#  Domain Adaptation for Learned Image Compression with Supervised Adapters

Alberto Presta∗ Gabriele Spadaro∗ Enzo Tartaglione†
Attilio Fiandrotti∗† and Marco Grangetto∗
∗University of Turin †LTCI, Télécom Paris, Computer science department
Institut Polytechnique de Paris Turin, Italy Paris, France
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In Learned Image Compression (LIC), a model is trained at encoding and
decoding images sampled from a source domain, often outperforming traditional
codecs on natural images; yet its performance may be far from optimal on
images sampled from different domains. In this work, we tackle the problem of
adapting a pre-trained model to multiple target domains by plugging into the
decoder an adapter module for each of them, including the source one. Each
adapter improves the decoder performance on a specific domain, without the
model forgetting about the images seen at training time. A _gate network_
computes the weights to optimally blend the contributions from the adapters
when the bitstream is decoded. We experimentally validate our method over two
state-of-the-art pre-trained models, observing improved rate-distortion
efficiency on the target domains without penalties on the source domain.
Furthermore, the gate’s ability to find similarities with the learned target
domains enables better encoding efficiency also for images outside them.

##  1 Introduction

Learned image compression (LIC) has gained considerable interest as it may
achieve equal or even better compression efficiency than standardized codecs
[1]. In this context, a parametric autoencoder is trained to project an image
into a latent representation that is quantized and entropy-coded as a
compressed bitstream. On the decoder side, the latent representation is
decoded and projected back to the pixel domain, recovering the encoded image.
The models suffer however a limited ability to adapt to images from a domain
different than that of the training samples; standard video codecs include ad-
hoc coding tools to deal with such contents [2], as a testament to their
relevance. A straightforward approach to domain adaptation consists of fine-
tuning the model over samples from the target domain. However, that comes at
the risk of _catastrophic forgetting_ , i.e. a performance loss on the source
domain. In [3] image-specific adapter modules are plugged into the encoder,
however, this method requires training the adapters at encoding time and
delivering their parameters to the decoder for each single image, jeopardizing
its practical deployability.
In this work, we propose a novel method for LIC domain adaptation based on
blending the contributions of domain-specific adapters plugged at the decoder
side. Instead of training a separate adapter for each image [3], an adapter
module is plugged into the decoder for both each target domain(s) and the
source one. The adapters’ contributions are blended according to the weights
computed by a _gate_ network that infers the domain of the image to encode.
The adapters and the gate network are plugged into a pre-trained model that
needs neither retraining nor refinement. The adapters and the gate are jointly
trained and each adapter specializes in a specific domain, whether it is one
of the target domains or the source one.
Our approach improves rate-distortion performance on the target domains,
avoiding catastrophic forgetting on the source one. Furthermore, the adapters
enhance image reconstruction even in classes that were never seen at training
time. Besides being effective, adapters do not modify parameters related to
the original pre-trained model; this ensures that even if these components
were not available during decoding, the image would be reconstructed without
compromising the performance of the original pre-trained model.

##  2 Background and related works

This section provides some background on LIC and reviews the relevant
literature regarding domain adaptation in such a context.

###  2.1 Learned image compression

In LIC, a convolutional parametric autoencoder is trained end-to-end at
compressing an image [1]. In a nutshell, the encoder receives an image
$\mathbf{x}$ as input and projects it to a latent representation $\mathbf{y}$
that is quantized into $\mathbf{\hat{y}}$ and then entropy coded, yielding a
compressed representation in the form of a bitstream. At the receiver side, a
decoder projects this representation back to the original dimension,
recovering an approximation of the original image $\mathbf{\hat{x}}$. The
entire model is trained by optimizing a Rate-Distorsion loss function $\lambda
D+R$, where $D$ is the reconstruction error, $R$ is the rate term, and
$\lambda$ is a hyperparameter that regulates the trade-off between the two. In
seminal works like [4], the latent representation is modeled as a fully
factorized distribution extracted either with an auxiliary neural network or
in an analytical way [5]. [6] improved this approach by adding a _scale
hyperprior_ that aims to find spatial correlation within an image, using
Gaussian priors. In [7], a local context model based on mask convolution is
used to enhance entropy estimation, while [8] channel-wise contexts are
exploited to decrease computing time. In the same wave, works like [9, 10]
make use of local and window-based attention, while [11, 12] exploited a
mixture of CNN and transformer. In general, the last models outperform
conventional codecs on natural images; however, when transitioning to other
specific domains, the emerging expected performance gap with traditional
codecs narrows.

###  2.2 Task-domain adaptation for learned image compression

To the best of our knowledge, task-domain adaptation for LIC has not been
exhaustively studied. In [13], the parameters related to the generalized
divisive normalization layer [14] and the entropy model are fine-tuned, with
the addition of a small number of custom channel-specific parameters; despite
this strategy, the authors were unable to eradicate the problem of forgetting.
Another possible approach to perform efficient domain adaptation is through
the adapters [15], consisting of small modules added in the pre-trained model.
In that sense, we identify [3] as the closest work to our method: Tsubota  _et
al._ exploit tiny adapters introduced on the second attention module in the
decoder to adapt the model to a single image. Specifically, for each image,
they optimize the latent representation rate-distortion-wise, and then they
tune the adapter parameters minimizing a loss function that combines
distortion and parameter rate. Despite its effectiveness, this work has some
drawbacks. First, the parameters of the adapters are part of the bitstream
(since they are specific for each image), which causes transmission overhead.
Second, Tsubota  _et al._ do not perform actual domain adaptation, but rather
single-image content adaptation. Lastly, the encoding phase aligns with the
optimization of the latent representation and adapter parameters, making the
latter quite computationally expensive in terms of time and resources. On the
other hand, we specialize adapters for multiple domains, either target or
source, and we blend them to improve the final reconstruction, eliminating
thus the need to encode adapter weights and simplifying the encoding phase,
which remains unchanged compared to the model without adapters.

##  3 Proposed method and architecture

In this section, we describe our method and architecture for domain adaptation
exemplifying the state-of-the-art Zou  _et al._ [10] model. Yet, our method is
architecture-agnostic as we experimentally show later on over the Cheng  _et
al._ [9] model. In a nutshell, we plug $K+1$ residual adapters into the
decoder model pre-trained on a given source domain, i.e. $K$ adapters are
meant for novel target domains and one is for the legacy source domain. The
adapter on the source domain could be skipped but it turns out to be useful
for fine-tuning. At decoding, we exploited such adapters along with a gate
network $\varphi$ that yields a probability distribution over the $K+1$
domains $\mathbf{v}\\!\in\\![0,1]^{K+1}$, that is used to blend adapters’
outcomes.
The rest of this section is structured as follows. In Sec. 3.1 and Sec. 3.2 we
first discuss the structure of the adapters and the gate, respectively. Next,
in Sec. 3.3, we present the policy used to blend adapters’ outputs. Finally,
in Sec. 3.4, we describe the complete workflow of the resulting architecture
along with the associated cost function and training process.

###  3.1 Domain adaptation at the decoder

In our architecture, one adapter $\mathbf{Ad}_{:}^{k}$ is plugged into the
decoder for each of the $K+1$ domains of interest, i.e. $K$ target domains
plus the source one; Each adapter is composed by three modules
$\mathbf{Ad}_{:}^{k}=\\{Ad^{k}_{0},Ad^{k}_{1},Ad^{k}_{2}\\}$, each one
including one convolutional layer with $3\times 3$ kernels. As shown in Fig.
1(a), the module $Ad^{k}_{0}$ is plugged into the second Window Attention
Module (WAM) block of the decoder, as in [3], to enhance the attention towards
the more detailed parts of the image. Namely, the convolutional layer receives
in input the output of the WAM block, preserving the number of channels and
spatial dimensions. The remaining $Ad^{k}_{1}$ and $Ad^{k}_{2}$ modules are
plugged at the output of the last two transpose-convolutional layers
respectively, as in Fig. 1(b). For the sake of similarity, these two modules
have the same structure as the respective layers they are plugged into,
consisting of a deconvolutional layer that doubles the spatial dimensionality
of the input feature maps. With respect to comparable literature where adapter
modules are plugged into the encoder, our choice of plugging adapters into the
decoder bears some advantages. First, the bitstream produced by the pre-
trained reference encoder can be reused as it is. Second, training the
adapters at the decoder avoids the cost of backpropagating the gradients to
the encoder. The overall decoder is shown in Fig. 1(c).

(a) _WAM adapters_

(b) _Tconv. adapters_

(c) _Decoder_

(d) _Gate network_

Figure 1: (a) Architecture of the WAM adapter $Ad^{k}_{0}$. (b) Architecture
of the deconvolutional adapters $Ad^{k}_{\\{1,2\\}}$. (c) Overall decoder. (d)
Gate network $\varphi$.

###  3.2 Gate Network

The Gate network $\varphi$ takes as input the latent representation
$\mathbf{y}$ and outputs a domain probability distribution. The gate
architecture is shown in Fig. 1(d) and includes one convolutional layer, a
ReLU activation, a MaxPooling layer for complexity control, and an adaptive
pooling layer that allows handling inputs of arbitrary size. A linear layer
then yields $K+1$ classification logits that are normalized into the
probability distribution $\mathbf{v}=(v_{0},..,v_{K})$ by a Softmax layer and
delivered to the decoder. Such distribution amounts to $K+1$ real-valued
coefficients that are compressed and embedded in the bitstream beside the
compressed latent representation $\mathbf{\hat{y}}$. While the compression of
such coefficients is out of the scope of this work, their rate can be safely
assumed negligible to the compressed latent representation. For example, for a
sample image from the classic Kodak dataset, the weight to encode $\mathbf{v}$
is on the order of $10^{-4}$ compared to the compressed latent representation.
The gate could be in principle plugged into the decoder, sampling the
compressed latent representation $\mathbf{\hat{y}}$. However, in this work we
refrain our experiments from sampling the uncompressed latent representation
$\mathbf{y}$, leaving the effect of quantizing the latent representation for
our future endeavors. Furthermore, using $\mathbf{y}$ as the input instead of
the original image allows us to leverage the latent representation already
extracted by the encoder, thus simplifying the gate’s task.

