For instance, two audio devices
playing the same audio need to be synchronous for a good user experience. 3D
soundscape generation requires introducing precise mutual delays in multiple
speakers. We focus on estimating time delay in audio signals. We use it for
round trip latency estimation in audio playing and recording devices. Let
$x_{1}(t)$ be the reference signal to be played and $x_{2}(t)$ be the recorded
signal. $x_{2}(t)$ is a noisy, delayed and damped replica of $x_{1}(t)$

$\displaystyle x_{2}[t]=\alpha x_{1}[t-\tau]+w[t]$ (1)

where $\alpha$ is an unknown attenuation factor, the transmitted signal is
distorted by additive noise $w[t]$ and $\tau$ is the time delay between the
two signals.

The efforts for enabling machines to estimate the time delay of signals date
back to the second half of the twentieth century [7, 8]. The classical
techniques for time delay estimation, such as those based on Cross-Correlation
[9, 10] or Generalized Cross-Correlation Phase Transform (GCC-PHAT) [7] are
not robust to the environment and are sometimes limited to producing estimates
only from a discrete set. Furthermore, excess noise levels in certain
intervals lead to spurious peaks in GCC which lead to poor estimates [11]. The
Frequence-Sliding GCC (FS-GCC) algorithm [12] tries to resolve the issue by
using sliding windows to analyze the cross-power spectrum phase, and thus,
capturing the estimate-features in frequency bands. Elizabeth R. et al. [13]
performs onset detection by locally approximating a relevant autonomous linear
state-space model (LSSM) to estimate temporal delays based on the precedence
effect. An interesting deep learning based approach [14] has also been
proposed as an alternative to the FS-GCC; it consists of a U-Net fully
convolutional denoising autoencoder that is fed by the extracted
representations from the FS-GCC itself. The results show that Deep Neural
Networks are capable of understanding the noisy and reverberating environments
much better than classical signal processing techniques. This is supported by
many other deep learning based methods [15, 16, 17, 18]. Most of the existing
learning-based studies for tasks on audio data utilize extracted feature-
representations like Spectrograms, MFCCs, etc [19, 20, 21]. Using spectral
domain features may harm the least count of the estimation method, leaving
them unsuitable for tasks requiring high precision.

In this work, we focus on high-precision time delay estimation in audio,
although the method can be applied to other signals too. Time delay can be
measured either from general audio signals, e.g., speech, music, etc. or from
specifically synthesized sounds, such as pulse, chirp or sinusoids. Pre-
calibration of mutual delay in devices is possible with synthesized sounds but
calibration with general sounds widens the range of applications. We propose
SyncNet, a robust and explainable estimation methodology for the delay
parameter $\tau$, given the reference and its noisy replica signals, using
semi-causal convolutional neural network with a task-specific correlating
objective function. To the best of our knowledge, this is the first deep
learning based method for mutual time delay estimation in signals.

##  2 Proposed Methodology

###  2.1 Causal Network

A causal convolution in a deep neural network ensures the resulting feature
representation at a time-step $t$ to not depend upon any future time-step
$t^{{}^{\prime}}$ $\forall$ $t^{{}^{\prime}}>t$. It is known that networks
with causal convolutions converge faster for long sequences relative to the
Recurrent Neural Networks as there is no recurrence [22]. Oord AV et al. [22]
show the usage of dilated causal convolutions to increase the receptive field,
allowing the network to be efficient, with lesser layers.

In this paper, we use convolutional neural towers over specified time
intervals which perform inter-causal operations. These towers are a set of
convolutional layers at different levels as shown in Fig. 1a, which are
expected to extract relevant representations from the input sequence and
transform it into a more useful space with respect to the objective of the
network. For any time interval $(t_{1},t_{2})$, for each layer $l_{i}\forall$
$i\in\\{1,2,3,..h\\}$, the $j^{th}$ causal tower $p_{j}$ takes input from the
tower $p_{j-1}$ in the previous interval, $(t_{0},t_{1})$, such that
$t_{0}<t_{1}<t_{2}$ and $t_{1}-t_{0}=t_{2}-t_{1}$. Here, $p$ is the total
number of casual towers and $h$ is the number of layers in each of these
units. These towers collectively are expected to learn relevant
representations from the input sequence which evolve over time. For these
towers to instead process the input smoothly, with lesser loss of features in
between these time-intervals where corresponding towers are placed, SyncNet
uses them as overlapping windows. That is, choose the intervals to place
causal towers such that the time-interval of the $j^{th}$ tower,
$(t_{j},t^{{}^{\prime}}_{j})$ overlaps with that of the subsequent tower i.e.,
$(t_{j+1},t^{{}^{\prime}}_{j+1})$ or $t^{{}^{\prime}}_{j}>t_{j+1}$. Let these
samples overlapping between each of the intervals be defined as
$\delta=t^{{}^{\prime}}_{j}-t_{j+1}$. One may trivially obtain the following
relation between all such hyper-parameters of the network.

$l\approx\frac{L-s}{(1-\gamma)(p-1)}$ (2)

where $l=t^{{}^{\prime}}_{j}-t_{j}$ is the number of samples in each of the
time interval, $L$ is the total number of samples in the input sequence,
$\gamma=\delta/l$ is the proportion of overlapping samples in an interval, and
$s$ is the number of samples left at the end of the sequence, after a maximum
possible integer has been selected for $p$. For the $i^{th}$ layer in the
$j^{th}$ tower in this causal structure, the resulting sequence is defined as-

$\displaystyle y_{j,i}[n]$
$\displaystyle=x_{j,i}^{{}^{\prime}}[n]*h_{j,i}[n]=\sum_{c=0}^{l-1}{x_{j,i}^{{}^{\prime}}[c].h_{j,i}[c+n]}$
$\displaystyle=\sum_{c=0}^{l-1}{(y_{j-1,i-1}[c]+y_{j,i-1}[c]).h_{j,i}[c+n]}$
(3)

Here, $h_{j,i}[n]$ is the filter and
$x_{j,i}^{{}^{\prime}}[n]=y_{j-1,i-1}[n]+y_{j,i-1}[n]$ is the input fed
causally to the layer at $(j,i)$.

After the input data is passed through the first $h$ levels in this causal
structure, our study introduces anti-causal forward passing to deduce emerging
relations in the neighborhoods of any interval. With this, the network can
leverage to use either set of layers more dominantly, allowing for a much
intensive deduction of representations from the input sequence. The
architecture of this semi-causal convolutional neural network for 1-D
sequences is depicted by Fig. 1b. Any block in this part of the architecture
is fed by input from the towers on previous, same and next time-intervals.

Figure 1: SyncNet: A semi-causal convolutional neural network for audio data.
(a): Causal Convolution Tower, (b): SyncNet

###  2.2 Objective Function

The cross-correlation function of two signals $x_{1}(t)$ and $x_{2}(t)$, where
$E[.]$ denotes the expectation is defined as-

$R_{x_{1}x_{2}}(\tau)\overset{def}{=}E[x_{1}(t)x_{2}(t-\tau)]$ (4)

The argument $\tau=\tau_{0}$ which maximizes $R_{x_{1}x_{2}}(\tau)$
corresponds to the estimate of temporal delay in two signals. Carter et al.
[7] and Song et al. [23] showed that using cross-correlation function either
directly or upon some filtered variant of the signal results into acceptable
performance as a time delay estimator.

Let $\hat{y_{1}}(t)$ and $\hat{y_{2}}(t)$ be the predicted sequences for the
input noisy-delayed and reference signals, $x_{1}(t)$ and $x_{2}(t)$
respectively by the network. The cross-correlation function of both sequences
can be denoted by $\hat{R}_{y_{1}y_{2}}(\tau)$. The network should learn to
transform the input sequences such that $\hat{R}_{y_{1}y_{2}}(\tau)$ peaks at
the actual delay in time. The use of cross-correlation function to form a
training objective has not been explored before, as far as we know. The
closest we know is the concept of maximizing cross-correlation for static
features, not time sequences, in the case of deep canonical correlation
analysis of multi-modal data [24].

We formulate the loss function as a regression loss. We aim to match the cross
correlation sequence to a Gaussian sequence, that peaks at the true time
delay. In case of periodic signals, their cross-correlation function will also
be periodic, leading us to use a sequence of Gaussians.

