11institutetext: University of California at Berkeley, USA 22institutetext:
International Computer Science Institute, Berkeley, USA

# DIME: An Online Tool for the Visual Comparison of Cross-Modal Retrieval
Models

Tony Zhao 11 Jaeyoung Choi 22 Gerald Friedland 11

###### Abstract

Cross-modal retrieval relies on accurate models to retrieve relevant results
for queries across modalities such as image, text, and video. In this paper,
we build upon previous work by tackling the difficulty of evaluating models
both quantitatively and qualitatively quickly. We present DIME (Dataset,
Index, Model, Embedding), a modality-agnostic tool that handles multimodal
datasets, trained models, and data preprocessors to support straightforward
model comparison with a web browser graphical user interface. DIME inherently
supports building modality-agnostic queryable indexes and extraction of
relevant feature embeddings, and thus effectively doubles as an efficient
cross-modal tool to explore and search through datasets.

###### Keywords:

Cross-modal Retrieval CBIR Evaluation

##  1 Introduction

With the exponentially increasing amount of multimedia content on the Internet
of different modalities, cross-modal retrieval becomes increasingly important
for indexing and searching through either large multi-modal datasets or
between datasets of different modalities. Such cross-modal models allow
retrieval across modalities such as text, image, video, etc. [1][2]. Standard
evaluation metrics such as mean average precision (mAP) allow quantitative
comparison between models, however, visual inspection of retrieval results
allows qualitative evaluation of the models which provides much practical
insight into the behavior of the models. Demands for interactive search-and-
retrieval tools that can accommodate large-scale datasets have been rapidly
increasing in the multimedia community and many such tools [3][4][6] have been
released in this regard, or to showcase the underlying cross-modal retrieval
models.

In this paper, we present DIME (Dataset, Index, Model, Embedding) that builds
upon related work in creating systems to index and explore large datasets of
images as well as other modalities. For instance, systems such as ImageX [3]
share similar functionality such as ranking images by their similarity to a
query image or classification concept instead of search through filtering.
Most general-purpose content-based multimedia retrieval stacks such as Vitrivr
[4] support two major workflows. Within these workflows, DIME adds several key
features that we found to be very useful in cross-modal retrieval research,
which are either lacking or missing in previous works.

First, the extraction workflow is used to process different media objects such
as images, audio, and video to generate relevant features. DIME adds some
minor efficiency improvements and functionality to this workflow such as
binarization and bit-packing of embeddings. These improvements allows for
memory-efficient extraction of feature vectors and removes the need to
repeatedly process datasets while testing models.

Second, the query workflow is used when users query a dataset through an
interface. This involves extracting the relevant features of the query and
dataset look-up before ranking results by their measured similarity. DIME also
adds some minor improvements to this workflow by adding metadata and
diagnostic information on top of the database look-up.

Finally, DIME improves the flexibility of these systems by supporting a
completely new workflow that we found to be highly useful for cross-modal
retrieval research. Since feature-based retrieval relies on accurate models,
DIME implements the model workflow, which is used to compare models against
each other both quantitatively with metrics as well as qualitatively with the
visual presentation of results that are returned by queries.

On top of these new features, DIME automates the process of building modality-
agnostic queryable indexes from corresponding datasets and models as well as
presenting information useful for model comparison. Lastly, we implemented a
graphical user interface in the web browser to support online and remote
functionality.

Figure 1: DIME provides information between models such as a graph of the
distribution of similarity (Euclidean distance) among the results for the
query (an image in this figure). The relevant tags are the results returned
from a dataset lookup of the text-modality index specific to the relevant
model. Figure 2: DIME provides diagnostic metadata as well as other minor
functionality such as using results as queries

##  2 DIME: Dataset, Index, Model, Embedding

DIME is comprised of its titular parts: dataset, index, model, and embedding.
We define model as an object with a callable function. This function has
specified input dimensions and output dimensions respectively. A dataset is
defined as an iterable of vectors. DIME takes a dataset and a model to extract
embeddings for that dataset from that specific model. Finally, it constructs a
queryable index from those embeddings to support search and exploration.

DIME receives datasets along with any preprocessors and a specific output
dimension. Any pair of model and dataset with matching input and output
dimensions respectively, can then be built into an index. Users can specify if
these embeddings should be binarized, which will greatly reduce the amount of
memory and time needed to build and load indexes. Once the index has finished
building, the entire processed dataset can support queries that pass through
modality-appropriate processing.

Database lookup is supported by efficient similarity search. A query such as
text or image is first vectorized through appropriate preprocessors to create
a query vector. This query vector is then passed through a selected model to
extract the query feature embedding. This query feature embedding is then
compared against the embeddings in an index using k-nearest neighbors search
to calculate similarity (euclidean distance) between the query feature
embedding and the embeddings of the dataset. This efficient similarity search
algorithm is implemented with FAISS [5]. The closest $n$ neighbors are
returned as the results of the database look-up as well as diagnostic
metadata.

##  3 Conclusion and Future Work

To simplify and expedite a tedious process of comparing models for cross-modal
retrieval, we present DIME, an online tool that builds upon previous work as a
modality-agnostic search engine which automates the construction of queryable
indexes from datasets of different modalities.

Besides aesthetic and user interaction improvements, our goal is to make
improvements on the quantitative diagnostics of performance and accuracy. A
wider array of metrics and tests can be added to help researchers assess and
compare models. Likewise, a multitude of diagnostic information and metrics
can be implemented for datasets as well.

Ultimately the end goal of DIME is to remove tedious but currently necessary
steps needed to train and compare cross-modal retrieval models. Ideally, a
researcher will only need to focus on designing models. In future iterations
of the tool, we would like to support automation of training, as the tool
currently only supports usage and comparison of pre-trained models.

## Acknowledgment

Parts of this work was performed under the auspices of the U.S. Department of
Energy by Lawrence Livermore National Laboratory under Contract DE-
AC52-07NA27344 and was supported by the LLNL-LDRD Program under Project No.
17-SI-003. Computation resources used in this work were partially supported by
AWS Cloud Credits for Research. Any findings and conclusions are those of the
authors, and do not necessarily represent the views of the funders.