(a) _blending policy._

(b) _Overall LIC model architecture with $\varphi$._

Figure 2: (a) Adapters blending policy based on weighted sum. Index 0 refers
to the source domain adapter. (b) Blueprint of a LIC model with the gate.

###  3.3 Adapters blending policy

The outputs of the adapters are blended at the decoder according to the
distribution coefficients $\mathbf{v}$ embedded in the bitstream. Namely, the
probability distribution $~{}{\mathbf{v}\\!=\\!(v_{0},..,v_{K})}$ among
domains of interest, i.e. target and source, is used to compute a weighted
average between the adapters outputs. Let $\mathbf{y}_{j}$ be the input of the
$j$-th layer of the decoder with the adapter ($j=\\{0,1,2\\}$), adapters-
enhanced output $\mathbf{\overline{y}}$ is calculated as

$\mathbf{\overline{y}}=\Psi_{j}(y_{j})+l_{j}(y_{j}),$ (1)

where $l_{j}$ represents the $j$-th layer of the decoder, and the addition
operator is element-wise sum. Then we have

$\Psi_{j}(y_{j})=\sum_{k=0}^{K}v_{k}\cdot Ad_{j}^{k}(y_{j}),$ (2)

where $Ad_{j}^{k}$ is the $j$-th level adapter of the $k$-th domain. The
blending policy just described is shown in Fig. 2(a), where we refer to the
pre-trained layer network without adapter, as the main layer.

###  3.4 Training the adapters and the gate

This section first describes the workflow of our method and then how both
adapters and the gate are trained given a model pre-trained on the source
domain, e.g. natural images. The training set $X_{t}$ is composed of $T$
images uniformly distributed among the $K$ target domains with the
corresponding domain labels $d_{t}$, with $t\\!=\\!1,...,T$. As shown in Fig.
2(b), an image $\mathbf{x}_{t}$ is passed to the encoder $g_{a}$ producing the
latent representation $\mathbf{y}$, which is first fed as input to the _gate
network_ $\varphi$ that yields a probability distribution over the $K+1$
domains $\mathbf{v}\\!\in\\![0,1]^{K+1}$. The probability vector $\mathbf{v}$
is then sent to the decoder beside the quantized latent representation
$\mathbf{\hat{y}}$, which is encoded through the channel-wise entropy
estimation module [8], and then it is used to weight the output of each
adapter. Once decoded back, $\mathbf{\hat{y}}$ is passed to our adapter-based
decoder $g_{s}+\mathbf{Ad}_{:}^{K}$ , returning the reconstructed image
$\mathbf{\hat{x}_{t}}$.
Once initialized from a Gaussian distribution, both the adapters
$\mathbf{Ad}_{:}^{k}$ and the gate $\varphi$ are jointly trained for all
domains of interest at once, freezing the other parameters of the model. The
cost function minimized at training time is the sum between the reconstruction
error $D$ and a _domain mismatch term_ , that should force $\varphi$ to learn
different domains from $\mathbf{y}$. By casting this as a classification
problem, the objective is to minimize the cross-entropy ($\operatorname{CE}$)
between the distribution $\mathbf{v}$ derived from $\varphi$ and the true
labels $d_{t}$. Summing up, the minimized cost function is

$\mathcal{L}=\gamma\cdot\operatorname{MSE}(\mathbf{x}_{t},\mathbf{\hat{x}}_{t})+\operatorname{CE}(d_{t},\mathbf{v})$
(3)

where $\gamma$ is a hyperparameter that regulates the trade-off between these
two terms, and $\operatorname{MSE}$ is the _mean squared error_ minimizing the
distortion. Notice that the second term in (3) does not represents the rate of
$\mathbf{\hat{y}}$, typically minimized in the traditional RD loss function.
Since we do not refine the encoder, the compressed latent representation
$\mathbf{\hat{y}}$ remains untouched as well as its rate, allowing us to drop
the rate component $R$ from the minimized cost function.

##  4 Experiments and Results

This section presents the results obtained with our method over the Zou  _et
al._ [10] and Cheng  _et al._ [9] models; concerning the latter, we added the
deconvolutional adapters in the last three layers of the decoder, used in the
same way as for Zou  _et al._ [10]. We experiment with adapting the pretrained
models over two different target domains gauging the performance on the target
as well as the source domain. Our implementation111 our code is available
online at https://github.com/EIDOSLAB/LIC-Domain-Adaptation-with-supervised-
Adapters. leverages on the CompressAI library [16].

###  4.1 Experimental setup

We considered a total of $K\\!=\\!2$ target domains, namely the Sketch and
Comic which are added to the source domain used for pretraining the model,
e.g. natural images. Towards training the gate and the adapters, we had to
combine different sources due to the lack of high-resolution datasets for
domain adaptation in LIC. For the natural domain, we used the OpenImages
dataset [17]; for the sketch domain, we used ImageNet-Sketch [18]; for the
comic domain, we used BAM [19]. Namely, we randomly sampled 4k images from
each dataset, resulting in a total of $T\\!=\\!12$k training images. During
training, we froze all the parameters but those related to the adapters and
the gate, i.e. the pre-trained encoder and decoder are not refined. We trained
gate and adapters for $400$ epochs using Adam with an initial learning rate of
$10^{-4}$, halving it when reaching a plateau with the patience of 15 epochs
and batch size of $16$ and with $\gamma=0.5$ in (3). At inference time, we
assess the performance of our method over four distinct test datasets. The
target sketch and comic domains are represented by 100 random images from
ImageNet-Sketch and BAM that are not present in the training set. The source
domain is represented by the Kodak [20] and CLIC [21].The performance of our
method is reported in terms of RD curves, BD-Rate, and BD-PSNR. Concerning
complexity, the adapter modules include about $3.8$M parameters, i.e. $\sim
5.5$% of Zou  _et al._ decoder complexity. Similarly, the gate is composed of
$184$k parameters, i.e. $\sim 0.28$% of Zou  _et al._ encoder complexity.

(a) _Sketch_

(b) _Comic_

Figure 3: Rate-PSNR plots for our Adapter-based method vs Reference for Zou
_et al._ (blue) and Cheng  _et al._ (orange), considering Sketch (a) and Comic
(b).

Figure 4: BD-Rate/PSNR for different domains.

_Dataset_ | _Domain_ | _Zou _et al.__ | _Cheng _et al.__
---|---|---|---
|  | BD-Rate | BD-PSNR | BD-Rate | BD-PSNR
_Kodak_ | _Natural_ | 0.0012 | $\sim 0$ | -1.46 | 0.06
_clic_ | _Natural_ | 0.038 | $\sim 0$ | -1.75 | 0.08
_Sketch_ | _Sketch_ | -2.45 | 0.1718 | -11.55 | 0.69
_Comic_ | _Comic_ | -4.93 | 0.28 | -19.15 | 1.25

Figure 5: Avg. probabilities predicted by $\varphi$.

Figure 6: (a) BD-Rate/PSNR for different domains, considering both Zou  _et
al._ and Cheng  _et al._ as anchor, with our models as reference. (b) Avg.
probabilities $\mathbf{v}$ predicted by the Gate $\varphi$, starting from Zou
_et al._ for the second-best quality model.

###  4.2 Rate-Distortion Performance

Fig. 3 shows the RD performance of our method for the Zou  _et al._ and Cheng
_et al._ models over the target sketch and comic domains. Our method visibly
improves the performance for both the target domains to the pre-trained model.
Table 6 summarizes the two models’ performance in terms of BD-Rate and BD-PSNR
gains over the four test datasets. Our method strikes rate reductions of 5%
and 10% over the comic and sketch domains. Yet, we achieve some gain also over
the source domain(Kodak and Clic datasets) for Cheng  _et al._ , proving our
domain adaptation method does not incur in any _catastrophic forgetting_. The
BD-Rate reduction is remarkable Especially for Cheng  _et al._ : we
hypothesize this model is less able to generalize outside the source domain
than Zou  _et al._ , hence our measured gains. Fig. 6 shows how the gate
network blends the outputs of the adapters across different domains for Zou
_et al._. For all domains the adopted training policy induces the exploitation
of all the available adapters to increase the performance since; even though
the gate is capable of identifying the predominant domain for an image, it
still utilizes the others to a lesser extent. Fig. 7 shows samples of decoded
images: our method preserves best the fine-grained details over the target
domains.

(a) Original Image

(b) Zou  _et al._ PSNR: 25.62

(c) Prop. PSNR: 25.90

Figure 7: Reconstruction of a sample taken from BAM [19] (from Comic domain).
(a) Original, (b) Zou  _et al._ , (c) Proposed. Both (c) and (d) have bpp
equal to 0.31.

_Dataset_ | _Reference_ | _Blending method_
---|---|---
|  | Proposed | Top1 | Oracle
_Kodak (Natural)_ | 35.947 | 35.945 | 35.948 | 35.950
_Clic (Natural)_ | 36.618 | 36.613 | 36.614 | 36.618
_Sketch_ | 37.925 | 38.112 | 38.005 | 38.109
_Comic_ | 36.833 | 37.161 | 37.138 | 37.141

Table 1: PSNR on different test sets considering different blending policies.