$R(\tau)=\sum_{n=0}^{g-1}{\frac{1}{\sigma_{n}\sqrt{2\pi}}\exp{-\frac{{(\tau-\mu_{n})}^{2}}{2\sigma_{n}^{2}}}}$
(5)

where $\mu_{n}=T_{0}+nT$ is the $n^{th}$ onset, and $\mu_{0}=T_{0}$ is the
actual delay in time and $T$ is the time period of the reference signal (if
periodic). For this context and simplicity, we can set $\sigma_{n}=\sigma$
$\forall$ $n\in\\{0,1,2,...(g-1)\\}$. Let $\mathcal{L}$ be a distance metric
between $R(\tau$) and $\hat{R}_{y_{1}y_{2}}(\tau)$. Since the number of peaks
are likely to be far lesser in number than other points in the target
sequence, we up-weighted $\mathcal{L}$ at $\tau=\tau_{0}+nT$ by $u$ and
correspondingly down-weighted for other indices by $d$. The values of these
constants are algebraically computed as-

$\displaystyle d\leftarrow 1-\frac{g+1}{\lambda}\;\;\text{and}\;\;u\leftarrow
d+\frac{N^{{}^{\prime}}}{\lambda}$ (6)

Here, $\lambda=10^{a}$ for the smallest number $a$ such that
$\lfloor\frac{N^{{}^{\prime}}}{10^{a}}\rfloor=0$ and $N^{{}^{\prime}}$ is the
number of samples in $\hat{R}_{y_{1}y_{2}}(\tau)$. This allows the network to
attend to the peaks more likely. We define $\mathcal{L}$ as-

$\displaystyle\mathcal{L}(.)\overset{def}{=}l_{1}\mathcal{L}_{1}(.)+l_{2}\mathcal{L}_{2}(.)+l_{3}\mathcal{L}_{3}(.)$
(7)

Here,
$\mathcal{L}_{1}=\sum_{i=1}^{N^{{}^{\prime}}}{{(R_{i}-\hat{R}_{i})}^{2}}$ is a
simple MSE function,
$\mathcal{L}_{2}=\sqrt{\sum_{i=1}^{N^{{}^{\prime}}}{{(\log{R_{i}^{\wp}}-\log{\hat{R}_{i}^{\wp}})}^{2}}}$
is the root-mean-log error function,
$\mathcal{L}_{3}=\sum_{i=1}^{N^{{}^{\prime}}}{R_{i}^{\wp}(\log{R_{i}^{\wp}}-\hat{R}_{i}^{\wp})}$
is the KL-Divergence loss and $l_{i}\forall i\in\\{1,2,3\\}$ are the
corresponding weights associated to each term. $R^{p}(\tau)$ and
$\hat{R}^{p}(\tau)$ are the pooled correlation sequences, i.e. the most
activated for every $p$ set of samples in $R(\tau)$ is chosen as $i^{th}$
sample in $R^{p}(\tau)$. Note that, $\mathcal{L}_{1}$ is applied onto the
correlation sequences, thus helps with matching its shape with target,
whereas, $\mathcal{L}_{2}$ and $\mathcal{L}_{3}$ are operating over the pooled
sequences. One must be cautious while choosing a value for $p$ as a least-
bound for the error made by network is expected by nearly
$\wp/N^{{}^{\prime}}$. Thus, the estimated parameters, where $\ast$ represents
the cross-correlation operation are-

$\displaystyle\hat{\theta}=\underset{\theta\in\Theta}{min\
}\mathcal{L}(R(\tau),\hat{y}_{1}(t)\ast\hat{y}_{2}(t))$ (8)

##  3 Evaluation

###  3.1 Dataset Used

We evaluate the proposed method on a variety of datasets - including both real
recordings as well as synthetically generated audio.

We prepare a new dataset, called MTic. A reference audio signal of periodic
tics with a time period of $1$s is played on an phone speaker and is recorded
by the microphone of the same phone. Tics are chosen for experiments because
they are localized in time and are musically relevant (as in metronome). The
recordings are done with a number of phones and in a variety of acoustic
background conditions. The time delay in the recorded signal is $<0.9$s. There
are total 170 audio recordings sampled at $16kHz$, each of a duration of
almost $10$ seconds. They all correspond to the same reference audio of $10$s
duration.

Furthermore, to increase the sample size, we synthetically generate 450 audio
files from the same reference signal with delays chosen uniformly between
$(0s,0.9s]$ and noise injections with signal-to-noise ratio varying in
$[-15dB,20dB]$.

To validate the robustness of this study over real world signals such as
speech and music as well. We experiment with speech files from the LibriSpeech
dataset [25]; speech is non periodic. We also experiment with the
accompaniment music from MIREX 2012 dataset [26], which we call MBeats
dataset. We randomly sample around 50 audio files from both and generated 25
synthetic audio files for each of these files, with injected delays and noise
with SNR randomly sampled from $[-15dB,20dB]$.

The MTic dataset as well as the codes to reproduce the results in this paper
are available at https://github.com/madhavlab/2022_syncnet.

###  3.2 Experiments

We have conducted several experiments to validate the robustness of SyncNet.
Both reference and signal-of-interest need to be transformed into embeddings,
trained for the task-at-hand. For this, both the signals are parallely passed
separately through the same network.

Table 1: Network architecture studied. Here $k$ represents the $k^{th}$ anti-causal layer Network Part | Layer
---|---
|  Conv1D + Relu ($h_{1}$ and $h_{3}$)
Causal $\times 1$ |  Conv1D + BatchNorm1D + Relu ($h_{2}$ and $h_{4}$)
|  Conv1D + Relu: ($h_{5}$)
|  Conv1D + Relu ($h^{{}^{\prime}}_{k1}$)
Anti-Causal $\times 5$ |  Conv1D ($h^{{}^{\prime}}_{k2}$)
|  BatchNorm1D + Relu ($h^{{}^{\prime}}_{k3}$)
Figure 2: Normalized-Loss for SyncNet during training epochs Table 2: Hyperparameters for proposed methodology. CT denotes ’Causal Tower/s’ and AC denotes ’Anti-Causal Structure’ Hyperparameter |  Description |  Value
---|---|---
$l$ |  Samples per tower |  3432
$p$ |  Number of CT |  50
$\gamma$ |  $\delta/l$ |  0.125
$h$ |  Depth of CT |  5
$h^{{}^{\prime}}$ |  Depth of AC |  5
$f^{h}_{i}$ |  Filters in CT |  {60,72,54,54,54}
$f^{h^{{}^{\prime}}}_{i}$ |  Filters in AC |  {54,50,32,16,16}
$\eta$ |  Initial Learning Rate |  9e-4

Pooling layers are not used. Batch-Normalization [27] layers are used in
between and layers are activated using the ReLU function [28] as in Table 1.
It is natural to choose intervals such that the receptive field grows and more
samples of the input signal are captured by the causal towers per interval. We
have experimented with various values for $p$ with the objective to have
lesser samples in $\hat{R}(\tau)$, lesser learnable-parameters and higher
receptive field, which led us to set it at $50$. The pool size ($\wp$) has
been set as 63. The values for other hyperparameters discussed previously can
be realized from table 2. We trained the network on MTic until the losses
converge, for a maximum 200 epochs. Mean squared error between the estimated
value of delay and the true delay value is chosen as the evaluation metric.

For baseline, we train a deep 1-D audio fed-ConvNet (AfC), a Mel-Spectrogram
fed ConvNet (MSfC) with similar layers and Cross-Entropy Loss function. A log
of training loss decay for all three networks for first 140 epochs has been
plotted in figure 2.

Table 3: Mean squared error in time delay estimation, and the corresponding standard deviation (within brackets), obtained for all the methods on the three datasets Method |  MTic |  LibriSpeech |  MBeats
---|---|---|---
Cross-Correlation |  0.093(0.091) |  0.102(0.098) |  0.098(0.095)
GCC-PHAT |  0.064(0.077) |  0.081(0.069) |  0.073(0.082)
AfC |  0.071(0.085) |  0.091(0.087) |  0.066(0.083)
MSfC |  0.054(0.082) |  0.063(0.075) |  0.059(0.086)
SyncNet |  0.018(0.075) |  0.028(0.072) |  0.022(0.083)
Figure 3: Violin plots for absolute error for GCC-PHAT, MfCN and SyncNet on
all 3 datasets. Vertically lower distribution indicates better performance.

We compare the performance of the proposed SyncNet based method with four
baselines – two neural-networks discussed and the two standard signal
processing based methods, namely, the Cross-Correlation method and the classic
GCC-PHAT method for time delay estimation.

Table 3 shows the mean squared error (MSE) and its standard deviation (SD) for
all the methods on the three datasets – MTic, Librispeech and MBeats. We can
see that SyncNet consistently outperforms all the baseline methods over all
the datasets in terms of the average performance. However, to dive deeper into
the performance of the three top performing methods, we draw the violin plots
to see the distribution of errors across all test samples. Figure 3 shows
these violin plots. It can be seen that the estimation errors for the proposed
SyncNet method are distributed more towards the lower end as compared to other
methods, supporting the improvements brought in it.