## References

  * [1] Zhen, L., Peng H., Wang, X., Peng, D.: Deep Supervised Cross-modal Retrieval, CVPR 2019
  * [2] Wang, K., Yin Q., Wang W., Wu S., and Wang L.: A comprehensive survey on cross-modal retrieval, 2016.
  * [3] Hezel N., Barthel K. U., Jung K.: ImageX - Explore and Search Local/Private Images, MMM 2018
  * [4] Gasser R., Rossetto L., Schuldt, H.: Multimodal Multimedia Retrieval with vitrivr, ICMR 2019
  * [5] Johnson, J., Douze, M. and Jégou, Hervé.: Billion-scale similarity search with GPUs, arXiv preprint arXiv:1702.08734, 2017.
  * [6] Amato, G., Falchi, F., Gennaro, C. and Rabitti, F.: YFCC100M-HNfc6: A Large-Scale Deep Features Benchmark for Similarity Search, SISAP, 2016.


Grotunits=360

# Phase space transport in a symmetric Caldera potential with three index-1
saddles and no minima

Matthaios Katsanikas  Research Center for Astronomy and Applied Mathematics,
Academy of Athens, Soranou Efesiou 4, Athens, GR-11527, Greece.
School of Mathematics, University of Bristol, Fry Building, Woodland Road,
Bristol, BS8 1UG, United Kingdom.
<EMAIL_ADDRESS>Makrina Agaoglou  Instituto de Ciencias
Matemáticas, CSIC, C/Nicolás Cabrera 15, Campus Cantoblanco, 28049 Madrid,
Spain
<EMAIL_ADDRESS>Stephen Wiggins  School of Mathematics, University
of Bristol,
Fry Building, Woodland Road, Bristol, BS8 1UG, United Kingdom.
<EMAIL_ADDRESS>Ana M. Mancho  Instituto de Ciencias Matemáticas,
CSIC, C/Nicolás Cabrera 15, Campus Cantoblanco, 28049 Madrid, Spain
<EMAIL_ADDRESS>
((to be inserted by publisher))

###### Abstract

We apply the method of Lagrangian Descriptors (LDs) to a symmetric Caldera-
type potential energy surface which has three index-1 saddles surrounding a
relatively flat region that contains no minimum. Using this method we show the
phase space transport mechanism that is responsible for the existence and non-
existence of the phenomenon of dynamical matching for this form of Caldera
potential energy surface.

##  1 Introduction

In Carpenter [1985] Carpenter introduced a two dimensional (2D) potential
energy surface (PES) which has been referred to as the ‘’Caldera” due to its
topography resembling a collapsed volcano. In terms of its critical points the
original Carpenter model was characterized by one central minimum surrounded
by four index-1 saddles that control the exit and entrance to the Caldera
region or to infinity Collins _et al._ [2014]. These types of potentials can
be encountered in many organic chemical reactions (see, for example, the
introduction of Collins _et al._ [2014]; Katsanikas & Wiggins [2018]) and PESs
describing four-armed barred galaxies Athanassoula _et al._ [2009]. The
dynamical properties associated with Caldera-type potentials have been studied
in many papers, such as Katsanikas & Wiggins [2018, 2019]; Katsanikas _et al._
[2020c, b]; Geng _et al._ [2021a, b]; Katsanikas & Wiggins [2022].

In this paper we study the dynamics associated with a Caldera PES with a
different critical point configuration. It has only three index-1 saddles
surrounding the Caldera region which contains a minimum. Two of the saddles
are upper saddles, much like the original Caldera, and the remaining saddle is
a lower saddle located in the center of the region. Nevertheless, the lower
saddle still controls the exit of trajectories from the lower part of the
Caldera region, as we will show. This form of the Caldera potential arises
from the original Caldera PES when the critical points undergo a pitchfork
bifurcation, as described in Geng _et al._ [2021a, b].

Dynamical matching is an important phenomenon in organic chemical reactions
Collins _et al._ [2014]; Katsanikas & Wiggins [2018]. This phenomenon is
encountered in the classical form of Caldera potential energy surface (i.e.,
with one central minimum and four index-1 saddles). What dynamical matching
means in this case is that trajectories that are initiated in the region of
the upper index-1 saddles, go straight across the caldera and exit through the
region of the opposite lower saddle (see for example Katsanikas & Wiggins
[2018]). In this paper we will study this phenomenon for the Caldera potential
energy surface with three index-1 saddles Geng _et al._ [2021a, b].

This paper is outlined as follows. In section 2 we describe the model that we
use in this paper. In section 3 we analyze the phase space mechanism that is
responsible for the existence and non-existence of dynamical matching in the
case of a Caldera potential energy surface with three index-1 saddles (see
section 3). We present our conclusions in section 4. The method of Lagrangian
descriptors plays an essential role in our analysis of phase space structure
and the aspects of this method that we require are described in Appendix A.

##  2 Model

The model that we consider is the two dimensional (2D) symmetric Caldera
potential energy surface (PES) introduced in Collins _et al._ [2014]:

$\displaystyle V(x,y)$
$\displaystyle=c_{1}r^{2}+c_{2}y-c_{3}r^{4}\cos(4\theta)$ (1)
$\displaystyle=c_{1}(x^{2}+y^{2})+c_{2}y-c_{3}(x^{4}+y^{4}-6x^{2}y^{2}),$

where $(x,y)$, $(r,\theta)$ describe the position in Cartesian and polar
coordinates, respectively, and $c_{1},c_{2},c_{3}$ are parameters.

The corresponding 2 degree-of-freedom (2 DoF) Hamiltonian is:

$H(x,y,p_{x},p_{y})=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+V(x,y),$ (2)

where $p_{x}$ denotes the conjugate momentum corresponding to $x$ and $p_{y}$
denotes the conjugate momentum corresponding to $y$. Moreover, we will assume
that $m$ is $1$. Therefore, the equations of motion are:

$\displaystyle\dot{x}=\frac{\partial H}{\partial p_{x}}=\frac{p_{x}}{m},$ (3)
$\displaystyle\dot{y}=\frac{\partial H}{\partial p_{y}}=\frac{p_{y}}{m},$
$\displaystyle\dot{p_{x}}=-\frac{\partial V}{\partial
x}(x,y)=-(2c_{1}x-4c_{3}x^{3}+12c_{3}xy^{2}),$
$\displaystyle\dot{p_{y}}=-\frac{\partial V}{\partial
y}(x,y)=-(2c_{1}y-4c_{3}y^{3}+12c_{3}x^{2}y+c_{2}).$

In this paper we will study the case where $c_{1}=0.4,c_{2}=3,c_{3}=-0.3$. For
these parameter values there are three index-1 saddles. Two upper saddles,
that we will refer to as upper left hand (LH) saddle and upper right hand (RH)
saddle, with value of energy 2.321, and one lower, that we will refer to as
the lower saddle, with value of energy -2.402. The positions of the
equilibrium points and the corresponding values of energy are given in table
2.