###  4.3 Comparison with other blending policies

Next, we assess the impact of the policy used to blend the adapters modules
outputs. We recall that with our proposed method, the outputs of the adapters
are weighted according to the probability distribution predicted by the gate
network $\varphi$. We now consider two other blending approaches, namely
_top1_ and _oracle_. The former considers only the output of the adapter
corresponding to the top-scoring class predicted by the gate, dropping the
contributions from other adapters. The latter is a hypothetical scheme that
relies on the assumption that the image class label is known at testing time,
discarding any input from the gate. We highlight that these policies are used
not only at inference but also during the training phase. Tab. 1 shows the
results in terms of PSNR considering the second highest quality model in Fig.
3. The _proposed_ policy outperforms the other two for both target domains,
meaning that exploiting all the adapters with the right blending maximizes the
image quality. On the other hand, the oracle policy yields the best image
quality over natural images, i.e. the source domain. However, the oracle
policy relies on the hypothesis that the image domain is known for each image
to encode, which is unrealistic in most cases.

###  4.4 Unseen domains

We evaluate our methods for domains unseen both when pretraining the Zou  _et
al._ model and at the time of training the adapters and the gate over the two
target domains. We considered different types of images from different
datasets like BAM [19], DomainNet [22], and IIIT-AR-13K [23]. Tab. 2 shows
that for every domain our method yields better encoding efficiency, even if
sometimes minimally. However, for some domains the gains are remarkable; for
example, the 20% gain for _Quickdraw_ is likely to be attributed to its
semantic similarity to the _Sketch_ domain. This hypothesis is supported by
the probability distribution generated by the gate network $\varphi$, where
approximately 90% of the image likelihood is assigned to the adapter
associated with the Sketch domain. For its possible practical application, an
important result is the BD-Rate equal to $-2.45$ on [23], representing
graphical and documents images: in this case, we see how $\varphi$ associated
this domain more with the Sketch class, which is intuitive concerning the
contents of this dataset. In general, The addition of domain-specific adapters
also brings further performance improvement, and therefore knowledge, in other
typologies of images, thanks to increased generalization capacity and the
capacity of the gate to spot correlations with known domains.

_Dataset_ | _Bjontegaard Metrics_ | _Predicted domain distribution_
---|---|---
| BD-Rate | BD-PSNR | Natural | sketch | comic
_Infograph_ [22] | -0.44 | 0.007 | 48 | 3 | 49
_Bam-Drawing_ [19] | -2.789 | 0.216 | 12 | 42 | 46
_Quickdraw_ [22] | -21.574 | 1.95 | 0 | 93 | 7
_watercolor_ [19] | -0.61 | 0.02 | 49 | 4 | 17
_clipart_ [22] | -1.32 | 0.065 | 15 | 0.07 | 78
_IIIT-AR-13K_ [23] | -2.45 | 0.20 | 15 | 50 | 35

Table 2: _First two columns_ : Bjontegaard Metrics for unseen domains,
considering [10] as reference. _Second three columns_ : average $\mathbf{v}$
considering highest quality model.

##  5 Conclusions and future works

This study addressed the problem of domain adaptation in learned image
compression. Our method outperforms reference pre-trained state-of-the-art
models on the target domains without forgetting the source one (Sec. 4.2),
showing also (Sec. 4.3) that aggregating the adapters brings a slight
performance improvement. In Sec. 4.4 we showed that our method offers
advantages also over unseen domains.This method could benefit from
improvements like unsupervised gate training to remove the need for predefined
adapter classes. Additionally, domain adaptation at the encoding stage could
enhance performance by modifying the entropy estimation.

## Acknowledgements

This work was partially funded by the Hi!PARIS Center on Data Analytics and
Artificial Intelligence.

##  6 References

## References

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# Spacecraft floating potential measurements for the _Wind_ spacecraft

Lynn B. Wilson III NASA Goddard Space Flight Center, Heliophysics Science
Division, Greenbelt, MD, USA.  Chadi S. Salem Space Sciences Laboratory,
University of California, Berkeley, CA 94720-7450, USA.  John W. Bonnell Space
Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA.

###### Abstract

Analysis of 8,804,545 electron velocity distribution functions (VDFs),
observed by the _Wind_ spacecraft near 1 AU between January 1, 2005 and
January 1, 2022, was performed to determine the spacecraft floating potential,
$\phi{{}_{sc}}$. _Wind_ was designed to be electrostatically clean, which
helps keep the magnitude of $\phi{{}_{sc}}$ small (i.e., $\sim$5–9 eV for
nearly all intervals) and the potential distribution more uniform. We observed
spectral enhancements of $\phi{{}_{sc}}$ at frequencies corresponding to the
inverse synodic Carrington rotation period with at least three harmonics. The
2D histogram of $\phi{{}_{sc}}$ versus time also shows at least two strong
peaks with a potential third, much weaker peak. These peaks vary in time with
the intensity correlated with solar maximum. Thus, the spectral peaks and
histogram peaks are likely due to macroscopic phenomena like coronal mass
ejections (solar cycle dependence) and stream interaction regions (Carrington
rotation dependence). The values of $\phi{{}_{sc}}$ are summarized herein and
the resulting dataset is discussed.

plasmas — (Sun:) solar wind

††journal: ApJS

Rough Draft 50pt

##  1 Background and Motivation

The solar wind is a nonequlibrium, kinetic, ionized gas comprised of electrons
and multiple ion species (with multiple charge states) from hydrogen up
through iron (e.g., Bame et al., 1968; Bochsler et al., 1985; Gloeckler et
al., 1998; Lepri et al., 2013). The solar wind is not in equilibrium because
it weakly collisional (e.g., Maruca et al., 2013; Salem et al., 2003),
allowing for the consistent presence of finite heat fluxes and the
temperatures of species $s^{\prime}$ and $s$ are not equal, i.e.,
$\left(T{{}_{s^{\prime}}}/T{{}_{s}}\right){{}_{tot}}$ $\neq$ 1, for
$s^{\prime}$ $\neq$ $s$ (see Appendix app:Definitions for parameter
definitions) (e.g., Wilson III et al., 2018, 2019a, 2019b).

Spacecraft propagating through space experience a net charging, called the
spacecraft potential or $\phi{{}_{sc}}$ (e.g., Besse & Rubin, 1980; Garrett,
1981; Geach et al., 2005; Génot & Schwartz, 2004; Grard, 1973; Grard et al.,
1983; Lai et al., 2017; Lavraud & Larson, 2016; Pedersen, 1995; Pulupa et al.,
2014; Salem et al., 2001; Scime et al., 1994; Scudder et al., 2000; Whipple,
1981). On average, when the spacecraft is in sunlight there is a net-zero
current balance between the ambient plasma and the spacecraft. The current
sources are thermal currents from ambient plasma species, $j{{}_{th,s}}$, bulk
flow velocity currents if in regions like the solar wind, $j{{}_{o,s}}$,
photoelectron currents due to the photoelectric effect, $j{{}_{ph}}$, and
currents from emission of secondary particles from impact ionization
$j{{}_{2nd,s}}$. The magnitude of $\phi{{}_{sc}}$ depends upon the
electromagnetic cleanliness of the spacecraft, the space environment, and
whether the spacecraft is in sunlight or the shadow of some object. Generally,
however, the better the electromagnetic cleanliness the smaller in magnitude
$\phi{{}_{sc}}$ will be, on average. An accurate measure of $\phi{{}_{sc}}$ is
critical for properly calibrating thermal electron detectors and calculating
accurate electron velocity moments (e.g., Geach et al., 2005; Génot &
Schwartz, 2004; Lavraud & Larson, 2016; Pulupa et al., 2014; Salem et al.,
2001; Scime et al., 1994).

Modern spacecraft equipped with electric field probes can directly measure
$\phi{{}_{sc}}$ versus time, e.g., the Magnetospheric Multiscale (MMS) mission
(e.g., Ergun et al., 2016; Lindqvist et al., 2016). This requires a constant
measurement of the quasi-static, DC-coupled electric potential versus time.
Many earlier spacecraft were much more restricted in onboard memory, telemetry
rates, and receiver electronics. Thus, it is often required that researchers
infer the spacecraft potential by relying on discrepancies between two
measurements of the same quantity111Often it is the difference between the
total electron density derived from the quasi-thermal noise upper hybrid line
and the measured density from a particle instrument. (e.g., see Salem et al.,
2001, 2023) or directly measuring the separation between ambient and
photoelectrons with something like a thermal electron detector (e.g., Phillips
et al., 1993). The former technique requires both instruments be very well
calibrated while the latter technique requires the instrument minimum energy,
$E{{}_{min}}$, be smaller than $\phi{{}_{sc}}$ for all $\phi{{}_{sc}}$ $>$ 0
eV.

Figure 1: Example electron energy distribution function (EDF) from the _Wind_
3DP instrument. The multi-colored thin lines show the phase space density
[$\\#\ cm^{-3}\ km^{-3}\ s^{+3}$] of each of the 88 solid angle bins versus
energy, $E$ [$eV$]. The photoelectrons are seen to the left of the vertical
magenta line marking the spacecraft floating potential, $\phi{{}_{sc}}$. The
thick green and black lines show simplified model EDFs for the photoelectrons
(following method in Wilson III et al., 2022) with parameter ranges shown in
the legend of the plot.