It is important to note that the proposed SyncNet method is not limited by the
length of the signals, and performs consistently with signals of variable
lengths, maintaining its higher precision. Also, it accepts signals of
different sampling rates.

The estimators based on Signal Processing methods [7, 8] are considered more
explainable, and thus hold higher potential for fine tuning and debugging [9,
10, 13]. Deep Learning based techniques, on the other hand, mostly lack
explainability. Nevertheless, SyncNet transforms a pair of input sequences
into a pair of sequences that are correlated to the extent that a simple
signal processing metric, such as cross-correlation, can be used to estimate
their mutual time delay. This makes the network explainable for its
predictions, as one remains informed of the causality of predictions made by
the network. As an example, Figure 4 shows a reference signal from MTic
database as input to SyncNet, the output of SyncNet and its cross-correlation
with that of the corresponding signal-of-interest. A clear peak at the actual
delay can be observed in the correlation plot.

Figure 4: Input delayed-noisy signal, its transformed variant and pooled
correlation sequence estimated. The correlation peaks correctly at the actual
time delay, i.e., 73 samples.

##  4 Conclusions

We introduce a novel correlation-objective function based deep neural network
with causally convolving modules followed by anti-causal convolution sets
which enhance the performance of time delay estimation. With optimal choice of
hyperparameters, the network also grows its receptive field. SyncNet
outperforms all of the classic approaches in precise estimation of time
delays. We are working towards applying this method to problems such as round
trip latency estimation for audio playback and recording on mobile devices.
This synchronized audio can be used by an online service provider to learn the
distortion as well as time delay characteristics of these devices in the wild.

##  5 Acknowledgements

This work was supported by IMPRINT-2C grant from SERB, Government of India,
Project no. IMP/2019/400.

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# Serendipity observations of far infrared cirrus emission in the Spitzer
Infrared Nearby Galaxies Survey:
Analysis of far-infrared correlations**affiliation: This work is based on
observations made with the _Spitzer Space Telescope_ , which is operated by
the Jet Propulsion Laboratory, California Institute of Technology, under a
contract with NASA.

Caroline Bot11affiliation: California Institute of Technology, Pasadena CA
91125, USA 55affiliation: Observatoire Astronomique de Strasbourg, 67000
Strasbourg, FRANCE and George Helou11affiliation: California Institute of
Technology, Pasadena CA 91125, USA and François Boulanger22affiliation:
Institut d’Astrophyisque Spatiale, 91405 Orsay, FRANCE and Guilaine
Lagache22affiliation: Institut d’Astrophyisque Spatiale, 91405 Orsay, FRANCE
and Marc-Antoine Miville-Deschenes22affiliation: Institut d’Astrophyisque
Spatiale, 91405 Orsay, FRANCE and Bruce Draine33affiliation: Princeton
University Observatory, Princeton, NJ08544, USA and Peter Martin44affiliation:
Canadian Institute for Theoretical Astrophysics, Toronto, Ontario, M5S 3H8,
Canada<EMAIL_ADDRESS>
###### Abstract

We present an analysis of far-infrared dust emission from diffuse cirrus
clouds. This study is based on serendipitous observations at 160$\mu$m at high
galactic latitude with the Multiband Imaging Photometer (MIPS) onboard the
Spitzer Space Telescope by the Spitzer Infrared Nearby Galaxies Survey
(SINGS). These observations are complemented with IRIS data at 100 and
60$\mu$m and constitute one of the most sensitive and unbiased samples of far
infrared observations at small scale of diffuse interstellar clouds. Outside
regions dominated by the cosmic infrared background fluctuations, we observe a
substantial scatter in the 160/100 colors from cirrus emission. We compared
the 160/100 color variations to 60/100 colors in the same fields and find a
trend of decreasing 60/100 with increasing 160/100. This trend can not be
accounted for by current dust models by changing solely the interstellar
radiation field. It requires a significant change of dust properties such as
grain size distribution or emissivity or a mixing of clouds in different
physical conditions along the line of sight. These variations are important as
a potential confusing foreground for extragalactic studies.

ISM:clouds — infrared:ISM

##  1 Introduction

The InfraRed Astronomical Satellite (IRAS) showed for the first time that
extended infrared emission was present at high galactic latitude, far from
star forming regions (Low et al. 1984). In these diffuse regions, clouds are
optically thin to stellar radiation and the radiation field is relatively
uniform which results in very limited variations of dust equilibrium
temperature (Boulanger et al. 1996; Arendt et al. 1998; Lagache et al. 1998;
Schlegel et al. 1998). These high latitude cirrus also show a tight
correlation between their infrared emission (100$\mu$m to 1mm observed by
DIRBE and FIRAS) and the HI column density (Boulanger et al. 1996) and the
dust emission is well characterized with a constant dust emissivity per
hydrogen atom ($\tau/N_{H}=10^{-25}(\lambda/250)^{-2}cm^{2}$) close to the
value expected from models of interstellar dust grains (Draine & Lee 1984). At
shorter wavelength, the smaller dust grains emission is characterized by a
ratio of $I_{60\mu m}/I_{100\mu m}\sim 0.2$ (Laureijs et al. 1991; Abergel et
al. 1996; Boulanger et al. 2000). All in all, dust emission from local cirrus
is then seen as rather homogeneous and simply characterized on large scales.
However, little is known about the dust properties (e.g. optical properties
for absorption and emission, distribution,…) in these high latitude clouds, at
resolutions smaller than the DIRBE beam (0.7o).

Smaller scale analysis of infrared colors have been done on individual regions
and show clear variations of dust properties. Laureijs et al. (1996) and
Abergel et al. (1994) observed a decrease of $I_{60\mu m}/I_{100\mu m}$ toward
dense clouds. Bernard et al. (1999) studied the far infrared emission at the
arcminute scale in the Polaris flare with IRAS, ISOPHOT and PRONAOS (200 to
600$\mu$m) in a region where extended emission from cirrus is detected as well
as a denser structure. The spectrum of the extended cirrus indicates a low
dust temperature associated with a low 60/100 $\mu$m ratio. This was also
observed in the Polaris flare toward moderately dense regions ($A_{V}\sim 1$)
and in a denser filament in the Taurus complex (Cambrésy et al. 2001; Stepnik
et al. 2003). It might be explained by the formation of large dust aggregates
through the adhesion of small dust particles onto the surface of larger
grains, leading to a change of dust emissivity properties. In the dense
regions the very small grains seem to have disappeared almost completely.
However, all these observations were restricted to individual regions, most of
which are much denser than the diffuse local interstellar medium seen at high
galactic latitudes.

By comparing near-infrared extinction and extinction deduced from far-infrared
dust emission in the whole anticenter hemisphere, Cambrésy et al. (2005)
observed a discrepancy between the two quantities in regions above 1 mag. This
effect is also interpreted by a change of dust emissivity due to the presence
of fluffy grains and the grain-grain coagulation scenario was therefore
extended to larger regions.

Kiss et al. (2006) analyzed the far-infrared emission properties in a large
sample of interstellar clouds observed with ISOPHOT with respect to extinction
in regions of the order of 100 arcmin2. They find variations of the far
infrared dust emissivities in the coldest (12K$<T_{d}<$14K) and densest
regions that are consistent with a dust grain growth scenario. But they also
observe changes of the dust emissivities in the warmer regions
(14K$<T_{d}<$17.5K) and interpret them as an effect of mixing along the line
of sight of components with different temperatures or a change of the dust
grain size distribution. However, a fraction of their sample was chosen on the
basis of high brightnesses in the IRAS bands and could therefore be biased
toward regions with enhanced small grain emission.

Extinction measures toward high galactic latitude sightlines show a
substantial fraction of low $R_{V}=A_{V}/E(B-V)$ values, also indicative of
enhanced relative abundances of small grains. However, with the lack of longer
wavelength measurements at small scales, it is difficult to relate these
variations to possible changes in the dust grain properties.