Stationary points of the Caldera potential for $c_{1}=0.4,c_{2}=3$ and
$c_{3}=-0.3$ (”RH” and ”LH” are the abbreviations for right hand and left hand
respectively).  Critical point x y E Lower saddle 0.000 -1.194 -2.402 Upper LH
saddle -1.204 0.840 2.321 Upper RH saddle 1.204 0.840 2.321

The potential energy surface (PES) of our model is depicted in Fig. 1:

Figure 1: The 3D representation of the PES for $c_{1}=0.4,c_{2}=3,c_{3}=-0.3$

The dynamical significance of the index-1 saddles is that for energies above
that of the saddles, there exist unstable periodic orbits (UPOs) in phase
space. This is a consequence of the Lyapunov subcenter theorem Moser [1958];
Kelley [1967]. These UPOs, and their stable and unstable manifolds, are the
essential phase space structures for understanding dynamical matching.

##  3 Phase Space Mechanism For the Existence and Non-existence of Dynamical
Matching

We begin by explaining what we mean by dynamical matching in this potential
energy surface with three index-1 saddles. In the ‘’usual” Caldera potential
energy surfaces with two upper saddles and two lower saddles dynamical
matching means that trajectories entering the Caldera region from the upper
right saddle (resp. upper left saddle) move directly across the Caldera and
exit by crossing the lower left saddle (resp. lower right saddle). For the
Caldera potential energy surface with three index-1 saddles there is only a
single lower index-1 saddle in the center of the lower regions. However, one
can see from Fig. 1 that there is an exit region to the left (left exit
region, LER) of the saddle and an exit region to the right (right exit region,
RER) of the saddle. In this case, dynamical matching corresponds to
trajectories that enter the Caldera from the upper right saddle (resp. upper
left saddle), move directly across the Caldera, and exit from the LER (resp.
RER).

Both the existence and non-existence of dynamical matching in this model can
be explained by heteroclinic connections between the unstable manifolds of the
upper UPOs and the stable manifold of the UPO associated with the lower saddle
(lower UPO) and the geometry of the unstable manifold of the lower UPO.

We will study the geometry of these manifolds by using the method of
Lagrangian descriptors (LDs), described in Appendix A. In order to study the
manifolds near the upper saddles we apply the method of LDs in the 2D slice
$y=0.84$ with $p_{y}>0$ (for a value of energy $E=2.821$) since this is the
$y$-coordinate of the upper index-1 saddles (see the panels A and B of Fig.
2).

This the geometric structures on this slice reveal the unstable manifolds of
the unstable periodic orbits of the upper index-1 saddles. We will focus on
the right hand side of this slice (see panels C and D of Fig. 2) in order to
study in detail the invariant manifolds that emanate from the unstable
periodic orbits of the right upper index-1 saddle. In particular, we will
describe the phase space mechanism for the case of the upper right index-1
saddle since for the case of the upper left saddle we have similar mechanism
as a result of the symmetry of the potential.

Next we extract the invariant manifolds of the panels C and D of Fig. 2 using
the gradient of LDs (see details in Katsanikas _et al._ [2020b]). We can see
the result in Fig. 3. In this figure, we see many lobes between unstable and
stable invariant manifolds. This is the unstable manifold of the UPO
associated with the upper right saddle and the stable manifold associated the
UPO of the lower saddle. We can verify this as follows. In the right corner of
this figure we depict a position of a trajectory inside a lobe. If we
integrate this trajectory backwards and forwards, the result will be the exit
of this trajectory from the caldera through the region of the upper right
index-1 saddle and the region of the lower right exit (see Fig. 4),
respectively. This means that this lobe corresponds to a lobe between
intersections of the unstable invariant manifold of the unstable periodic
orbits of the UPO of the upper right index-1 saddle with the stable invariant
manifold of the UPO family of the lower index-1 saddle (that is located at the
center). The trajectories that are initially at the region of the right upper
saddle follow the unstable invariant manifolds of the UPO associated with the
right upper saddle until they are trapped in the lobes between the
heteroclinic intersections of the unstable invariant manifolds of the right
upper index-1 saddle with the stable invariant manifolds of the UPO of the
lower index-1 saddle. Then the trajectories follow the stable invariant
manifolds to the area of these unstable periodic orbits (the central area of
the Caldera). This is the reason that the trajectory that corresponds to the
black point of Fig. 3, moves towards the central area forward in time (as
indicated from the arrows in Fig. 4). This explains the first part of the
mechanism, the transport of the trajectories from the region of the upper
index-1 saddle to the central area of the caldera.

The second part of the mechanism explains the transport of the trajectories
from the area of the lower index-1 saddle (central area of the caldera) to one
of the lower exits from the Caldera (for example the lower right exit from the
Caldera for the trajectory of Fig. 4). In order to see this we use the method
of LDs in the slice $y=0$ with $p_{y}>0$ to detect the invariant manifolds
emanating from the central area of the caldera and, in particular, the
invariant manifolds of the lower index-1 saddle. In Fig. 5, we depict, using
LDs, the unstable and stable invariant manifolds of the central area. The next
step is to extract the invariant manifolds of the panels A and B of Fig. 5
using the gradient of LDs (see details in Katsanikas _et al._ [2020b]). We can
see the result in Fig. 6. In this Figure, we see the unstable and stable
manifolds start from the point of the periodic orbit of the lower index-1
saddle (that is located at the center). The trajectories that approach the
periodic orbit are on the stable manifolds (the motion of these trajectories
is indicated by the violet arrows in Fig. 6). The trajectories that move away
from the periodic orbit are on the unstable manifolds (the motion of these
trajectories is indicated by the green arrows in Fig. 6. This means that the
trajectories in the first part of the mechanism approach the neighborhood of
the periodic orbit of the lower index-1 saddle (that is located in the central
area of the caldera) will leave from this neighborhood following the unstable
manifolds in one of the two directions that are indicated by the green arrows
in Fig. 6. In each of these two directions, we take one initial condition (see
the black points in Fig. 6) and we integrate it forward in time. In this way,
we want to check the fate of the trajectories that will follow the one or the
other direction of the unstable manifolds. When we integrate these initial
conditions, we observe in the panel A of Fig. 7 that the trajectory that
corresponds to the black point on the right side from the center (where the
periodic orbit of the lower index-1 saddle is located) exits through the
region of the lower right exit. On the contrary, when we integrate the other
initial condition (the black point on the left side from the center of Fig. 6)
the corresponding trajectory exits through the region of the lower left exit
(see panel B of Fig. 7). We see that the two directions of the unstable
manifolds that are emanated from the unstable periodic orbit of the lower
index-1 saddle guide the trajectories directly to different exit regions. This
means that the trajectories that are in the neighborhood of the unstable
periodic orbits of the lower index-1 saddles in the central area of the
caldera leave this area following the unstable invariant manifolds of these
periodic orbits. If they follow the one direction of the unstable invariant
manifolds exit through the one lower exit region and if they follow the other
direction, they exit through the other lower exit region. This indicates the
important role of the periodic orbits of the lower index-1 saddle that control
through the direction of the unstable invariant manifolds the exit of the
trajectories that come from the region of the upper index-1 saddles to the
central area of the caldera.