For most spacecraft operating in the solar wind, there are few (if any) energy
bins that satisfy $E$ $\leq$ $\phi{{}_{sc}}$, so the proper shape or form of
the velocity distribution function (VDF) for the photoelectrons is not well
known (e.g., Phillips et al., 1993). The simplest approach is to model them as
a single, isotropic Maxwellian-Boltzmann distribution (e.g., Grard, 1973;
Halekas et al., 2020; Wilson III et al., 2022). There is some evidence that
the photoelectron distribution in the solar wind is not a single Maxwellian
but may be at least two populations (Salem et al., in preparation), consistent
with magnetospheric observations (e.g., Pedersen, 1995). However, it is beyond
the scope of the current study to determine the shape of photoelectron VDF but
it is a topic of active investigation.

This paper is outlined as follows: Section sec:DefinitionsDataSets introduces
the datasets and methodology; Section sec:SolarWindStatistics presents some
preliminary long-term solar wind statistics; and Section
sec:DiscussionandConclusions discusses the results and interpretations with
recommendations for future measurement/instrumentation requirements.
Appendices are also included to provide additional details of the parameter
definitions (Appendix app:Definitions), some additional one-variable
statistics (Appendix app:ExtraStatistics), and a detailed description of the
public dataset generated from the analysis in this study (Appendix
app:PublicDataset).

##  2 Data Sets and Methodology

###  2.1 Data Sets

Nearly all data presented herein were measured by the _Wind_ spacecraft
(Wilson III et al., 2021) near 1 AU in the solar wind, specifically the
thermal electron detector EESA Low, part of the 3DP instrument suite (Lin et
al., 1995). The data are taken from both burst and survey modes, which has
cadences of $\sim$3 seconds and $\sim$24–78 seconds, respectively222All
products discussed herein start from the _Wind_ 3DP level zero files found at
http://sprg.ssl.berkeley.edu/wind3dp/data/wi/3dp/lz/.. All EESA Low
distributions have spin period durations or total integration times (i.e.,
$\sim$3 seconds for entire mission). All VDFs have 15 energy bins and 88 solid
angle look directions with typical resolutions of $\Delta\ E$/$E$ $\sim$ 20%
and $\Delta\ \phi$ $\sim$ 5∘–22.5∘ depending on the poloidal anode (i.e., the
ecliptic plane bins have higher angular resolution than the zenith). All in
situ measurements presented herein are shown in the spacecraft frame of
reference.

We also present observations of solar emission lines from the Thermosphere
Ionosphere Mesosphere Energetics Dynamics (TIMED) spacecraft, specifically the
Solar EUV Experiment (SEE) (Woods et al., 2005). The data are used as a proxy
for solar cycle and variations in ionizing photon and photoelectron flux
versus time.

We examined all electron energy distribution functions (EDFs) between January
1, 2005 and January 1, 2022 corresponding to a total of 8,804,545 EDFs. We
chose this time range because the start is well after _Wind_ moved to the
first Sun-Earth Lagrange point permanently and the detector $E{{}_{min}}$ was
below $\sim$6 eV. The lower bound on $E{{}_{min}}$ ranged from $\sim$3.2–5.2
eV for all EDFs examined. It is critical that $E{{}_{min}}$ $<$
$\phi{{}_{sc}}$ otherwise we can only define an upper bound estimate for
$\phi{{}_{sc}}$ from the thermal electron measurements.

###  2.2 Photoelectron Identification

All EDFs were plotted to a computer screen and viewed, “by eye”, in rapid
succession333The plots were viewed more like a high frame rate movie than a
series of individual images to make the effort feasible. to determine some
overarching properties of the electrons. Several important features were
observed from this effort. The first is that $\phi{{}_{sc}}$ $<$ 30 eV for
nearly every EDF of the 8,804,545 EDFs examined herein. Thus, we imposed an a
priori maximum energy, $E{{}_{max}}$, for any $\phi{{}_{sc}}$ solution of 30
eV444Previous solar wind estimates/measurements of $\phi{{}_{sc}}$ in the
solar wind typically fall below $\sim$10–15 eV (e.g., Geach et al., 2005;
Pedersen, 1995; Salem et al., 2001).. Second, we observed that the EDFs had
four characteristic shapes/profiles (discussed in more detail below) that
allowed us to simplify our approach.

To provide the reader with some context, Figure fig:ExamplePhotoelectrons
shows an example EDF with example model predictions for the photoelectrons.
The electron data are shown as the thin, multi-colored lines while the
photoelectron model and $\phi{{}_{sc}}$ solution are shown with thick, color-
coded lines. The data are shown in units of phase space density for ease of
modeling since these are the natural units of a Maxwell-Boltzmann
distribution. The details of the model are discussed in Wilson III et al.
(2022). The model parameters are shown in the figure legend in the upper
right-hand corner of the plot (see Appendix app:Definitions for symbol
definitions). The purpose of Figure fig:ExamplePhotoelectrons is to provide
the reader with context as to which parts of the EDF are photoelectrons versus
ambient solar wind electrons.

One may ask how we know the data falling between the thick green and black
lines are photoelectrons and not some cold, ambient solar wind population. We
can measure the total electron density, $n{{}_{e}}$, from the upper hybrid
line (e.g., Meyer-Vernet & Perche, 1989) observed by the _Wind_ WAVES
instrument (Bougeret et al., 1995). We can invert the frequency of this line,
knowing the quasi-static magnetic field, to calculate the electron plasma
frequency and thus the electron density. Doing this gives us $n{{}_{e}}$
$\sim$ 7.95 cm-3. For comparison, a typical value for the photoelectron number
density, $n{{}_{ph}}$, is $>$200 cm-3 and the mean kinetic energy of the
photoelectrons, or their temperature, $T{{}_{ph}}$, is $\sim$1 eV (e.g.,
Grard, 1973; Grard et al., 1983).

To further illustrate why the electrons between the thick lines are not part
of the ambient solar wind, we can numerically integrate the EDF following the
usual approach of adjusting the distribution by $\phi{{}_{sc}}$ then
calculating the velocity moments (e.g., see Wilson III et al., 2019b, 2022).
We find the integrated velocity moments are $n{{}_{e,int}}$ $\sim$ 7.01 cm-3
and $T{{}_{e,int}}$ $\sim$ 22.5 eV. If we do not remove the photoelectrons and
do not adjust the energies prior to integration, we find $n{{}_{e,bad}}$
$\sim$ 39.2 cm-3 and $T{{}_{e,bad}}$ $\sim$ 7.34 eV. Since the upper hybrid
line density is $\sim$7.95 cm-3, it is clear that the electrons below
$\sim$7.41 eV are photoelectrons.

Figure 2: Illustrative example of an electron energy distribution function
(EDF) showing the photoelectrons and the ambient electrons in the solar wind
near 1 AU. The red solid line shows the median of the energy flux,
$f{{}_{e}}\left(E\right)$ [$eV^{+1}\ cm^{-2}\ s^{-1}\ sr^{-1}\ eV^{-1}$], at
each energy over all 88 solid angle bins versus energy, $E$ [$eV$], on a log-
log scale. The vertical lines indicate values or ranges of values for the
estimated $\phi{{}_{sc}}$ of this EDF (see labels). The magenta lines (Method
1, see text for definition) show the region bounding the region of positive
curvature (note left-hand bound covered by green dashed line). The orange
lines show the region bounding the transition from negative to positive slope
(Method 2). The green lines show the region bounding the minimum-to-maximum
curvature transition (Method 3). Finally, the blue line shows the point of
minimum in $f{{}_{e}}\left(E\right)$ (Method 4).

Figure fig:ExampleEDF shows an example EDF to illustrate how we identify the
region in energy filled by photoelectrons versus those with ambient electrons.
The data in Figure fig:ExampleEDF (solid red line) show the median at each
energy of all the thin, multi-colored lines (i.e., the 88 solid angle bins)
from Figure fig:ExamplePhotoelectrons but in units of energy flux555Energy
flux was used when calculating $\phi{{}_{sc}}$ as the changes in
$f{{}_{e}}\left(E\right)$ tend to be more pronounced than in units of phase
space density.. For brevity, we will call this $f{{}_{med}}\left(E\right)$.
There are also multiple vertical lines in Figure fig:ExampleEDF that will be
explained below. Figure fig:ExampleEDF is an ideal case where there is a clear
discontinuity in the data, below which one sees a sharply increasing
$f{{}_{e}}\left(E\right)$ with decreasing $E$. As explained in the discussion
above and referenced citations, this “kink” in the EDF roughly marks the point
corresponding to $\phi{{}_{sc}}$.

###  2.3 Floating Potential Calculation

The EESA Low detector returns measurements for 88 solid angle look directions
and 15 energy bins. To simplify and reduce computation requirements, we
construct a pitch-angle distribution (PAD) for each EDF where we average the
solid angle look directions within $\pm$22.5∘ of the parallel, perpendicular,
and anti-parallel directions (with respect to the quasi-static magnetic field
vector, $\mathbf{B}{{}_{o}}$). This reduces the number of computations for
each EDF by a factor of $\sim$30\. Note that when we examine an arbitrary EDF
and view all 88 solid angles, the profiles/shapes are nearly always the same.
That is, if one solid angle direction looks like the EDF in Figure
fig:ExampleEDF, the other 87 will as well.