The photometric data from the Spitzer Space Telescope enable us to have access
to sensitive observations up to 160$\mu$m. Among the large programs, the
Spitzer Infrared Nearby Galaxies Survey (SINGS) observed a sample of 75 nearby
galaxies in photometry with the IRAC and MIPS instruments. The fields observed
were chosen to be at high galactic latitude in order to limit the foreground
cirrus contamination in the study of the targeted galaxy. Because the region
observed was larger than the targeted galaxy, these serendipitous observations
are then ideal to study far infrared dust emission in a large sample of high
galactic latitude regions. We combine these new observations with IRIS data
(Miville-Deschênes & Lagache 2005) at 60 and 100$\mu$m, a reprocessing of IRAS
data including a better calibration of the infrared brightnesses and better
zodiacal light subtraction. The goal of this paper is to _study the infrared
colors of diffuse local dust emission on the scale of a few arcminutes._

##  2 The data

The Spitzer Infrared Nearby Galaxies Survey (SINGS Kennicutt et al. 2003)
observed in imagery with IRAC (Fazio et al. 2004) and MIPS (Rieke et al. 2004)
onboard Spitzer a sample of 75 nearby galaxies. While the IRAC images only
observed the galaxy itself, a significant part of the MIPS observations
(strips) encompass the surrounding sky. Since the SINGS observations were
chosen to be at high galactic latitude to limit the galactic foreground
contamination, the MIPS observations at 160$\mu$m provide a good opportunity
to study the low surface brightness diffuse infrared emission from high
galactic cirrus in a large number of fields at a resolution of $\sim 37"$.
These data are complemented with IRIS data (Miville-Deschênes & Lagache 2005)
at 100 and 60$\mu$m. The position of the fields on the sky are shown in Fig. 1
while their characteristics are summarized in Tab. 1.

Although SINGS observations were also done at 70$\mu$m with MIPS, the regions
observed are offset with respect to the galaxy targeted and only a small
fraction of the 70 and 160$\mu$m observations overlap outside the galaxy
itself, making them inappropriate for our galactic cirrus emission study.
$24\mu$m observations were also available, but they are dominated by point
sources emission as well as stronger zodiacal light. Once the point sources
removed and the regions at low ecliptic latitude are discarded, the 24$\mu$m
brightness have a low dynamic range in each field and no meaningful
correlation can be done with longer wavelength observations. This study was
therefore restricted to the comparison of 60, 100 and 160$\mu$m brightnesses.
The IRIS 25$\mu$m observations were however used together with longer
wavelength in order to remove point sources (like galaxies) more efficiently
in the observations.

###  2.1 Data treatment

The observations at 160$\mu$m were reduced using the GeRT
software111http://ssc.spitzer.caltech.edu/mips/gert/index.html on the raw MIPS
observations. Standard parameters were used for the reduction, but data where
flashes of the internal source led to a significant number of saturated pixels
which were removed. The removed data are most often positioned on the bright
center of the galaxy. Other saturated pixels removed from processing were due
to cosmic ray hits. These saturated flashes when not removed can bias the
sensitivity of the diffuse extended emission. This step in the reduction may
be not appropriate for the photometry of the galaxy, but significantly reduces
latents (stripes) in the outer regions we are interested in. Each region
targeted was observed twice. Discrepant fluxes at the same position between
the two observations are removed and the data are combined into a mosaic for
each region.

The MIPS 160$\mu$m maps and IRIS 60$\mu$m are convolved to the IRIS 100$\mu$m
resolution assuming gaussian beams with FWHM of 4.0’, 4.3’ and 37” for IRIS
60, 100$\mu$m and MIPS 160$\mu$m observations respectively.

For each MIPS strip, the galaxy and other point sources are detected in the
25, 60 and 100$\mu$m maps using the method described in Miville-Deschênes et
al. (2002). These point sources at the IRIS resolution (but with the MIPS
sampling) are then smoothed by a gaussian kernel with a full width half
maximum of $3\times 3$ pixels (at the MIPS original pixel size) to encompass
possible extended emission from these galaxies. The smoothed point sources are
then masked in all the maps. All maps are then projected on the IRIS grid to
avoid oversampling. The observation targeting the galaxy Holmberg IX was
removed from the sample since the emission in the whole strip is dominated by
the galaxy and its interaction features with nearby galaxies. We chose to
remove the observation containing the galaxy NGC3034 (M82) which was hampered
by saturation effects in the whole central region of the galaxy, affecting the
observation globally. The observations containing the galaxies NGC1266,
NGC2915 and M81 Dwarf B were also removed because the width of the region
observed were too narrow to be convolved meaningfully to the IRIS resolution.
We ended up with 70 fields of view222Although there are 75 galaxies in the
SINGS sample, some galaxies are in the same field of view: NGC3031 is in the
same observation as M81 dwarf B and NGC5195 was observed simultaneously with
NGC5194 at a resolution of 4.3’ observed at 60, 100 and 160$\mu$m. Due to
uncertainties in the zodiacal light subtraction at 60$\mu$m that can dominate
the flux at the low surface brightnesses we sample, we limited the sample at
60$\mu$m to the 9 observations at high ecliptic latitude ($|\beta|>15^{o}$).

A constant brightness of 0.78 MJy/sr is removed from the IRIS 100$\mu$m maps
to account for the cosmic infrared background (Lagache et al. 2000), i.e. the
emission from the distant unresolved galaxies (called hereafter CIB). The
exact level of CIB emission has not yet been established at 60$\mu$m and the
MIPS observations can have offsets in the calibration of the brightness that
are not physical. To overcome the uncertainties (physical or instrumental) on
the zero levels in the different maps, we hereafter perform the analysis of
the data through the use of correlations (c.f. §3.1).

The errors on the surface brightness are taken to be 0.03 and 0.06 MJy/sr at
60 and 100$\mu$m respectively (Miville-Deschênes & Lagache 2005). At
160$\mu$m, we take a quadratic combination of a constant sensitivity limit of
0.12 MJy/sr 333the sensitivity of the observations is computed for a 16s
integration time per pixel using the SENS-PET tool,
http://ssc.spitzer.caltech.edu/tools/senspet/ and is divided by $\sqrt{N}$
where $N=49$ is the number of MIPS 160$\mu$m PSF inside an IRIS PSF at
100$\mu$m and a 2% uncertainty on the brightness (due to the uncertainty on
the calibration factor from instrumental units to surface brightness
(Stansberry et al. 2007)).

###  2.2 Sample selection

In low surface brightness regions, the variations of the infrared emission in
the observations can come from cirrus emission, fluctuations in the cosmic
infrared background or from noise. Since we want to study the variations of
cirrus emission only, we want to select observations in the SINGS sample that
are dominated by the dust emission variations.

The different contributions (cosmic infrared background, cirrus emission) to
the infrared emission have different power spectra that can help to
disentangle them. In particular, the cirrus power spectrum normalization
depends on the mean surface brightness while the contribution from background
galaxies does not. This dependence can be translated into a relationship
between the mean brightness and the standard deviation square,
$\sigma_{cirrus}^{2}$, in a region and depends on the size of the region
(Miville-Deschênes et al. 2007). For each of the observed field of view, we
computed the standard deviation at 100$\mu$m and the mean brightness at
100$\mu$m (minus the average CIB contribution at this wavelength) and then
plot the $\sigma^{2}$–$<B>$ relationship observed at 100$\mu$m in our sample.
To model $\sigma_{cirrus}$, we use the relationship derived by Miville-
Deschênes et al. (2007) below 10 MJy/sr, for a maximum scale length of 50’
(dotted line in Fig 2). We observe that our observations are consistent with
the model, with a large scatter as in the original relationship. This
dispersion is likely enhanced due to the fact that our fields of view are
elongated and the size of the region mapped varies between field.The
contribution from the CIB fluctuations can be described by two terms: a
Poisson noise that represents the galaxies distributed homogeneously with
respect to the resolution and a component with correlated spatial variations
corresponding to the clustering of galaxies on large scales. The contribution
from the clustering of infrared galaxies is predicted by using the Lagache et
al. (2003) model for galaxy evolution, with a bias parameter from Lagache et
al. (2007). The contribution from the Poisson noise to the $\sigma^{2}$
observed at 100$\mu$m is taken to be that measured by Miville-Deschênes et al.
(2002) since we used the same point source detection method. However, compared
to their study, we removed point sources applying the detection scheme at all
wavelength (25, 60 and 100$\mu$m). This enables us to mask faint galaxies at
100$\mu$m more efficiently and the Poisson noise in our measurements could be
lower than their measurement. Because we want to select the fields with the
least contribution from other sources (CIB) than cirrus to the observed
variations, this choice is therefore conservative.