A) B)
C) D)

Figure 2: Computation of variable-time LDs in the 2D slide $y=0.84$ using
$\tau=8$ and p = 1/2 for value of energy $E=2.821$. A) Forward integration
LDs; B) Backward integration LDs; C) Forward integration for the interval
$0.6<x<1.5$ and $-1<p_{x}<-0.4$ . D) Backward integration for the interval
$0.6<x<1.5$ and $-1<p_{x}<-0.4$.
Figure 3: The stable (violet) and unstable (green) invariant manifolds
extracted (in the 2D slice $y=0.84$ with $p_{y}>0$) from the gradient of the
Lagrangian descriptors for the interval $0.6<x<1.5$ and $-1<p_{x}<-0.4$. We
depict using black color the initial condition of a trajectory.
Figure 4: The contours of the Caldera potential and a trajectory in the
configuration space forward (with blue color) and backward (with red color) in
time (the initial condition of this trajectory is depicted by a black point in
Fig. 3). The arrows denote the direction of the trajectory forward in time.

A) B)

Figure 5: Computation of variable-time LDs in the 2D slice $y=0$ using
$\tau=5$ and p = 1/2 for value of energy $E=2.821$. A) Forward integration
LDs; B) Backward integration LDs;
Figure 6: The stable (violet) and unstable (green) invariant manifolds
extracted (in the 2D slice $y=0$ with $p_{y}>0$) from the gradient of the
Lagrangian descriptors for the interval $-2.5<x<2.5$ and $-2.5<p_{x}<2.5$. We
depict using black color the initial conditions of two trajectories on the
unstable manifold. The green and violet arrows present the directions of the
unstable and stable invariant manifolds respectively that are emanated from
the unstable periodic orbit of the lower index-1, that is located at the
center.

A) B)

Figure 7: The contours of the caldera potential and two trajectories in the
configuration space forward in time (with green color). The initial conditions
of these trajectories correspond to the two black points in the right side
from the center in Fig. 6 (for panel A) and in the left side from the center
in Fig. 6 (for panel B).

##  4 Conclusions

In this paper, we investigated the phase space mechanism of transport that is
responsible for the creating and breaking the phenomenon of dynamical matching
(that was observed in Geng _et al._ [2021a, b]) for the form of a Caldera
potential energy surface with only three index-1 saddles and no minimum. The
first part of the mechanism is based on heteroclinic intersections of the
unstable invariant manifolds of the unstable periodic orbits of the upper
index-1 saddles and the stable invariant manifolds of the unstable periodic
orbits of the lower index-1 saddle. This explains the transport from the
region of the upper index-1 saddles to the region of the lower index-1 saddle.
The second part of the mechanism is based on the direction of the unstable
invariant manifolds of the unstable periodic orbits of the lower index-1
saddle. The trajectories leave the central area of the caldera following the
unstable invariant manifolds of the unstable periodic orbits of the lower
index-1 saddle. This happens in two directions. One direction guides the
trajectories to the region of the lower left exit region and the other to the
region of the lower right exit region. The unstable periodic orbits of the
lower index-1 saddle control the transport from the central region of the
caldera to the lower exit regions.

AcknowledgmentsThe authors acknowledge the financial support provided by the
EPSRC Grant No. EP/P021123/1 and MA acknowledges support from the grant
CEX2019-000904-S and IJC2019-040168-I funded by: MCIN/AEI/
10.13039/501100011033.

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##  5 appendix

The Lagrangian Descriptors is a diagnostic tool that can reveal the phase
space structures. The first time that this technique was used was in the paper
Madrid & Mancho [2009], were was aiming to study transport and mixing in
geophysical flows. In the last years this technique has been broadly used not
only in fluid mechanics Lopesino _et al._ [2015]; Balibrea-Iniesta _et al._
[2016]; Mendoza & Mancho [2010]; Mancho _et al._ [2013]; García-Garrido _et
al._ [2018]; Balibrea-Iniesta _et al._ [2019]; Lopesino _et al._ [2017] but
also in the area of chemical reaction dynamics Agaoglou _et al._ [2019, 2020,
2019]; Katsanikas _et al._ [2020a]; Crossley _et al._ [2021]. The LDs method
works in the following way: In order to reveal the phase space structures in a
given slice using the method of LDs the steps that we need to follow are very
simple. First we choose the slice and we define in this slice a grid of
initial conditions. Then we integrate this initial conditions forward and
backward in time for a given integration time $\tau$ and while we are
integrating we accumulate along these trajectories a positive quantity defined
from the vector field that determines the dynamical system that we are
studying. If you integrate trajectories forwards in time, the LD function is
going to detect the stable manifolds while the backwards in time integration
will detect the unstable manifolds. The scalar output obtained from the method
will highlight the location of the invariant stable and unstable manifolds
intersecting this slice, which are detected at points where the values of LDs
display an abrupt change.

There are several definitions for the M function. In this work we are using
the p-norm definition of the method that relies on variable time integration,
where $p\in(0,1]$ (the reader can find more information about the application
of variable time LDs to Caldera potentials in Katsanikas _et al._ [2020b]). We
have fixed the value of p to be $p=1/2$. This definition of the M function is
preferable here due to the nature of the Caldera’s potential energy surface
(PES), that is an open potential. That can lead to an increasingly fast pace
escape of the trajectories.