To keep the approach simple and focus on finding accurate but quick solutions
for $\phi{{}_{sc}}$, we use four methods to determine $\phi{{}_{sc}}$, which
is a rough boundary between photoelectron and ambient electrons. Note that
each of the four $\phi{{}_{sc}}$ methods illustrated in Figure fig:ExampleEDF
will have at least three values. The four methods are as follows:

Method 1

find the range of energies where $\tfrac{d^{2}f}{dE^{2}}$ $>$ 0, or the
positive curvature region (dashed magenta lines in Figure fig:ExampleEDF);

Method 2

find the range of energies where $\tfrac{df}{dE}$ transitions from negative to
positive, i.e., find the local minimum of $f{{}_{e}}\left(E\right)$ (dashed
orange lines in Figure fig:ExampleEDF);

Method 3

find the range of energies bounding the minimum and maximum values of
$\tfrac{d^{2}f}{dE^{2}}$, i.e., region of minimum and maximum curvature
(dashed green lines in Figure fig:ExampleEDF); and

Method 4

find the local minimum in $f{{}_{e}}\left(E\right)$ between $E{{}_{min}}$ and
$E{{}_{max}}$ (solid blue line in Figure fig:ExampleEDF).

Figure 3: Characteristic EDFs representing the four shapes observed herein,
shown as the red solid lines. All $f{{}_{e}}\left(E\right)$ were normalized to
the peak magnitude prior to calculation for this figure. Type A is the ideal
case, illustrated in Figure fig:ExampleEDF. Type B is the next best shape but
often shows a shoulder or saddle-point rather than a local minimum, increasing
the uncertainty of the $\phi{{}_{sc}}$ estimates. Type C1 has $\phi{{}_{sc}}$
$<$ $E{{}_{min}}$ and the lowest energy bin has $\tfrac{df}{dE}$ $\sim$ 0\.
Type C2 also has $\phi{{}_{sc}}$ $<$ $E{{}_{min}}$ but the lowest energy bin
has $\tfrac{df}{dE}$ $>$ 0\. Both Type C1 and C2 are not considered useful
shapes as we can only constrain the upper bound on $\phi{{}_{sc}}$. The green
line in each panel represents the median of all EDFs for each Type. The
blue(magenta) line in each panel represents the corresponding 5th(95th)
percentile of all EDFs of each type.

###  2.4 Assigning Type Labels

Figure fig:FourShapes shows the EDF profile of the four characteristic shapes
we defined in this study (all lines have been normalized by their respective
peak values). During the examination of the EDFs, we observed two useful
shapes that satisfy $E{{}_{min}}$ $<$ $\phi{{}_{sc}}$ (i.e., Types A and B in
Figure fig:FourShapes) and two not so useful shapes/profiles that satisfy
$E{{}_{min}}$ $>$ $\phi{{}_{sc}}$ (i.e., Types C1 and C2 in Figure
fig:FourShapes). We chose, at random, one example EDF for each of the four
characteristic shapes. We then calculate $f{{}_{med}}\left(E\right)$ (i.e.,
same as done for Figure fig:ExampleEDF) to construct an example type, shown as
the red lines in each panel of Figure fig:FourShapes. We then compared these
example types to all 8,804,545 EDFs to assign the type labels accordingly
(method for assignment discussed below). After the EDFs were assigned a type
label, we needed to further validate that the assignments are correct. We
calculated one-variable statistics of $f{{}_{med}}\left(E\right)$ at each
energy for all EDFs in each type category666One can think of this similar to a
superposed epoch analysis but performed on EDFs. It allows us to statistically
examine the EDFs from the resulting labeling scheme without checking each EDF
“by eye” to ensure they exhibit the proper shape.. For instance, for all EDFs
labeled Type A, there will be N values of $f{{}_{med}}\left(E{{}_{j}}\right)$
at energy $E{{}_{j}}$. We can calculate the one-variable statistics of
$f{{}_{med}}\left(E{{}_{j}}\right)$ to get things like the mean ($\bar{X}$),
median ($\tilde{X}$), 5th percentile ($X{{}_{5\%}}$), 95th percentile
($X{{}_{95\%}}$), etc. (see Appendix app:Definitions for symbol definitions).
Then for, say, $X{{}_{5\%}}$ we can construct a line of 15 points (one for
each energy) for this specific one-variable statistic value. We can do this
for any of the one-variable statistic values. We chose the values of
$X{{}_{5\%}}$, $\tilde{X}$, and $X{{}_{95\%}}$, which are shown as the blue,
green, and magenta lines, respectively, in Figure fig:FourShapes. As one can
see, the profile of the blue, green, and magenta lines for each EDF type
follows that of the characteristic example used to assign the type label.
Thus, the assignments are validated as being statistically correct.

We needed a quick method for identifying all the EDFs with an associated type
label. This is necessary to determine which of the 8,804,545 values of
$\phi{{}_{sc}}$ we can trust. The quickest approach is to calculate the cross-
correlation coefficient (CC) between each characteristic example
$f{{}_{med}}\left(E\right)$ (shown as red lines in Figure fig:FourShapes) and
the $f{{}_{med}}\left(E\right)$ for all 8,804,545 EDFs. We do this in the
following steps:

  1. 1.

normalize all 8,804,545 $f{{}_{med}}\left(E\right)$ by their respective peak
values to avoid vertical offsets when comparing to each characteristic example
$f{{}_{med}}\left(E\right)$ (also normalized by peak value);

  2. 2.

calculate 8,804,545 CCs for each characteristic example
$f{{}_{med}}\left(E\right)$ allowing for seven energy offsets (i.e., allows
each EDF to shift left and right in energy), generating a [N,7]-element array
of CCs for each characteristic example $f{{}_{med}}\left(E\right)$;

  3. 3.

find the maximum CC by energy offset then by characteristic example
$f{{}_{med}}\left(E\right)$ to determine the associated shape/type for each
EDF; and

  4. 4.

then ensure that the slope777Note that this slope requirement supersedes the
CC magnitude ranking. of the first few energy bins for Types A, B, and C2
match that expected from the characteristic example
$f{{}_{med}}\left(E\right)$ (i.e., Types A and B must have a negative slope,
Type C2 must have a positive slope).

We did not bother with further constraints or complications because as we will
discuss below, there are many fewer Type C1 and Type C2 EDFs than Type A or B.
The one-variable statistics of all EDFs for each type shown as the blue,
green, and magenta lines in Figure fig:FourShapes indicate that our type label
assignments are statistically correct.

The one-variable statistics of all CC values for each type are shown below in
the following form $X{{}_{5\%}}$–$X{{}_{95\%}}$($\bar{X}$)[$\tilde{X}$] (see
Appendix app:Definitions for symbol definitions)

Type A

0.9915–0.9998(0.9965)[0.9969];

Type B

0.9918–0.9998(0.9968)[0.9975];

Type C1

0.9864–0.9996(0.9957)[0.9974];

Type C2

0.9832–0.9997(0.9933)[0.9944];

Type C1 & C2

0.9833–0.9997(0.9938)[0.9955].

Figure fig:CCHistograms shows the one-dimensional histograms of the CCs for
each type, illustrating they have distinct distributions. For both Types A and
B, the distribution is a narrow peak with $\sim$95% of all CCs above
$\sim$0.991 and very weak (if any) tails to lower CC magnitudes. Both Type C1
and C2 had much broader peaks and stronger tails extending to lower CC
magnitudes (especially C1). This further validates the identification scheme
discussed above.

Figure 4: Histograms of the CC values for each type are shown as percentages
of the total number of counts for each type.

Thus, our labeling scheme identified, from all 8,804,545 EDFs, that there are:

Type A

4,014,873(45.60%) EDFs;

Type B

4,213,734(47.86%) EDFs;

Type C1

127,514(1.45%) EDFs;

Type C2

448,424(5.09%) EDFs; and

Type C1 & C2

575,938(6.54%) EDFs.

Given the small number of Type C1 and C2 EDFs, we did not further enhance the
differentiation algorithms. The statistical shapes shown in Figure
fig:FourShapes illustrate that our algorithm works and that we have properly
classified the EDFs. Therefore, we found no need to refine further.

We define a discrete quality flag for each set of $\phi{{}_{sc}}$ as follows:

QF $=$ 4

All EDFs of Type A;

QF $=$ 2

All EDFs of Type B; and

QF $=$ 0

All EDFs of Type C1 or C2.

Again, this is is to simplify the process and use of the dataset. It is to
help the user determine whether to use or not use any given $\phi{{}_{sc}}$
solution. We did not feel it was useful to delineate the quality flags further
as all Type Cs are one-sided constrained and all Type As are good. Type B
solutions are reliable and most are good but a few had weaker saddle points
than others, thus we used a less confident QF than for Type A.

##  3 Solar Wind Statistics

In this section we present some preliminary statistical analysis of our
$\phi{{}_{sc}}$ estimates.

Table tab:AllEDFs shows the one-variable statistics of $\phi{{}_{sc}}$ for all
8,804,545 EDFs, i.e., not distinguishing by EDF type888We also list the one-
variable statistics for the methods separated by EDF type in Appendix
app:ExtraStatistics.. Note that the $\tilde{X}$ values for Method 2 and Method
3 nicely bound the values for Method 4. Method 1 has a much broader range of
values but this is not surprising as Method 1 can be ill-constrained except
for EDF Types A and C2. The main takeaway here is that the spacecraft
potential for the _Wind_ spacecraft near 1 AU in the solar wind typically
satisfies 5 eV $\lesssim$ $\phi{{}_{sc}}$ $\lesssim$ 13 eV, consistent with
previous studies (e.g., Geach et al., 2005; Pedersen, 1995; Salem et al.,
2001).

Figure fig:2DHist shows the 2D histograms of the $\phi{{}_{sc}}$ estimates for
Method 1–Method 4 (shown in order). The histogram bin sizes are four weeks for
the horizontal (time) axis and $\sim$2 eV (i.e., 16 uniformly sized bins
between 0 eV and 30 eV) for the vertical axis. White bins indicate no data
whereas colors are quantified in the logarithmic scale to the right. Note that
blue and purple are only a few tens of counts to single digits whereas the
yellow-to-red portion are in the thousands. Thus, the latter are statistically
significant while the former have low signal-to-noise ratios.