Combining all contributions (represented in Fig. 2) to the observed
variations, we determine that a cut at 2.5 MJy/sr corresponds to
$\sigma_{cirrus}/\sigma_{CIB}=1.5$ (so that the total infrared fluctuations
$\sigma_{tot}=\sqrt{\sigma_{cirrus}^{2}+\sigma_{CIB}^{2}}$ are less thatn 20%
larger than from cirrus fluctuations alone). The regions with a mean 100$\mu$m
brightness above this threshold will therefore be dominated by variations of
cirrus emission. In each field, we computed the mean brightness at 100$\mu$m
as well as the standard deviation at 60, 100 and 160$\mu$m (c.f. Tab.1). By
keeping only the fields above the 2.5 MJy/sr cut, the sub-sample we will study
in this paper is composed of 15 fields with 100 and 160$\mu$m brightnesses,
among which 9 can be studied as well at 60$\mu$m ($|\beta|>15^{o}$, see sec.
2.1).

Stellar reddenings obtained from the analysis of the Sloan Digital Sky Survey
data enable us to put an upper limit of 1.2 mag on the extinction in these
fields444Using $N(H)/A_{V}$ (Bohlin et al. 1978) and $B_{100}/N(H_{I})\approx
6.67\times 10^{-21}MJy/srcm^{2}$, this upper limit implies
$B_{100}<15.2MJy/sr$, which is fully consistent with the brightness observed
in our sample. This confirms that the variations in the infrared cirrus
emission studied in each region comes from diffuse clouds according to the van
Dishoeck & Black (1988) classification.

##  3 Results

###  3.1 Cirrus emission at 160 and 100$\mu$m

In each field of the selected sample, we plot the point to point correlation
between the brightnesses observed at 100 and 160$\mu$m (represented in Fig 3
and 4) and apply a linear fit taking into account the errors at both
wavelengths. This enables us to obtain for each field a slope corresponding to
the ratio $B_{160}/B_{100}$ unbiased by variations of the zero point level
(residuals from the zodiacal light subtraction, absolute value of the CIB).
The correlation coefficient and the slope derived in each region are
summarized in Tab. 2.

Large scale observations of high galactic latitude emission of cirrus with
COBE were well characterized by a modified black body with a dust temperature
of 17.5K and an emissivity index proportionnal to $\nu^{2}$ (Boulanger et al.
1996). Using this law, we estimate the large scale 160/100 color for cirrus to
be of $B_{160}/B_{100}=2.0$ (taking into account color corrections). This
ratio is represented in the correlation plots (Fig. 3 and 4) to guide the eye.
While some correlations between $B_{100}$ and $B_{160}$ are in agreement with
the $B_{160}/B_{100}=2.0$ obtained on large scales, clear deviations are also
seen (4 fields out of 15 have a slope discrepant at a 5-6$\sigma$ level with
respect to the value of 2.0. The most extreme case is the field of NGC2976,
with a fitted slope on the $B_{160}$ versus $B_{100}$ correlation that is
10$\sigma$ away from the 2.0 standard value).

In Fig. 5, we compare the obtained ratios $B_{160}/B_{100}$ to the mean
surface brightness at 100$\mu$m in each field (black points). We observe a
large dispersion in the 160/100 colors that can not be explained by the error
on the data or the fitting process. At 100 and 160$\mu$m, the interstellar
emission is dominated by the emission from big dust grains at thermal
equilibrium with the radiation field (Désert et al. 1990) and the
$B_{160}/B_{100}$ ratio is therefore related to that characteristic dust
temperature. Taking a standard emissivity of dust per hydrogen atom in Hi from
Boulanger et al. (1996) and an emissivity index of 2, the 160/100 color
variations we are probing can therefore be related for illustrative purposes
to temperatures between 15.7 and 18.9K for column densities ranging from
$N_{H}=3\times 10^{20}$ to $2\times 10^{21}$cm-2. We note that these
variations are consistent _on average_ with the large scale estimate (blue
solid line), confirming that the fields used in this study are sampling the
cirrus emission observed on large scale.

For a given grain size and composition, this characteristic temperature
depends on the local radiation field strength and spectrum which depends on
the presence and distance of nearby heating sources and on the extinction. In
the framework of this model, the presence of large variations in the
160/100$\mu$m ratio observed in our sample would suggests the presence of
large variations in the heating of grains at small scales (variations by a
factor of 3 of the intensity of the incident radiation field). This can be
surprising since at low FIR surface brightness and at high latitude the
interstellar radiation field might be expected to be homogeneous. We looked at
the far-infrared color temperature maps derived from DIRBE observations by
Lagache et al. (1998) and Schlegel et al. (1998). The regions we are studying
appear to be reasonably representative of the high latitude cirrus given the
small number statistics. For the sightlines covered by our sample, the FIR
color variations seen in the DIRBE data are compatible with the variations
that we observe. Our study is indeed more sensitive than previous observations
and therefore able to probe color variations smaller than the uncertainties in
the previous studies.

The shape of the optical spectrum heating the grains could also affect the
far-infrared colors: the radiation field could become gradually harder with
position off the galactic plane (Mattila 1980). Using the cirrus model from
Efstathiou & Rowan-Robinson (2003) with different stellar populations heating
the clouds, we checked that changes in the shape of the optical radiation
field is unlikely to affect the 160/100 and 60/100 colors of cirrus by more
than 20%.

The dust equilibrium temperature depends however also on the structure of the
grains. Grain aggregates for example cool more efficiently. The temperature
variations observed in the diffuse medium could therefore be either due to
variations of the intensity of the interstellar radiation field or to changes
in the grain structure.

We compared our findings with different studies of far infrared emission from
the literature: the quiescent high galactic latitude clouds from del Burgo et
al. (2003), the large sample from archival ISOPHOT data by Kiss et al. (2006),
the two regions in a high latitude cirrus MCLD 123.5+24.9 observed by Bernard
et al. (1999), and the quiescent filament in the Taurus molecular complex from
Stepnik et al. (2003). Because other observations were obtained with different
instruments, we have to interpolate the brightnesses at various far infrared
wavelengths to estimates at 100 and 160$\mu$m. To do so, we took the dust
temperatures determined in each study with a brightness at 100 or 200$\mu$m
and used a modified black body law with a spectral index. The power index is
either taken from the study itself (if it was computed) or is fixed to a
standard value of 2. For each region, we also compute the mean 100$\mu$m
brightness as observed by IRIS and subtract a mean CIB contribution as for our
observations. Despite large scatter, we observe a trend between $<B_{100}>$
and the 160/100 color that is consistent with the idea that denser regions are
colder. However, the effect of selection biases of these studies remains
unclear. The comparison of our results with that from the literature (Fig. 5)
shows that previous studies in the far-infrared have been targeting higher
$B_{160}/B_{100}$ and $<B_{100}>$, i.e. denser and colder clouds. Due to our
better sensitivity, our observations fill the gap at low $B_{100}$ and low
$B_{160}/B_{100}$.

For the first time, we observe interstellar dust emission at low surface
brightness in an unbiased way (in the observing strategy) with a high
sensitivity. These observations show that former studies on dust properties at
FIR wavelength at small scale, have been biased toward colder and denser
clouds. Our study shows that the 160/100 brightness ratios of high galactic
cirrus clouds at small scales are consistent on average with the observations
on large scales. However, these 160/100 colors show a wide dispersion that
could arise from variations in the heating of the clouds or from change of the
dust grain structure. In order to investigate further the origin of the
160/100 variations, we extend the comparison to the 60$\mu$m data.

###  3.2 Comparison to the 60$\mu$m data

To investigate the origin of the 160/100 color variations observed in the
diffuse cirrus, we compare the 160 and 100$\mu$m data to the 60$\mu$m
emission. The sample for this part of the study is however reduced to fields
with an ecliptic latitude above 15o in order to avoid artifacts due to the
uncertainties in the zodiacal light subtraction in the IRIS data. For each
field, we determine a 60/100 color by using the same correlation technique as
above. The correlations in each region are shown in Fig. 6 and the obtained
$B_{60}/B_{100}$ are summarized in Tab. 2.

Fig. 7 presents the $B_{60}/B_{100}$ ratio obtained in each field with respect
to the $B_{160}/B_{100}$ ratio. Here again, large variations are observed in
the 60/100 colors that can not be explained by the uncertainties in the data
or in the analysis. As for the 160/100 colors, the 60/100 brightness ratios
are consistent on average with the ”reference values” (the pink cross and the
blue star in the figure) obtained for high latitude emission on large scales
(Boulanger & Perault 1988; Boulanger et al. 1996; Arendt et al. 1998).
Furthermore, there is a trend of decreasing $B_{60}/B_{100}$ with increasing
$B_{160}/B_{100}$.