Let’s consider the following dynamical system with time dependence:

$\dfrac{d\mathbf{x}}{dt}=\mathbf{v}(\mathbf{x},t)\;,\quad\mathbf{x}\in\mathbb{R}^{n}\;,\;t\in\mathbb{R}\;$
(4)

where $\mathbf{v}(\mathbf{x},t)\in C^{r}(r\geq 1)$ in $\mathbf{x}$ and it is
continuous in time. Given an initial condition $x_{0}$ at time $t_{0}$, take a
fixed integration time $\tau>0$ and $p\in(0,1]$. The method of LDs in the
p-norm definition is as follows:

$M_{p}(\mathbf{x}_{0},t_{0},\tau)=\sum_{k=1}^{n}\bigg{[}\int^{t_{0}+\tau}_{t_{0}-\tau}|v_{k}(\mathbf{x}(t;\mathbf{x}_{0}),t)|^{p}\;dt\bigg{]}=M_{p}^{(b)}(\mathbf{x}_{0},t_{0},\tau)+M_{p}^{(f)}(\mathbf{x}_{0},t_{0},\tau)\;,$
(5)

where $M_{p}^{(b)}$ and $M_{p}^{(f)}$ its backward and forward integration
parts:

$\begin{split}M_{p}^{(b)}(\mathbf{x}_{0},t_{0},\tau)&=\sum_{k=1}^{n}\bigg{[}\int^{t_{0}}_{t_{0}-\tau}|v_{k}(\mathbf{x}(t;\mathbf{x}_{0}),t)|^{p}\;dt\bigg{]}\;,\\\\[5.69046pt]
M_{p}^{(f)}(\mathbf{x}_{0},t_{0},\tau)&=\sum_{k=1}^{n}\bigg{[}\int^{t_{0}+\tau}_{t_{0}}|v_{k}(\mathbf{x}(t;\mathbf{x}_{0}),t)|^{p}\;dt\bigg{]}\;,\end{split}$
(6)

The formulation of the p-norm definition that we apply to this model is the
following:

$M_{p}(\mathbf{x}_{0},t_{0},\tau)=\sum_{k=1}^{n}\bigg{[}\int^{t_{0}+\tau_{x_{0}}^{+}}_{t_{0}-\tau_{x_{0}}^{-}}|v_{k}(\mathbf{x}(t;\mathbf{x}_{0}),t)|^{p}\;dt\bigg{]}$
(7)

where

$\tau_{x_{0}}^{\pm}=min\\{\tau_{0},|t^{\pm}|_{|x(t^{\pm};x_{0}\notin R)}\\}$

Note that the integration time $\tau_{0}$ is fixed and that $t^{+}$ is the
time that the trajectory exits the interaction region $R$ in forward time
whereas $t^{-}$ is for backward time.


# Event-scale Internal Tide Variability via X-band Marine Radar

###### Abstract

A combined radar remote sensing and in situ data set is used to track packets
of nonlinear internal waves as they propagate and shoal across the inner shelf
($\sim$40 $m$ \- 9 $m$). The dataset consists of high space-time resolution
(5$m$, 2$min$) radar image time series collected over a 10 $km$ radial
footprint, with over a dozen synchronous and co-located moorings measuring
temperature, salinity, and velocity throughout the water column. The internal
bores and higher-frequency internal waves that make up the internal tide are
tracked in the radar image time series and, therefore, provide continuous
cross-shore speed and angle estimates as the waves propagate across the inner
shelf. We compare radar-estimated speeds to those estimated from moorings and
confirm a cross-shore shoaling profile that deviates from linear theory. We
additionally use combined remote sensing and in situ data to perform a
detailed analysis on four consecutive internal tides. These analyses reveal
intra-packet speed variability, tide-tide influences, reflected internal
waves, and an instance of internal wave polarity reversal observed in the
radar and moorings.

##  1 Introduction

The inner shelf is a dynamically rich region of the ocean characterized by
nonlinear processes that bridge the shoreline, through the surf zone, to the
shelf’s edge. Nonlinear internal waves (NLIWs) are a predominant feature of
the inner shelf and have strong horizontal (O(1 $m/s$)) and vertical (O(0.1
$m/s$)) velocities that drive large vertical displacements (O(10s $m$)) of the
pycnocline. NLIW implications can include the generation of intense nearshore
mixing and the transport of nutrients and pollutants (Kumar et al., 2021).
Additionally, the presence of NLIWs can strongly influence acoustic
propagation and the operation of sonar systems, which may inhibit navigation,
object detection, and communication with other vessels. While there is a large
body of research that includes modeling, laboratory measurements, and field
observations describing internal waves in deep water, less is known about NLIW
shoaling and transformation as the waves propagate through the shallow ($<$50
$m$) coastal waters of the inner shelf.

In deep water, NLIWs exist as waves of depression, meaning that their
amplitude is a downward depression of the pycnocline. As they propagate
shoreward into shallower waters, they undergo nonlinear processes that may
influence their speed and shape characteristics. For instance, if not fully
dissipated during the shoaling process due to depth-induced breaking, the
NLIWs may undergo a polarity reversal from waves of depression to waves of
elevation (e.g., Orr and Mignerey (2003); Scotti et al. (2008); Shroyer et al.
(2009)). This wave transformation occurs as a NLIW of depression reaches a
pycnocline that is at or below the mid water column (i.e., the critical
depth). In shelf waters, the relative layer depths change due to changing
total water depth. Generally, the wave of depression forms a dispersive tail
containing waves of elevation and waves of depression. The lead wave of
depression is eventually destroyed, thus leaving a lead wave of elevation in
its place (Shroyer et al., 2009). This phenomenon has been documented in both
laboratory (Helfrich and Melville, 2006) and field observations (Orr and
Mignerey, 2003; Shroyer et al., 2009). In addition, numerous models have been
derived to describe NLIW shoaling. Weakly nonlinear equations include the
Korteweg-de Vries (KdV) and extended KdV (eKdV) (Lee and Beardsley, 1974;
Djordjevic and Redekopp, 1978; Choi and Camassa, 1999). Fully nonlinear
solutions are also possible through the Dubriel-Jacotin-Long (DJL) equation
Long (1953). NLIW speed is a balance between nonlinear and dispersive
processes, and evolves with shoaling water depths.

Traditional in situ methods for observing NLIWs utilize thermistor chains and
ADCPs (e.g., (Orr and Mignerey, 2003; Colosi et al., 2018)), turbulence
profilers (e.g. (Becherer et al., 2020)), and/or acoustics (e.g. (Moum and
Smyth, 2006)). Vertically-profiling in situ sensors can also provide a local
characterization of NLIW shapes at the point of deployment. In addition, by
using triangulation of NLIW arrival times from several sensors, wave speed and
angle can be determined Alford et al. (2010); McSweeney et al. (2020a).
However, these speeds and angles represent averages over the triangulation
array and these arrays are spaced at relatively low spatial resolution due to
the constraints and sparsity of mooring deployments. Therefore, high
resolution cross- and alongshore variability is difficult to capture using
point measurements alone.