Recall that Method 1–Method 3 result in a range of values, i.e., two per EDF.
Therefore, to construct these histograms for Method 1–Method 3, we calculate
the median over the three PAD directions first, then take the average of the
range of values. For Method 4, we take the median from the three pitch-angles
and then construct the histogram. The bottom two panels show solar irradiance
measurements over the wavelength range of 0.1–7.0 nm (Diode 1) and $\sim$58.4
nm (He I) observed by the TIMED SEE experiment.

Table 1: All EDF Types

$\phi{{}_{sc}}$ [eV] |  $X{{}_{min}}$ aaminimum |  $X{{}_{5\%}}$ bb5th percentile |  $\bar{X}$ ccmean |  $\tilde{X}$ ddmedian |  $X{{}_{95\%}}$ ee95th percentile |  $X{{}_{max}}$ ffmaximum
---|---|---|---|---|---|---
All 8,804,545 EDFs, Lower Bound
Method 1 | 1.00 | 5.18 | 5.05 | 5.18 | 5.18 | 17.3
Method 2 | 1.00 | 5.42 | 7.98 | 7.09 | 12.1 | 27.1
Method 3 | 1.00 | 5.18 | 5.24 | 5.18 | 5.93 | 16.6
Method 4 | 4.91 | 5.93 | 8.03 | 7.41 | 12.7 | 29.7
All 8,804,545 EDFs, Upper Bound
Method 1 | 4.91 | 16.6 | 28.5 | 29.7 | 29.7 | 29.7
Method 2 | 4.91 | 5.93 | 8.88 | 7.75 | 13.3 | 29.7
Method 3 | 4.91 | 5.93 | 12.7 | 12.7 | 18.1 | 29.7
Method 4 | 4.91 | 5.93 | 8.92 | 7.41 | 18.1 | 29.7

Note. — For symbol definitions, see Appendix app:Definitions.

Note that Method 1 has, by construction, the largest range of values and often
is bounded by our the instrument $E{{}_{min}}$ and our assumed $E{{}_{max}}$
values. For EDFs of Type C1 and C2, the lower bound for Method 1–Method 3 is
often 1 eV999This was a default minimum we chose based upon the physically
justified (and empirically motivated) assumption that the spacecraft will not
charge negative in the solar wind in sunlight.. The midpoint over the range of
energies allowed is 15 eV. This is why there is a high density of counts near
15 eV for Method 1. Since these are constructed using four week bins, each
column can have multiple values, but this does not imply structure in the
photoelectron part of any individual EDF. That is, finite counts in multiple
rows of the histograms indicate that during any given four week interval
(i.e., one column), $\phi{{}_{sc}}$ varied a lot.

For EDFs of Type A, Method 2–Method 4 are consistent with Method 2 and Method
3 bounding Method 4. This is largely why these three methods show a strong
peak below $\sim$10 eV for all times. For EDFs of Type B, Method 2 and Method
4 are in the best agreement while Method 3 has difficulty limiting the bounds
of $\phi{{}_{sc}}$. Note that the repeated value of $\sim$4.91 eV in Table
tab:AllEDFs corresponds to $\sim$95% of most $E{{}_{min}}$ detector values.
This was a default guess solution in the software when it was clear that
$E{{}_{min}}$ $>$ $\phi{{}_{sc}}$ (i.e., for EDFs of Type C1 and C2). For
reference, previous work has shown that typical $\phi{{}_{sc}}$ magnitudes for
_Wind_ in the solar wind are below $\sim$10 eV (e.g., Salem et al., 2001;
Wilson III et al., 2019a), consistent with the $\phi{{}_{sc}}$ estimates for
Method 2–Method 4 shown in Figure fig:2DHist.

The $\phi{{}_{sc}}$ estimates in Figure fig:2DHist also show an interesting
solar cycle variation, consistent with other studies (e.g., Salem et al., in
preparation). We focus on the peaks in the yellow-to-red color range as they
have statistically significant counts. The histograms for Method 2 (second
panel from top) show a bifurcated structure101010That is, there are two peaks
in the same time bin. in $\phi{{}_{sc}}$ that varies in time (Method 4 shows
this as well, but to a weaker extent). The first bifurcated peak occurs before
2006, which corresponds to the tail end of solar cycle 23 (e.g., see trend in
Diode 1 panel). The second bifurcated peak starts just before 2012 and extends
to nearly 2016, which correponds to the peak in sunspot numbers (SSNs) for
solar cycle 24 (also illustrated in trend in Diode 1 panel). Note that solar
cycle 24 had a double-peaked structure in SSNs, with the first peak just
before 2012 and the second in early 2014. The last three years of the data
shown in Figure fig:2DHist are the start of solar cycle 25 corresponding to
solar minimum, thus only a weak bifurcated peak is starting to show. Note that
there is some hint of a third peak for Method 2 and Method 4 in Figure
fig:2DHist.

There are two environments known to enhance $\phi{{}_{sc}}$ in the solar wind,
low $n{{}_{e}}$ regions and the sheath regions downstream of collisionless
shock waves (e.g., Wilson III et al., 2019a). The low $n{{}_{e}}$ regions have
lower ambient electron currents which increases the relative importance of the
escaping photoelectron current, thus increasing $\phi{{}_{sc}}$ to more
positive values. The sheath regions downstream of shock waves have much
stronger ion currents, which increase $\phi{{}_{sc}}$ to more positive values.
Interplanetary (IP) shock waves driven by coronal mass ejections (CMEs) (e.g.,
Jian et al., 2006a; Richardson et al., 2018) and/or stream interaction regions
(SIRs) (e.g., Jian et al., 2006b; Richardson et al., 2018) are more common
during solar maximum than solar minimum, thus show a clear solar cycle
variation. Thus, the bifurcated peak likely occurs because of IP shock waves.
As stated above, we are not implying that the solar irradiance changes versus
time directly result in larger $\phi{{}_{sc}}$. Rather, the solar irradiance
measures are proxies for solar cycle variations which directly influence the
rate of CME and SIR occurrence.

The magnitude of $\phi{{}_{sc}}$ is observed to increase downstream of IP
shocks (e.g., Wilson III et al., 2019a), which is likely due to increases in
the ion currents111111That is, the ion temperature typically increases much
more than the electron temperature in shock sheath regions, which increases
the ion thermal currents more than the electron. relative to the electron
thermal and photoelectron currents. This drives the spacecraft more
electrically positive as it reaches a new current balanced equilibrium. It may
also be related to enhanced secondary electron generation internal to the
instrument as seen on other spacecraft in shock sheaths (e.g., Gershman et
al., 2017), but this is beyond the scope of the present study.

Figure 5: 2D histograms of the $\phi{{}_{sc}}$ estimates for Method 1–Method 4
(shown in order). The thick black line in each panel shows the $E{{}_{min}}$
value versus time. The horizontal (time) axis is parsed into four week bins
while the vertical axis is parsed into 16 uniformly sized bins between 0 eV
and 30 eV (i.e., roughly 2 eV each). The color scale shows total counts in
each 2D histogram bin. The bottom two panels show solar irradiances [$W\
m^{-2}$] measured with two different detectors from the TIMED SEE experiment.
These are shown as proxies for solar cycle variation and ionizing photon
intensities versus time.

As discussed before, Method 1–Method 3 provide a range of values while Method
4 is a single value. However, all of these have three solutions for each of
the three pitch-angle ranges discussed in Section sec:DefinitionsDataSets. To
provide a range of values for Method 4, we take the minimum and maximum values
of $E$ from the three pitch-angles. We can then calculate one-variable
statistics for each of the methods for all EDFs or by type and provide a range
of values for each one-variable statistic variable (e.g., this is in reference
to the values shown in Table tab:AllEDFs).

Figure 6: The power spectral density (PSD) [$eV^{+2}\ days^{-1}$] of the fast
Fourier transform (FFT) of all $\phi{{}_{sc}}$ estimates for all four EDF
types, using Method 4 (note this is the median of the three PADs), versus
frequency [$days^{-1}$] shown in purple. The red line is the smoothed PSD,
using a running boxcar average with an eight point bin width. Overplotted are
four vertical lines corresponding to the frequency of the synodic Carrington
rotation period ($\sim$27.2753 days) and its first three harmonics.

The main takeaway from Table tab:AllEDFs and Figure fig:2DHist is that for the
majority of the mission, the following is satisfied 5 eV $\lesssim$
$\phi{{}_{sc}}$ $\lesssim$ 13 eV, with peak occurrence rates in the $\sim$6–9
eV range. In fact, most of the intervals where $\phi{{}_{sc}}$ $\gtrsim$ 9 eV
correspond to the sheath regions of interplanetary shocks (e.g., see
discussion in Wilson III et al., 2019a) because the ion thermal currents are
much larger than in the ambient solar wind.

Figure fig:FFTofphisc shows the PSD of all $\phi{{}_{sc}}$ estimates for all
four EDF types (using Method 4) versus frequency. We also superposed the
corresponding frequency of the synodic Carrington rotation period
($\sim$27.2753 days) and its first three harmonics. One can immediately see
that $\phi{{}_{sc}}$ peaks at this frequency and its harmonics. The most
likely explanation is due to the variations in $j{{}_{th,s}}$ and
$j{{}_{o,s}}$ caused by SIRs (e.g., Jian et al., 2006b; Richardson et al.,
2018), which typically occur at intervals defined roughly by the Carrington
rotation of the Sun. We examined both the synodic ($\sim$27.2753 days) and
sidereal ($\sim$25.38 days), periods but the one of relevance is the synodic.
An interesting observation is that these peaks are much less well defined when
we construct the PSD using only Types A and B and are completely missing if we
look at Type C1 or C2 or both C1 and C2. It is not immediately clear why the
peaks change in amplitude or disappear altogether when isolating the different
EDF types. However, this is beyond the scope of the current study.