To try to interpret this trend, we used two models of the dust grain emission
at different interstellar radiation fields: the Draine & Li (2007) model for
the Milky Way 555We took the model with a PAH fraction q${}_{PAH}=4.58\%$ but
we checked that this parameter does not influence significantly the results of
this study and the ”DUSTEM” model (updated model based on Désert et al.
(1990)). The models take into account the shape of the IRAS and MIPS/Spitzer
filters, the color corrections. For both models, the abundances of different
grain types are kept constant. The tracks obtained are compared to the data in
Fig. 7. The comparison shows that, if the variations of colors are due to
variations in heating of the grains, this would imply large changes in the
interstellar radiation field at high galactic latitude (from $U\approx 0.3$ to
1). Furthermore, the trend observed between the 60/100 and 160/100 colors is
not well reproduced with the current models by changing the interstellar
radiation field alone. In that respect, the Draine & Li (2007) model is
however closer to the observed trend than the DUSTEM model (only 3 fields are
more than 3$\sigma$ deviant from the expected curve), but for
$B_{160}/B_{100}<2.1$ all observations show systematically higher
$B_{60}/B_{100}$ than predicted by both models, while for
$B_{160}/B_{100}>2.1$ all data points have systematically lower
$B_{60}/B_{100}$ values than expected from both models.

Current dust models might be missing an additional dust grain type. Such an
addition might reproduce all color variations while changing the interstellar
radiation field alone. Another way to interpret the observed trend is that the
variations in the dust emission spectrum reflect spatial changes in the grain
properties – composition, structure or size distribution.

The equilibrium temperatures of dust grains is expected to decrease for
increasing grain sizes and small grains ($\leq 0.01\mu$m) undergo temperature
excursions following single-photon heating that enhances the 60$\mu$m
emission. Thus regions with fewer small grains may have lower 60/100 ratios.
The observed trend between the 60/100 and 160/100 infrared colors could be
reproduced by changing the dust grains size distribution or composition. For
example, enhancing the amount of small grains in regions with higher
interstellar radiation field (i.e. higher temperatures) and reducing it at low
equilibrium temperatures could reproduce the observed variations.

In the same way, regions with enhanced populations of large grains may have
increased 160/100 ratios. In that case, reproducing the observations could be
obtained with only modest variations in the starlight heating rate and shifts
in the grain size distribution (fewer small grains and increased sizes for the
larger grains at low temperatures, more small grains and smaller sizes for the
big grains at higher dust temperatures).

The 60/100 colors we observe therefore suggest changes of the dust properties
(dust size distribution and/or composition) from one region to the other.
These changes are related to variations in the 160/100 brightness ratio.
Whether the 160/100 color variations require a change of the starlight
intensity heating the clouds or result from the change of dust properties
alone is unclear. The interpretation of the color variations and of the trend
between the 160/100 and 60/100 colors is discussed further in the next
section.

##  4 Discussion

Despite its rather constant color distribution on large scale, the far
infrared emission from diffuse cirrus at high galactic latitude is observed to
host large color variations on small scales. These variations seem related to
each other (the 60/100 color decreases as the 160/100 color increases). In
this section, we will first check that these variations come indeed from
cirrus emission and are not related to the galaxies targeted with the
observations. Second, we will discuss possible interpretations for these large
color variations and the trend between them.

###  4.1 Extended disks in galaxies

Because the MIPS observations were taken to observe nearby galaxies, it is
legitimate to ask whether the infrared emission that we observe could be
associated with these targets. In particular, Hi observations have shown that
gaseous disks can extend much farther than the optical diameters. Dust grains
could be present in these outer parts of the galaxies and bias our
measurements.

To avoid this extended emission from the galaxies, we were careful in masking
regions larger than the detected emission (c.f. Sec. 2.1). Some of the
galaxies in the observations used in this study have been observed in Hi
observations through The HI Nearby Galaxies Survey (THINGS, Walter et al.
2005). We checked that the Hi diameters reported for these galaxies are
smaller than the region masked for our study. We are therefore confident that
the variations observed in the infrared emission between fields do not come
from the targeted galaxies, but rather from diffuse cirrus emission.

###  4.2 Physical conditions in cirrus clouds

A possibility to interpret the variations of far-infrared colors at high
galactic latitude is that we are sampling clouds in different physical
conditions and/or composition (different heating of the clouds, different dust
size distribution, …). The color changes would then be due to mixing along the
sightline of these different components.

An unbiased survey of H2 absorption in high galactic latitude clouds by FUSE
(Gillmon et al. 2006) has been interpreted as showing that some clouds have
been compressed. The dynamical history leading to this compression may involve
shock waves or strong turbulence, which could also lead to changes in the
grain size distribution by shattering in grain-grain collisions, possibly
explaining the regions of higher than average 60/100 and lower than average
160/100 colors.

One tantalizing possibility, in terms of mixing, is the presence of
Intermediate or high velocity clouds (IVCs and HVCs) along the sightline.
These cloud falling onto our galaxy could have very different dust properties
(e.g. Miville-Deschênes et al. 2005) and would bias the measured infrared
emission from more local cirrus clouds. We checked for the presence of
intermediate velocity clouds in the LAB Hi survey spectra (Kalberla et al.
2005) in the direction of the fields in our sample. For about half of the
sample, there is a intermediate velocity component seen in Hi in the
sightline. Two sightlines also have a high velocity component. However, no
conclusive trend between the fraction of the Hi in the IVCs and/or HVCs and
the infrared colors could be seen. This may be due to the lack of resolution
of the Hi observations ($\sim 0.6^{o}$), to the difficulty to disentangle IVCs
and the Milky Way in some regions or to the small size of our sample.

###  4.3 Variations in grain properties

The grain size distribution is the result of processes such as sputtering,
shattering and coagulation, and sightline-to-sightline variations in the
wavelength-dependence of optical and ultraviolet extinction toward stars have
already demonstrated regional variations in the grain properties. Whether the
observed variations in emission can be fully explained by variations in the
size-distribution alone, or whether other properties (e.g., composition or
porosity) are also involved is uncertain.

In denser clouds, variations of infrared colors (Stepnik et al. 2003; Kiss et
al. 2006) have been interpreted with a grain coagulation scenario, combining
changes of the size distribution with changes of the dust emissivity
properties. The trend observed between the 60/100 and 160/100 colors would be
consistent with this idea. In this scenario, most of the 160/100 color
variations would then be due to the change of emissivity of dust grains (due
to changes of their structure), while the 60/100 colors would change with the
incorporation or release of small grains in large dust aggregates. We could be
witnessing variations of dust properties due to variations in turbulent
motions in the diffuse interstellar medium. Alternatively, dust grains in the
diffuse medium could retain for some time the aggregate structure they had
previously acquired in denser regions. So we could be seeing a sequence of
regions corresponding to increasing time since their release from high density
environments.

Such changes in the dust size distribution and structure of grains would imply
related variations of the UV-optical extinction curves at high galactic
latitude. An extinction and reddening study of stars at high galactic latitude
behind translucent clouds (Larson & Whittet 2005) shows variations in the
extinction curves obtained with respect to the average curve for the diffuse
interstellar medium. In particular, 48% of their sightlines have $R_{V}<2.8$,
much lower that the diffuse ISM average of $3.05$. Such values are indicative
of enhanced abundances of small grains, and these regions could have a higher
that average 60/100 color. To test if the high 60/100 colors and low $R_{V}$
values are connected, we computed the 60/100 colors from IRIS data in a
similar fashion to this study for a $\sim 1\times 1^{o}$ region around each
sightline of the Larson & Whittet (2005) sample. We do not observe any
correlation however between the $R_{V}$ and 60/100 color, nor between $A_{V}$
and the 60/100 color. Unfortunately, no longer wavelength observations exist
for these regions and it would be important to determine the 160/100 colors in
these regions as well and study their dependancy with extinction properties.
This result is however a concern for the coagulation scenario as an
interpretation of the 60/100 color variations we observe.

Interpreting the trend observed between the 60/100 and 160/100 colors with a
change of dust optical properties and dust size distribution remains
hypothetical without further observations. In particular, Hi observations of
these diffuse sightlines, with a high resolution (at least similar to the IRIS
one), will be needed to determine the emissivity of dust per hydrogen atom and
test if variations are observed and correlated with the infrared colors.

It is presently not possible to interpret further the far-infrared color
variations in terms of physical condition changes, grain size distribution,
grain properties, etc. A larger number of observations of high latitude cirrus
could help to probe the spatial variations. Hi studies of these regions at
high resolution would also be crucial to probe possible variations of dust
grain emissivities, or to check whether the velocity structure of the cirrus
correlates with the 60/100/160 colors. Finally, extinction curves on
sightlines where cirrus emission properties have been determined would be most
useful to see whether extinction properties would correlate with the FIR
colors.