Alternately, an advantage of remote sensors is that they offer a synoptic
picture of the region of interest, essentially providing thousands of point
observations over regions of several kilometers. Synthetic Aperature Radar
(SAR) on satellites is a commonly used sensing tool for internal waves because
of its large spatial footprint (O(100$km$)) at relatively high spatial
resolution (O(10$m$)) (e.g. Apel and Gonzalez (1983); Gasparovic et al.
(1988); Da Silva et al. (1998); Liu et al. (2014)). NLIWs are imaged by radar
due to their associated surface currents. Generally, where NLIWs induce
convergent surface currents there is an increase in surface roughness at short
length scales, leading to heightened radar backscatter. Conversely, in regions
of surface divergence, the relatively smoother water surface returns very
little backscatter. These influences on backscatter intensity yield images of
NLIWs that contain alternating bright and dark bands. Such remotely sensed
observations provide snapshots of spatial complexity capable of revealing
inter-packet dynamics. For example, Hsu et al. (2000) observed phase shifts
due to wave-wave interactions in snapshots from satellite SAR, and Magalhaes
et al. (2021) used SAR to look at Mach-stem wave-wave interactions. However,
due to orbital constraints, most satellite sensors have a relatively sparse
sampling period for image collection (hours to days), thus making spatial
evolution difficult to track. On the other hand, terrestrial remote sensors
can sample more continuously.

As an alternative to SAR, marine radars (X-band) have a long history of being
utilized for NLIW imaging (Alpers, 1985; Chang et al., 2008; Ramos et al.,
2009; Celona et al., 2021). Marine radars can sample over spatial scales of
several kilometers with resolution on the order of several meters, and are
deployed on vessels or shore-based towers. A real utility of marine radars is
that they can image internal waves with high temporal resolution (O(1 min)) to
allow tracking in time, which provides the capability of high resolution NLIW
speed data and spatial evolution (e.g. McSweeney et al. (2020b); Celona et al.
(2021)). Recent work using marine radar has utilized the Radon Transform for
NLIW tracking, with much higher cross- and along-shore spatial information
than can be obtained with in situ measurements, but only focused on the lead
bore (Celona et al., 2021), or several homogeneously oriented waves (Ramos et
al., 2009). The complex NLIW packet dynamics, such as wave merging, wave-wave
interactions, and dispersion remains relatively unexamined via remote sensing.

In this paper, we concentrate on marine radar (X-band) observations collected
over two months on the south-central California coast during the fall of 2017.
Using space-time diagrams that demonstrate complex intra-tide NLIW behavior,
and focus on event-scale NLIW variability that has not been depicted by
previous in situ and remote sensing studies. In this work we will refer to
packets of NLIWs generated on tidal time scales as ”internal tides”, the first
or leading wave of an internal tide as the ”bore”, and the remaining non-
leading NLIWs in an internal tide as trailing waves.

Herein, we first provide a site description and an overview of the sampling
strategy. Then we detail the developed tracking scheme for estimating cross-
shore phase speed and angle of NLIWs, which is effective for both the large
amplitude leading bore of an internal tide, as well as the higher frequency
trailing waves that follow and make up the rest of the wave packet.
Observations are compared to measurements from an array of moorings and
confirms that the cross-shore speeds deviate from linear theory. Using the
space-time diagram approach, we have revealed instances of wave direction
reversal at shallow depths indicating evidence of bore reflection.
Additionally, three nearly-consecutive tides are examined in close detail,
revealing evidence of alongshore variability, intra-packet speed variability
(internal wave dispersion), and an instance of polarity reversal.

##  2 Study site & observations

This study utilizes data collected during the Inner Shelf Dynamics Experiment
(ISDE) that is further described in Lerczak et al. (2019); Kumar et al.
(2021). The ISDE was a multi-institution field experiment conducted from
September-November 2017 near Point Sal, California. The focus of the ISDE was
the transition zone between midshelf waters (O(100 $m$)) and the surf zone,
and utilized a broad range of moored, towed, land-based, and airborne
instrumentation. In this work, we focus on measurements of internal waves from
a tower-mounted, X-band marine radar, as well as a subset of the in situ
mooring data from the region offshore of Guadalupe, California, termed the
”Oceano array”. The radar footprint and locations of the moorings are shown in
Figure 1.

Figure 1: Example image from shore-based radar, with mooring locations
indicated. Circles indicate sites with both full water-column temperature and
current velocity data, whereas triangles note sites with only the temperature
data. This study site, which is just north of Pt Sal is indicated as a black
star in the lower left map of California. The location of the radar is
indicated with a red star onshore of the mooring array. The legend lists the
names of the moorings, where the number represents water depth. The “A” in the
mooring names indicates an upward-looking ADCP and “T” indicates a string of
temperature sensors.

The radar observations were collected using a commercial Koden X-Band marine
radar mounted atop a 100-ft tower. The tower was located at the Chevron Oil
restoration site in Guadalupe, California, directly onshore of the Oceano
Mooring array as indicated in Figure 1. Radar backscatter intensity images
were collected over a footprint of 12 $km$ radius with a sampling period of
approximately 1.2 $sec$. These raw images were then averaged over 64 rotations
($\sim$1.3 $min$) to eliminate signatures of surface gravity waves and instead
enhance the imaging of the lower frequency hydrodynamic features. Internal
waves were frequently imaged from the edge of the domain 12 $km$ offshore (50
$m$ water depth), and tracked as they propagated to shore towards the radar.
Given the high resolution in space ($O(1m)$) and time ($O(1min)$) of the radar
imagery, this data set provided the unique ability to estimate high-resolution
space and time properties of the internal wave packets as the waves propagate
across the shelf, with synchronous mooring data. For additional specifications
of the radar deployment, the reader is referred to (Haller et al., 2019).

The spatio-temporal overlap of the radar footprint and the Oceano mooring
array provided the opportunity to directly compare remotely sensed and
directly measured in situ observations of NLIWs. The moorings provided
corroboration of bore arrivals, observations of NLIW shape, estimation of
background stratification, and sparse estimations of NLIW speeds at the
mooring locations (McSweeney et al., 2020a). We utilized data from an array of
15 moorings deployed in 100$m$ to 9$m$ depth, which included both a string of
temperature sensors and an upward-looking ADCP. We focused primarily on the
Southern cross-shore axis of the mooring array (OC40S, OC32S, OC25SB, OC17S,
STR3B) because the radar imaging tended to be better in this region of the
domain due to winds being more frequently from the southwest. Further
description of the mooring instrumentation and data processing can be found in
McSweeney et al. (2020a). The temperature and velocity data provided in situ
knowledge of NLIW shape. In addition, we utilized bore arrival times
determined using a semi-automated detection of filtered pycnocline
displacements (McSweeney et al., 2020a). Generally, two internal bores were
detected within a semidiurnal tidal period (i.e., roughly every 6 hours).
Arrival times at the mooring sites were used to identify those specific bores
in the radar data. This is an important component of our analysis, as bore
identification in radar imagery is not always obvious. For instance, the
leading bore is not always the first or brightest wave imaged in an internal
tide. We additionally utilize the mooring-estimated bore speeds as comparison
data. Those speeds were determined by (McSweeney et al., 2020a) using a
triangulation method of arrival times as described in Thomas et al. (2016). In
this work, stratification data from the moorings are also used to estimate
theoretical linear internal wave phase speeds, as described in Section 3.3.2.