##  4 Discussion and Conclusions

We present measurements of the spacecraft floating potential for the _Wind_
spacecraft near 1 AU in the solar wind between January 1, 2005 and January 1,
2022 (i.e., 17 years). The analysis was performed as the first step in an
effort to properly calibrate the _Wind_ 3DP thermal electron detector, EESA
Low. We describe the resulting publicly available dataset and some preliminary
results of the long-term statistical trends.

The long-term, statistical estimate for $\phi{{}_{sc}}$ satisfies 5 eV
$\lesssim$ $\phi{{}_{sc}}$ $\lesssim$ 13 eV, with peaks in the $\sim$6–9 eV
range, consistent with previous measurements in the solar wind (e.g., Salem et
al., 2001). Note that part of the reason for the small magnitudes of
$\phi{{}_{sc}}$ is that _Wind_ was designed and built with strict requirements
for electromagnetic and electrostatic cleanliness (e.g., see Wilson III et
al., 2021, and references therein). This is an important consideration in
mission development and instrument calibration, as less stringent requirements
lead to larger $\phi{{}_{sc}}$, which will result in larger deviations between
the observed energy-angle of an incident particle versus its true energy-angle
(e.g., Barrie et al., 2019).

When we examined the statistical trends of $\phi{{}_{sc}}$ versus time, we
found a bifurcated peak that correlated with solar maximum121212A 1D cut
through the second peak for Method 1 in Figure fig:2DHist shows the count
rates of this higher peak vary in phase with the solar irradiance measures
shown at the bottom of Figure fig:2DHist, i.e., they are highly correlated..
This is most likely due to enhanced ion currents downstream of IP shocks
and/or regions of high ion temperature and low total electron density. We also
found peaks in the power spectral density at the frequency corresponding to
the synodic Carrington rotation period ($\sim$27.2753 days) and its first
three harmonics. The obvious candidates that depend upon Carrington rotation
and solar cycle are SIRs and CMEs.

A critical takeaway from this study is that future thermal electron instrument
design should require that $E{{}_{min}}$ $<$ $\phi{{}_{sc}}$ (for positive
$\phi{{}_{sc}}$) for the largest fraction of the measurements possible in any
given region of space. This is critical for two reasons. The first is that
when $E{{}_{min}}$ $<$ $\phi{{}_{sc}}$, the instrument is measuring all
ambient electrons within its field-of-view as they are accelerated to energies
of at least $e\phi{{}_{sc}}$ before entering the detector (for positive
$\phi{{}_{sc}}$). If $E{{}_{min}}$ $>$ $\phi{{}_{sc}}$, the instrument does
not measure all ambient electrons and, in fact, misses most of the velocity
distribution. This can result in highly inaccurate velocity moments and affect
the interpretation of any analysis performed on the data (e.g., Song et al.,
1997; Wilson III et al., 2022). There are ways to estimate $\phi{{}_{sc}}$
when $E{{}_{min}}$ $>$ $\phi{{}_{sc}}$ as previously discussed (e.g., see
Salem et al., 2001, 2023). However, it requires multiple assumptions and very
well calibrated instrumentation.

The second reason one should require that $E{{}_{min}}$ $<$ $\phi{{}_{sc}}$ is
that it gives at least one direct, independent measure of $\phi{{}_{sc}}$.
That is, the observed “kink-like” change in the EDF as illustrated in the
example in Figure fig:ExampleEDF does not depend upon corrections to the
instrument geometric factor or anode efficiencies. It only requires knowledge
of the energy bin values, which is one of the most well known parameters in a
particle detector. The ability to directly determine $\phi{{}_{sc}}$ from the
particle data enables missions that do not have a direct measure of
$\phi{{}_{sc}}$ from an electric field instrument to properly calculate
velocity moments and/or perform analysis on the electron velocity
distributions. It can also provide a sanity check to help calibrate electric
field measurements that directly measure $\phi{{}_{sc}}$.

As previous efforts have shown (e.g., Génot & Schwartz, 2004; Lavraud &
Larson, 2016; Song et al., 1997), properly correcting the electron
measurements for $\phi{{}_{sc}}$ is critical for calculating any velocity
moments and/or instability analysis. In fact, poor estimates of
$\phi{{}_{sc}}$ or low resolution instrumentation or instruments with
$E{{}_{min}}$ $>$ $\phi{{}_{sc}}$ can result in velocity moment errors from
$\sim$10–30% to over 100% (e.g., Song et al., 1997; Wilson III et al., 2022).
Therefore, it is essential that electron thermal particle detectors be
designed such that $E{{}_{min}}$ $<$ $\phi{{}_{sc}}$.

Analysis software (Wilson III, 2021) used herein can be found at:
https://github.com/lynnbwilsoniii/wind_3dp_pros.
The _Wind_ 3DP level zero data files are publicly available at:
http://sprg.ssl.berkeley.edu/wind3dp/data/wi/3dp/lz/.
Some of the results presented in this document rely on data measured from the
Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) Solar EUV
Experiment (SEE). These data are available from the TIMED SEE website at
https://lasp.colorado.edu/see/data/.
These data were accessed via the LASP Interactive Solar Irradiance Datacenter
(LISIRD) at:
https://lasp.colorado.edu/lisird/.
The resulting dataset of _Wind_ spacecraft floating potential measurements are
found at Wilson III et al. (2023). Work at the University of California
Berkeley was supported by NASA grant 80NSSC20K0708 and NSF grant 2203319.

##  Appendix A Definitions

In this appendix the symbols and notation used throughout will be defined. All
direction-dependent parameters we use the subscript $j$ to represent the
direction where $j$ $=$ $tot$ for the entire distribution, $j$ $=$ $\parallel$
for the the parallel direction, and $j$ $=$ $\perp$ for the perpendicular
direction, where parallel/perpendicular is with respect to the quasi-static
magnetic field vector, $\mathbf{B}{{}_{o}}$ [nT]. Below are the
symbol/parameters definitions:

  * one-variable statistics

    * –

$X{{}_{min}}$ $\equiv$ minimum

    * –

$X{{}_{max}}$ $\equiv$ maximum

    * –

$\bar{X}$ $\equiv$ mean

    * –

$\tilde{X}$ $\equiv$ median

    * –

$X{{}_{25\%}}$ $\equiv$ lower quartile

    * –

$X{{}_{75\%}}$ $\equiv$ upper quartile

    * –

$X{{}_{y\%}}$ $\equiv$ yth percentile

    * –

$\sigma$ $\equiv$ standard deviation

    * –

$\sigma^{2}$ $\equiv$ variance

  * fundamental parameters

    * –

$\varepsilon{{}_{o}}$ $\equiv$ permittivity of free space

    * –

$\mu{{}_{o}}$ $\equiv$ permeability of free space

    * –

$c$ $\equiv$ speed of light in vacuum [$km\ s^{-1}$] $=$
$\left(\varepsilon{{}_{o}}\ \mu{{}_{o}}\right)^{-1/2}$

    * –

$k{{}_{B}}$ $\equiv$ the Boltzmann constant [$J\ K^{-1}$]

    * –

$e$ $\equiv$ the fundamental charge [$C$]

  * plasma parameters

    * –

$n{{}_{s}}$ $\equiv$ the number density [$cm^{-3}$] of species $s$

    * –

$m{{}_{s}}$ $\equiv$ the mass [$kg$] of species $s$

    * –

$Z{{}_{s}}$ $\equiv$ the charge state of species $s$

    * –

$q{{}_{s}}$ $\equiv$ the charge [$C$] of species $s$ $=$ $Z{{}_{s}}\ e$

    * –

$T{{}_{s,j}}$ $\equiv$ the scalar temperature [$eV$] of the jth component of
species $s$

    * –

$V{{}_{Ts,j}}$ $=$ $\sqrt{\tfrac{2\ k{{}_{B}}\ T{{}_{s,j}}}{m{{}_{s}}}}$
$\equiv$ the most probable thermal speed [$km\ s^{-1}$] of a one-dimensional
velocity distribution

    * –

$\mathbf{V}{{}_{os}}$ $\equiv$ the drift velocity [$km\ s^{-1}$] of species
$s$ in the plasma relative to a specified rest frame

    * –

$\phi{{}_{sc}}$ $\equiv$ the scalar, quasi-static spacecraft potential [eV]
(e.g., Pulupa et al., 2014; Scime et al., 1994)

    * –

$E{{}_{min}}$ $\equiv$ the minimum energy bin midpoint value [eV] of a
particle detector

    * –

$E{{}_{max}}$ $\equiv$ the maximum energy allowed when finding $\phi{{}_{sc}}$
from the particle data

    * –

$j{{}_{th,s}}$ $=$ $q{{}_{s}}\ n{{}_{s}}\ V{{}_{Ts,tot}}$ $\equiv$ the thermal
current density [$\mu A\ m^{-2}$] due to the ambient plasma species $s$

    * –

$j{{}_{o,s}}$ $=$ $q{{}_{s}}\ n{{}_{s}}\ \lvert\mathbf{V}{{}_{os}}\rvert$
$\equiv$ current density [$\mu A\ m^{-2}$] due to ambient plasma species $s$
drifting relative to spacecraft

    * –

$j{{}_{ph}}$ $\equiv$ the photoelectron current density [$\mu A\ m^{-2}$]

    * –

$j{{}_{2nd,s}}$ $\equiv$ the current density [$\mu A\ m^{-2}$] due to
secondary particles of species $s$ emitted from the spacecraft due to impact
ionization

    * –

$f{{}_{s}}\left(E\right)$ $\equiv$ energy distribution function (EDF) of
particle species $s$

    * –

$f{{}_{med}}\left(E\right)$ $\equiv$ the median of $f{{}_{s}}\left(E\right)$
at each energy over all solid angles

##  Appendix B Extra Statistics

In this appendix we provide additional one-variable statistics of our
estimates for $\phi{{}_{sc}}$ separated by EDF types. Note that we only show
Types A and B here, as Types C1 and C2 have $E{{}_{min}}$ $>$ $\phi{{}_{sc}}$
and thus, we cannot properly measure $\phi{{}_{sc}}$ from the thermal electron
EDFs.