##  5 Conclusion

We performed an unbiased study of dust emission from high galactic latitude
cirrus using serendipitous Spitzer MIPS observations at 160$\mu$m from the
SINGS survey, complemented by IRIS data at 60 and 100$\mu$m. After an
appropriate post-reduction of the data and a removal of the targeted galaxy, a
sub-sample is selected so that the variations of the cirrus emission dominate
over the CIB fluctuations in each field.

We observe 160/100 colors in our fields that are consistent on average with
large scale studies. However, strong variations are also observed from field
to field. This paper extends former studies on dust properties at high
galactic latitude to more diffuse, fainter and warmer clouds. The 60/100 color
is also observed to vary significantly in the sample and there is a trend of
decreasing 60/100 with increasing 160/100 ratios. This trend is not completely
reproduced by current models taking only into account variations of the
radiation field strength and requires changes in the dust properties,
composition or size distribution.

The exact origin of these variations remains unknown, but the variations of
the 60/100/160 colors may reflect variations of the grains size distribution,
of grain properties in addition to heating variations. However, we can not
completely rule out the possibility that our fields contain emission from
matter at different heights above the Galactic plane, the juxtaposition of
multiple components in the fields could be affecting the infrared color
estimates.

All in all, we observe unexpected variations of far-infrared colors in the
supposed ”homogeneous” cirrus at high galactic latitudes. These variations are
not yet understood and further studies will be needed to test their origin. In
particular, studies on a larger area of sky is needed to confirm the
significance of these variations and their spatial distribution on the sky
could give new clues on their origin.

These infrared color variations will most probably be linked with variations
of the infrared colors at longer wavelengths. They therefore represent an
important point to study for the Planck and Herschel missions. Such longer
wavelength observations will enable us to determine precisely the temperature
and spectral index of the dust, and their variations, in high latitude
regions.

We would like to thank our referee M. Rowan-Robinson for his useful and
interesting comments and A. Efstathiou for his help with their cirrus model.
We are also grateful to L. Cambrésy for useful discussions and work on upper
limit for extinction in our fields. Facilities: Spitzer.

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Figure 1: Position of the SINGS fields (red circles and blue squares) overlaid
on the dust column density maps from Schlegel et al. (1998) centered around
the north galactic pole (left panel) and the south galactic pole (right
panel). The blue squares correspond to the fields selected for this study (the
variation in the infrared emission is dominated by the cirrus component). A
grid of galactic coordinates is overlayed.

Figure 2: Variations of the square of the standard deviation (related to the
power spectrum of the signal) measured in each field with the mean brightness
at 100$\mu$m. The observed values are compared with models for the different
contributions: the infrared galaxies clustering (dotted-dashed line), the
poisson noise (dashed line) and the cirrus variation (dotted line). This
enables us to define a cut in 100$\mu$m brightness (the vertical black line)
above which the cirrus variations dominate over CIB fluctuations.

Figure 3:  160–100 scatterplots for all SINGS observations with $<B_{100}>\geq
2.45$ MJy/sr. In each plot, a canonical slope of 2.0 is represented by a
dashed line (corresponding to a temperature of 17.5K). A linear fit is
performed on the correlation and the best fit is represented with a solid
line. The value of the slope obtained is written in the legend.

Figure 4:  idem as Fig. 3

Figure 5: Variations of the 160/100 surface brightness ratio with the mean
100$\mu$m surface brightness. The diamonds represent the SINGS observations.
The blue line denotes a typical ratio of 2.0.

Figure 6:  60–100 scatterplots for all SINGS observations with $<B_{100}>\geq
2.5$ MJy/sr. In each plot, a canonical slope of 0.2 is represented by a dashed
line. A linear fit is applied and the best fit is represented with a solid
line. The value of the slope obtained is written in the legend. The green
point and its bold uncertainties represent the value chosen to represent the
local background in the observation.