##  3 Methods

We use X-Band radar imagery to estimate the continuous cross-shore speed and
angle of NLIWs as they propagated across the inner shelf. This section details
the developed image processing technique for internal bore and high frequency
solitary wave tracking and outlines the use of mooring observations in
conjunction with the remote sensing observations. The stratification measured
by the moorings is used to to calculate theoretical linear wave speed, and to
look in detail at wave shape.

###  3.1 X-Band radar

To estimate NLIW speed and angle from the radar, we utilize space-time (aka
Hovmöller) diagrams (Hovmöller, 1949) where intensity is extracted along a
spatial transect through time. These diagrams are a commonly used technique
throughout coastal oceanography for analysis of remote sensing data (e.g.
(Holman et al., 1993; Chickadel et al., 2003; Catalán et al., 2014)), and have
been previously used to image internal wave propagation (Ramos et al., 2009).
Herein, we use the space-time diagrams to track NLIWs and estimate propagation
speed and angle through phase relationships between nearby transects. Multiple
NLIWs are traced throughout an internal tide and individual waves in the
space-time diagrams are linked to in situ observations to reveal their
subsurface profiles.

To generate the space-time diagrams that can be compared directly to mooring
measurements, six cross-shelf transects aligned with the axes of the mooring
array are used for speed and angle estimations (Figure 2a). The transect that
crosses the southern arm of the mooring array (white in Figure 2a) is the
primary transect used for analysis due to both its close proximity to moorings
OC40S, OC32S, OC25SB, OC17S, and STR3B and the fact that radar imaging was
better in the southern region of the radar footprint due to predominantly
southwesterly winds. While the transects are roughly aligned with the wave
propagation direction, the slope of features in the space-time diagrams
represent the component of speed in the transect direction, rather than the
true vector speed. Therefore, the estimated speeds in transect coordinates are
corrected using wave angle to find the true wave speed. Wave angle is found
using the phase relationship of the lead bore between the six transects.

Figure 2b shows a sample space-time diagram extracted along the white transect
(third from the bottom in panel a), with distance offshore on the vertical
axis and time on the horizontal axis. The bright, curvilinear streaks are the
imaged NLIWs propagating along the transect through time. In order to
normalize the range-dependent roll-off of backscatter intensity that occurs
with marine radars, the space-time diagram is lowpass filtered from the raw
extracted background intensity using a forward-backward filter as described in
Gustafsson (1996) to achieve a zero-phase digital filter in the along-transect
dimension with a window size of 400$m$. Next, this range-filtered space-time
matrix is subtracted from the original matrix, yielding the range-normalized
space-time diagram used for analysis.

The orientation of the space-time diagram is such that the waves can be
considered to propagate from the upper left corner to the lower right corner
of the diagram. One can initially detect that the slope of the waves decreases
as the waves approach shore, indicating a slowing in speed as the waves shoal.
The lead bore of this packet of internal waves is identified by grey dots in
Figure 2b. These represent the arrival times of the lead bore at the moorings
in the southern axis of the array, which will be discussed in more detail in
the next subsection. By using the arrival times of the lead bore in the
moorings, we are able to determine with certainty which streak in the space-
time diagram is the lead bore, which can be difficult to determine using the
radar data alone.

Figure 2: a) Sample radar image; internal waves are visible as bright and dark
bands parallel to shore. Mooring locations are given as white dots. Six
colored lines are transects used for space-time diagram extraction. b) Space-
time diagram extracted from the white transect in panel a. Arrival times at
the moorings overlapping the white transect are shown as grey dots, aligned
with the lead internal bore.

For the tracking of individual waves in the space-time diagram, an edge
detection and identification scheme was devised. First, a standard Canny edge
detector (Canny, 1986) was applied to the image. While the edge detection
alone is effective in providing locations of sharp intensity gradients, these
gradients are not always continuous in coordinate space. Thus, an edge linking
tool (Kovesi, ) was used to link coordinates of consecutive edge points in
order to form continuous curvilinear features. Each coordinate group
corresponded to a different curvilinear feature in the space-time diagram. The
edge groups are demonstrated as colored lines in Figure 3a. For consistently
imaged waves, a single group encompassed the entire length of the wave; for
features with imaging gaps, the wave was represented by several (typically
less than three) edge groups. From the linked edge coordinates, we manually
determined which edge group corresponded with which wave. Once all edge groups
corresponding to a single wave were identified, a locally weighted regression
was fitted to the edge group indices to create a smooth trace of the wave. The
lead bore is traced with a bold white line in Figure 3, Many of the high
frequency waves preceding and following the lead bore were also traced.

Figure 3: a) Space-time diagram showing edge link tool for identifying edges
of waves in the image. b) Space-time diagram with multiple traced internal
waves. In both panels, grey dots indicate arrival times of the lead bore in
the Oceano mooring data.

Traces of waves in the space-time diagram are in coordinates of range along
transect, $r$, and time, $t$. From these values, the slope of the traces are
estimated, yielding a value for $c_{t}$, or speed in the transect coordinates.
$c_{t}$ is estimated using a finite difference scheme:

$c_{t_{i}}=\frac{r_{i+1}-r_{i}}{t_{i+1}-t_{i}}$ (1)

The lead bore was traced in all six transects. By using the phase relationship
of the detected bore among groups of transects, wave angle, $\theta_{w}$, was
determined. Specifically, to correct speed along the white transect, the
transect just to the north (green) and just to the south (yellow) are used.
Actual wave speed is then found, $c_{w}=c_{t}\cos(\theta_{w})$. Estimated
values of $\theta_{w}$ ranged from 265 to 290 degrees (compass angle waves are
coming from). These angles yield up to a 15 degree offset from the transect
direction, or up to 3 percent adjustment in speed values, which is small
though still utilized in the speed profiles for best results.