Table 2: Type A EDFs

$\phi{{}_{sc}}$ [eV] | $X{{}_{min}}$ | $X{{}_{5\%}}$ | $X{{}_{25\%}}$ | $\bar{X}$ | $\tilde{X}$ | $X{{}_{75\%}}$ | $X{{}_{95\%}}$ | $X{{}_{max}}$
---|---|---|---|---|---|---|---|---
All 4,014,873 Type A EDFs, Lower Bound
Method 1 | 1.00 | 1.00 | 5.18 | 4.96 | 5.18 | 5.18 | 5.18 | 17.3
Method 2 | 1.00 | 1.00 | 5.67 | 6.54 | 7.09 | 7.09 | 8.48 | 17.3
Method 3 | 1.00 | 1.00 | 5.18 | 5.14 | 5.18 | 5.93 | 5.93 | 12.7
Method 4 | 4.91 | 5.13 | 5.93 | 7.02 | 7.41 | 7.41 | 9.27 | 29.7
All 4,014,873 Type A EDFs, Upper Bound
Method 1 | 4.91 | 5.13 | 29.7 | 27.7 | 29.7 | 29.7 | 29.7 | 29.7
Method 2 | 4.91 | 5.13 | 6.20 | 7.37 | 7.75 | 7.75 | 9.27 | 19.0
Method 3 | 4.91 | 5.13 | 12.7 | 13.2 | 12.7 | 12.7 | 18.1 | 29.7
Method 4 | 4.91 | 5.13 | 7.09 | 7.28 | 7.41 | 7.41 | 9.27 | 29.7

Note. — Symbol definitions are the same as in Table tab:AllEDFs.

Table tab:TypeAEDFs shows one-variable statistics for only Type A EDFs while
Table tab:TypeBEDFs shows one-variable statistics for only Type B EDFs. Note
that we have added the 25th and 75th percentiles to these tables for extra
detail.

Table 3: Type B EDFs

$\phi{{}_{sc}}$ [eV] | $X{{}_{min}}$ | $X{{}_{5\%}}$ | $X{{}_{25\%}}$ | $\bar{X}$ | $\tilde{X}$ | $X{{}_{75\%}}$ | $X{{}_{95\%}}$ | $X{{}_{max}}$
---|---|---|---|---|---|---|---|---
All 4,213,734 Type B EDFs, Lower Bound
Method 1 | 1.00 | 5.18 | 5.18 | 5.18 | 5.18 | 5.18 | 5.18 | 6.48
Method 2 | 1.00 | 8.48 | 8.87 | 9.96 | 9.27 | 11.6 | 12.7 | 27.1
Method 3 | 1.00 | 5.18 | 5.18 | 5.31 | 5.18 | 5.18 | 5.93 | 16.6
Method 4 | 5.13 | 5.93 | 6.48 | 9.26 | 9.27 | 9.70 | 18.1 | 29.7
All 4,213,734 Type B EDFs, Upper Bound
Method 1 | 5.13 | 29.7 | 29.7 | 29.3 | 29.7 | 29.7 | 29.7 | 29.7
Method 2 | 5.13 | 9.27 | 9.70 | 10.9 | 10.1 | 12.7 | 13.9 | 29.7
Method 3 | 5.13 | 5.93 | 6.48 | 12.5 | 12.7 | 18.1 | 18.1 | 29.7
Method 4 | 5.13 | 5.93 | 6.48 | 10.7 | 9.70 | 12.7 | 18.1 | 29.7

Note. — Symbol definitions are the same as in Table tab:AllEDFs.

##  Appendix C Public Dataset

In this section we describe the variables that are included in the public
dataset (Wilson III et al., 2023) and list where to find them.

Below we describe the variables in the public dataset and we use generic
characters to define the number of elements in an array. We define $N$ as the
total number of EDFs (e.g., $\sim$8,804,545 for entire dataset, but will vary
by year), $P$ as the number of pitch-angle bins (i.e., 3), and $R$ as the
number of bounds or limits for any given estimtae (i.e., 2). The array
dimensions will be shown in brackets, with each dimension separated by a
comma. The public dataset will include the following variables:

Unix

[N,R]-Element (numeric) array of Unix131313Note that the times are not really
Unix times, as they include leap seconds from UTC time conversions. The UMN
software discussed in the Acknowledgements Section below handles this
accordingly. start and end times [seconds]

YMDB

[N,R]-Element (character string) array of UTC start and end times with format
‘YYYY-MM-DD/hh:mm:ss.xxx’

PosCurv

[N,P,R]-Element (numeric) array of $\phi{{}_{sc}}$ [eV] estimates calculated
from Method 1 (i.e., positive curvature region)

InflPnt

[N,P,R]-Element (numeric) array of $\phi{{}_{sc}}$ [eV] estimates calculated
from Method 2 (i.e., inflection point of $f{{}_{e}}\left(E\right)$)

MinMaxC

[N,P,R]-Element (numeric) array of $\phi{{}_{sc}}$ [eV] estimates calculated
from Method 3 (i.e., region of minimum and maximum curvature)

MinEflx

[N,P]-Element (numeric) array of $\phi{{}_{sc}}$ [eV] estimates calculated
from Method 4 (i.e., local minimum in $f{{}_{e}}\left(E\right)$)

MinEner

[N]-Element (numeric) array of the instrument $E{{}_{min}}$ values [eV]

MaxEner

[N]-Element (numeric) array of the $E{{}_{max}}$ values [eV] chosen a priori
to bound the solutions for $\phi{{}_{sc}}$

EDFType

[N]-Element (character string) array of EDF types (e.g., ‘A’ for Type A)

QFlag

[N]-Element (integer) array of quality flags based upon EDF types (e.g., 4 for
Type A)

Note that the values for each of these have not been altered from those
returned by the automated software performing these tests. For instance, we
have not adjusted results from Method 3 for EDFs of Type C1 or C2. We left the
original values in place and allow the user to alter the upper/lower bounds as
they see fit for their own purposes. In general, for EDFs of Type C1 or C2
with shapes matching that shown in Figure fig:FourShapes, $\phi{{}_{sc}}$ $<$
$E{{}_{min}}$ for all four methods. This is why they are given a QF of zero.
However, one should note that such EDFs and associated $\phi{{}_{sc}}$ are not
useless or meaningless. These times are intervals where we can confidently
define an absolute upper bound on $\phi{{}_{sc}}$. They also correspond to
times when $\phi{{}_{sc}}$ is very small, relative to the typical values shown
in Figure fig:2DHist and Table tab:AllEDFs.

The dataset will consist of a list of yearly ASCII files from January 1, 2005
to January 1, 2022. More years can be added as time progresses to expand the
dataset. This dataset is the first step towards generating a properly
calibrated electron velocity moment data product for public consumption. The
next step requires the knowledge of the total electron density from the upper
hybrid line, a dataset that is currently in production (e.g., see Martinović
et al., 2020, for details). The combination of accurate estimates from these
two quantities will allow us to update the anode calibrations141414In the unit
conversion process, this calibration enters through a modification to the
optical geometric factor for each energy-angle bin. It is important to note
that these calibrations do not affect the detector energy bin values, i.e.,
they will not affect $\phi{{}_{sc}}$ for the methods used herein. for the
_Wind_ 3DP thermal electron detector, EESA Low. Note that spacecraft charging
can also affect the trajectories of incident low energy particles (e.g., see
Barrie et al., 2019, for details). However, this would require an accurate
model of the _Wind_ spacecraft surface materials and the evolution of their
work functions over time, which is beyond the scope of this effort.

Finally, we should note that the dataset Salem & Pulupa (2023) provided by and
discussed in Salem et al. (2023) does not overlap with this new dataset and it
spans the interval when $E{{}_{min}}$ $>$ $\phi{{}_{sc}}$, early in the _Wind_
mission. Thus, Salem et al. (2023) relied on different techniques to estimate
$\phi{{}_{sc}}$ than the methods described herein. The measurements in Salem
et al. (2023) are also subject to smaller variations in the anode calibrations
than the measurements made later in the mission (i.e., the data herein). This
was shown in a recent study (e.g., see appendices of Wilson III et al., 2019a)
in some example anode correction calculations for the time period examined by
Salem et al. (2023). Thus, they were able to estimate $\phi{{}_{sc}}$
numerically from the difference between the total electron density calculated
from the upper hybrid line and that calculated by numerically integrating the
velocity distribution. They iteratively modify $\phi{{}_{sc}}$ until the
numerically integrated electron density matches that from the upper hybrid
line. This is a well known and valid approach when $E{{}_{min}}$ $>$
$\phi{{}_{sc}}$, assuming the instrument anodes and energy-angle geometric
factors are properly calibrated.

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