Figure 7: Color-color plot showing the evolution of the 160/100 and 60/100 brightness ratios with each other for all the the regions where $<B_{100}>\geq 2.5$ MJy/sr. Two reference values from the literature representing the colors measured on large scales are overplotted. The observed colors are compared to the trend obtained with two dust models: the Draine et al. (2007) model (dark solid line) and the DUSTEM model (blue dotted line). In both case, the incident interstellar radiation field intensity is the only parameter varying (the proportion of grain types are kept constant).  Table 1: Characteristics of the observations: galactic and ecliptic coordinates, average 100$\mu$m cirrus brightnesses in each field as well as standard deviations at 60, 100 and 160$\mu$m field | (l,b) | ($\lambda$,$\beta$) | Area (deg2) | $\sigma_{60}$ |  $<B_{100}>$ (MJy/sr) | $\sigma_{100}$ | $\sigma_{160}$
---|---|---|---|---|---|---|---
NGC0337 | ( 126.983 , -70.4576) | ( 10.0714 , -12.6317) | 0.048 | —- | 5.75 | 0.713 | 1.951
NGC0584 | ( 148.863 , -68.1957) | ( 17.8130 , -15.3840) | 0.100 | 0.057 | 2.38 | 0.177 | 0.387
NGC0628 | ( 138.043 , -46.3226) | ( 27.4637 , 5.05541) | 0.115 | —- | 3.09 | 0.287 | 0.809
NGC0855 | ( 143.962 , -32.1481) | ( 39.9931 , 13.3583) | 0.056 | —- | 3.33 | 0.111 | 0.399
NGC0925 | ( 144.527 , -25.8230) | ( 44.7625 , 17.7346) | 0.124 | 0.040 | 3.09 | 0.101 | 0.316
NGC1097 | ( 227.959 , -65.0380) | ( 26.0183 , -43.9238) | 0.113 | 0.037 | 1.08 | 0.112 | 0.250
NGC1291 | ( 247.760 , -57.5207) | ( 27.3597 , -55.9323) | 0.154 | 0.018 | 0.68 | 0.104 | 0.212
NGC1316 | ( 240.631 , -56.8186) | ( 32.3476 , -53.2688) | 0.197 | 0.042 | 1.00 | 0.093 | 0.189
NGC1377 | ( 212.367 , -52.3728) | ( 44.7413 , -38.7417) | 0.042 | 0.029 | 1.65 | 0.071 | 0.246
NGC0024 | ( 40.4333 , -80.1196) | ( 351.074 , -23.9783) | 0.099 | 0.043 | 1.14 | 0.073 | 0.130
NGC1404 | ( 237.002 , -53.9016) | ( 38.2431 , -52.8271) | 0.069 | 0.032 | 0.59 | 0.059 | 0.101
NGC1482 | ( 213.765 , -48.2291) | ( 50.1099 , -39.5384) | 0.039 | 0.040 | 2.44 | 0.087 | 0.212
NGC1512 | ( 248.603 , -48.4042) | ( 40.9466 , -61.8125) | 0.176 | 0.042 | 0.49 | 0.075 | 0.132
NGC1566 | ( 264.199 , -43.5133) | ( 32.0129 , -73.2076) | 0.116 | 0.025 | 0.39 | 0.081 | 0.124
NGC1705 | ( 260.913 , -38.8932) | ( 50.3989 , -74.3634) | 0.043 | 0.033 | 0.42 | 0.094 | 0.121
NGC2403 | ( 150.172 , 28.7424) | ( 102.746 , 43.5117) | 0.261 | 0.049 | 1.62 | 0.132 | 0.286
HolmbergII | ( 143.899 , 32.2721) | ( 106.073 , 49.5466) | 0.077 | 0.028 | 1.34 | 0.091 | 0.300
M81DwarfA | ( 143.536 , 32.5857) | ( 106.476 , 49.8996) | 0.041 | 0.043 | 1.00 | 0.121 | 0.248
DDO053 | ( 148.991 , 34.4963) | ( 110.352 , 45.7107) | 0.077 | 0.048 | 1.57 | 0.113 | 0.324
NGC2798 | ( 178.927 , 43.7838) | ( 127.750 , 25.1694) | 0.051 | 0.040 | 1.04 | 0.086 | 0.166
NGC2841 | ( 166.458 , 43.6210) | ( 124.989 , 33.8516) | 0.101 | 0.043 | 0.74 | 0.061 | 0.140
NGC2976 | ( 143.612 , 40.4772) | ( 118.782 , 50.4361) | 0.076 | 0.100 | 2.59 | 0.594 | 1.951
HolmbergI | ( 140.502 , 38.2525) | ( 115.197 , 52.8405) | 0.079 | 0.036 | 1.18 | 0.185 | 0.659
NGC3049 | ( 226.610 , 44.4312) | ( 146.934 , -2.98701) | 0.027 | —- | 2.20 | 0.223 | 0.504
NGC3190 | ( 211.891 , 54.4481) | ( 147.733 , 10.7909) | 0.090 | —- | 1.61 | 0.094 | 0.195
NGC3184 | ( 178.537 , 54.9707) | ( 139.768 , 28.3651) | 0.102 | 0.031 | 0.93 | 0.089 | 0.149
NGC3198 | ( 170.656 , 54.2275) | ( 138.036 , 32.7413) | 0.050 | 0.039 | 0.52 | 0.074 | 0.157
IC2574 | ( 139.983 , 43.1538) | ( 123.351 , 52.9795) | 0.156 | 0.036 | 1.38 | 0.205 | 0.509
NGC3265 | ( 200.457 , 58.5603) | ( 147.821 , 18.3074) | 0.029 | 0.030 | 1.30 | 0.055 | 0.146
MRK33 | ( 156.610 , 52.2144) | ( 135.166 , 41.0632) | 0.032 | 0.032 | 0.86 | 0.212 | 0.424
NGC3351 | ( 232.652 , 56.1499) | ( 157.328 , 3.64920) | 0.074 | —- | 1.95 | 0.108 | 0.210
NGC3521 | ( 254.316 , 52.8287) | ( 166.849 , -5.14601) | 0.149 | —- | 2.81 | 0.328 | 0.718
NGC3621 | ( 280.580 , 25.9208) | ( 184.582 , -34.1228) | 0.180 | 0.111 | 4.12 | 0.444 | 0.900
NGC3627 | ( 240.246 , 64.2844) | ( 165.038 , 8.27119) | 0.097 | —- | 1.70 | 0.179 | 0.248
NGC3773 | ( 249.132 , 66.7215) | ( 169.474 , 8.73315) | 0.034 | —- | 1.84 | 0.102 | 0.125
NGC3938 | ( 153.689 , 68.7273) | ( 156.788 , 39.2954) | 0.075 | 0.065 | 1.01 | 0.194 | 0.328
NGC4125 | ( 130.164 , 50.9404) | ( 139.741 , 57.1897) | 0.095 | 0.037 | 0.81 | 0.174 | 0.285
NGC4236 | ( 127.310 , 46.9854) | ( 134.438 , 60.6054) | 0.267 | 0.035 | 0.66 | 0.098 | 0.170
NGC4254 | ( 267.627 , 75.3684) | ( 177.739 , 15.3315) | 0.075 | 0.054 | 2.37 | 0.161 | 0.370
NGC4321 | ( 267.852 , 77.0715) | ( 178.035 , 17.0118) | 0.119 | 0.053 | 1.37 | 0.160 | 0.214
NGC4450 | ( 270.395 , 78.8333) | ( 178.801 , 18.7003) | 0.079 | 0.032 | 1.36 | 0.119 | 0.338
NGC4536 | ( 291.454 , 65.1324) | ( 186.331 , 5.66591) | 0.102 | —- | 1.64 | 0.090 | 0.239
NGC4552 | ( 285.274 , 74.9059) | ( 182.409 , 14.8782) | 0.106 | —- | 2.28 | 0.210 | 0.688
NGC4559 | ( 193.095 , 85.8409) | ( 175.296 , 29.2435) | 0.124 | 0.029 | 1.05 | 0.082 | 0.184
NGC4569 | ( 285.793 , 75.9750) | ( 182.385 , 15.9551) | 0.101 | 0.060 | 2.65 | 0.174 | 0.470
NGC4579 | ( 287.803 , 74.2939) | ( 183.194 , 14.3801) | 0.071 | —- | 2.37 | 0.126 | 0.311
NGC4594 | ( 297.305 , 51.1460) | ( 193.066 , -6.96614) | 0.151 | —- | 3.12 | 0.141 | 0.298
NGC4625 | ( 131.096 , 75.1008) | ( 168.734 , 41.5922) | 0.047 | 0.052 | 0.88 | 0.121 | 0.239
NGC4631 | ( 145.167 , 83.5710) | ( 174.202 , 33.9219) | 0.171 | 0.054 | 1.02 | 0.182 | 0.248
NGC4725 | ( 271.647 , 88.6791) | ( 179.905 , 28.4957) | 0.079 | 0.037 | 0.70 | 0.102 | 0.161
NGC4736 | ( 124.738 , 75.5220) | ( 170.796 , 42.2419) | 0.187 | 0.047 | 0.67 | 0.192 | 0.293
DDO154 | ( 110.504 , 89.5114) | ( 179.918 , 30.2879) | 0.042 | 0.043 | 0.57 | 0.085 | 0.137
NGC4826 | ( 311.183 , 85.0014) | ( 183.181 , 25.6681) | 0.125 | 0.059 | 1.89 | 0.182 | 0.239
DDO165 | ( 121.301 , 49.1180) | ( 142.645 , 62.9864) | 0.058 | 0.051 | 1.14 | 0.171 | 0.301
NGC5033 | ( 101.051 , 79.6584) | ( 178.930 , 40.1141) | 0.128 | 0.033 | 0.54 | 0.096 | 0.149
NGC5055 | ( 107.903 , 73.9458) | ( 175.505 , 45.4687) | 0.163 | 0.052 | 0.75 | 0.178 | 0.267
NGC5194 | ( 106.114 , 68.6881) | ( 174.458 , 50.7106) | 0.233 | 0.087 | 1.23 | 0.364 | 0.712
Tololo89 | ( 318.732 , 27.7794) | ( 219.272 , -19.6125) | 0.051 | 0.186 | 5.26 | 0.544 | 0.978
NGC5408 | ( 316.804 , 20.0386) | ( 223.038 , -26.7581) | 0.056 | 0.163 | 5.55 | 0.761 | 1.702
NGC5474 | ( 101.503 , 59.8880) | ( 174.929 , 59.7255) | 0.061 | 0.027 | 0.48 | 0.086 | 0.185
NGC5713 | ( 350.094 , 52.5981) | ( 217.013 , 14.3062) | 0.052 | —- | 2.08 | 0.070 | 0.187
NGC5866 | ( 92.3464 , 52.7620) | ( 186.246 , 66.8587) | 0.105 | 0.029 | 0.62 | 0.097 | 0.210
IC4710 | ( 327.879 , -22.1180) | ( 273.178 , -43.4420) | 0.034 | 0.119 | 5.69 | 0.426 | 1.007
NGC6822 | ( 24.6969 , -17.9857) | ( 294.785 , 6.07295) | 0.185 | —- | 7.77 | 0.745 | 1.705
NGC6946 | ( 95.3945 , 12.0399) | ( 356.855 , 72.2046) | 0.086 | 0.125 | 9.57 | 0.722 | 1.948
NGC7331 | ( 93.0851 , -20.9465) | ( 356.050 , 39.1427) | 0.126 | 0.121 | 3.91 | 0.429 | 0.761
NGC7552 | ( 347.564 , -64.7400) | ( 330.605 , -34.6836) | 0.046 | 0.060 | 0.81 | 0.128 | 0.173
NGC7793 | ( 5.28269 , -76.5898) | ( 344.625 , -29.1746) | 0.119 | 0.046 | 0.95 | 0.126 | 0.184
Table 2: Infrared colors in the observations, i.e. the 160/100 and 60/100 brightness ratios obtained from a fit of the correlations in each field field | 160/100 color | 60/100 color
---|---|---
NGC0337 | $2.3\pm 0.2$ | —
NGC0628 | $2.6\pm 0.1$ | —
NGC0855 | $2.7\pm 0.5$ | —
NGC0925 | $2.4\pm 0.3$ | $0.12\pm 0.03$
NGC2976 | $3.0\pm 0.1$ | $0.15\pm 0.01$
NGC3521 | $1.81\pm 0.07$ | —
NGC3621 | $1.86\pm 0.04$ | $0.225\pm 0.005$
NGC4569 | $2.0\pm 0.2$ | $0.31\pm 0.03$
NGC4594 | $2.3\pm 0.1$ | —
TOL89 | $1.7\pm 0.1$ | $0.32\pm 0.02$
NGC5408 | $2.22\pm 0.04$ | $0.19\pm 0.01$
IC4710 | $2.0\pm 0.1$ | $0.31\pm 0.01$
NGC6822 | $2.23\pm 0.04$ | —
NGC6946 | $2.5\pm 0.1$ | $0.16\pm 0.006$
NGC7331 | $1.75\pm 0.07$ | $0.238\pm 0.009$