With knowledge of the lead bore location in time along each transect, it is
also possible to link the wave among the transects, and trace the wave in the
original radar images. Figure 4 shows the identified lead bore in spatial
coordinates over three time stamps. This demonstrates some along-shore
complexity in the bore’s shape.

Figure 4: Example of three radar frames with the lead internal bore traced in
white, as detected from a series of six space-time diagrams.

###  3.2 In situ

Temperature, salinity, and velocity profiles measured by the moorings reveal
subsurface NLIW shape, as well as the background stratification that serves as
the wave guide through which the NLIWs propagate. To estimate theoretical
linear internal wave phase speeds, the background stratification was estimated
at each mooring location by computing a sorted density product, which follows
Winters et al. (1995). The sorted product provided a representation of the
stratification over a 6 hour window without requiring averaging over the
packet of internal waves, which could potentially bias the stratification.
Using the sorted density, we calculated linear phase speed, $C_{0}$, from the
eigenvalue problem

$\frac{d^{2}\Phi}{dz^{2}}+\frac{1}{C_{0}}(N(z)^{2}-\omega^{2})\Phi=0$ (2)

where $\Phi(z)$ is the vertical density structure and a solution to the
eigenvalue problem, $\omega$ is the frequency taken as 12.42 hr, and $N(z)$ is
the Brunt-Väisälä, or buoyancy frequency, computed from the sorted density.
The eigenvalues $\frac{1}{C_{0}}$ were found, and only the mode one phase
speeds were considered.

The radar-estimated speeds were computed as a continuous cross-shore profile;
however, the density used for computing $C_{0}$ was only measured at the
mooring locations, or five discrete points along the cross-shore transect.
Thus, to compute a continuous cross-shore linear speed from the mooring
observations, we used a sigma-coordinate interpolation method to find a
background stratification along the transect. The sorted density profiles from
the five moorings were first scaled by the total depth at each mooring
location. A linear interpolation was then performed in the cross-shore between
the moorings (discretized to 10$m$), before re-scaling back to the original
depths. An example of this interpolated density product is shown in Figure 5.
A value for $C_{0}$ can then be computed from significantly more density
profiles throughout the transect. For our purposes we estimated theoretical
linear wave speed using density profiles at the red dots indicated along the
bathymetry profile - considerably more discrete locations than at the measured
density profiles alone.

Figure 5: Example of continuous cross-shore stratification found through sigma
interpolation of the sorted density product in the cross-shore direction
between the discrete locations of the moorings.

##  4 Results

###  4.1 Cross-shore shoaling speeds of internal bores

Using the described detection routine, we estimated the speeds and angles of
the lead bores for a total of seven internal tides between September 9-17. To
choose the bores used for analysis, we first generated space-time diagrams
along each of the six transects for every internal tide detected by the
moorings throughout the September-October experiment. The time frames for the
space-time diagram extraction were chosen by using start times an hour before
bore arrivals at OC40S and end times an hour after bore arrivals at STR3B. We
then chose the space-time diagrams for analysis that had both 1) consistent
imaging of NLIWs in the space-time diagram from offshore to onshore (Figure 2)
during the extracted time window, and 2) bore arrivals detected at all of the
transect-overlapping moorings (OC40S, OC32S, OC25S, OC17S, STR3B). This led to
seven total bores and their respective internal tides. Each of the seven
chosen tides were processed for lead bore speed and angle, where the angle was
used to correct transect-direction wave speeds to actual wave speeds.

Figure 6 is adapted from McSweeney et al. (2020a), with radar-estimated speeds
added to the figure. The locations of the moorings are shown as black dots and
labeled using their respective names (see Figure 1). Linear speed estimations
(Equation 2) are provided at the mooring locations using the density profiles
at the time of the time of the bore arrivals. Linear speeds for all of the
detected bores are shown as small grey dots. Linear speed speeds for only the
analyzed seven bores are plotted as small black dots, and the mean of those
seven shown as large black dots connected in the cross-shore via a black line.
The mooring-estimated speeds using bore arrival times are estimated at
locations between the moorings via the triangulation method and indicated by
red letters A-E. All of the mooring-estimated speeds are shown as small faded
red dots, and the speeds for the seven chosen bores are shown as small solid
red dots. The mean mooring speeds are shown as larger solid red dots and
connected via a solid red line. The cross-shore speeds of the seven radar-
analyzed bores are averaged and plotted as a solid blue line. The max and min
speeds of the seven bores is plotted as faded blue lines.

The cross-shore trend in radar-estimated speeds is very close to the
identified trend in mooring-estimated speeds. It can also be seen that the
range of radar-estimated speeds nearly encapsulates the full range of mooring-
estimated speeds for the seven analyzed bores. However, the linear estimated
speeds do not ever reach the maximum observed speeds, rather maintain a nearly
linear profile in the cross-shore. As discussed by McSweeney et al. (2020a),
the observed bore speeds are slower than linear theory offshore of 32 $m$
water depth, and faster than linear theory inside of the 32 $m$ depth.
Additionally, linear theory predicts that the bores will be slowing
consistently from 50 $m$ inshore. However, in the mooring and radar
observations, the bores maintain their offshore speed until they reach water
depths between 25-17 $m$ before beginning to slow.

Figure 6: Summary of lead bore speeds estimated from the radar space-time
diagram (blue) compared to mooring-estimated speeds (red) and linear theory
(black). The ranges of speeds for each source are shown in lighter colors.

###  4.2 Summary of Internal Bore Speeds

We chose three nearly-consecutive internal tides for a detailed analysis of
the combined radar and mooring observations due to their strong presence in
the radar imagery as well as their distinctness from one another in wave
packet compositions. A space-time diagram for the full duration of the three
tides is shown in Figure 7. They occurred between September 9-11, where tides
A and B are internal tides spaced by 6 hours, and tide C is an internal tide
12 hours after tide B (skipping one tide). Temporal delineations between the
three tides are plotted as diagonal cyan lines; these are chosen qualitatively
only to help the reader distinguish between the three tides. Tides A, B, and C
were well-imaged due to ample onshore wind (2-4 $ms^{-1}$), as shown in Figure
7. A wind relaxation event following Tide B prevented radar imaging of the
internal tide between B and C, before the wind picked back up again around the
timing of Tide C. The onshore wind component increased on Sep 10 around 16:00,
which can be seen in both the wind time series and the space-time diagram, as
the intensity increases along the full offshore distance. Thus a short later
portion of the internal tide between B and C can be seen.

The radar signature of mooring OC40S is visible in the space-time diagram as a
horizontal streak roughly 5000 $m$ offshore, demonstrating that the extracted
radar transect used to create the space-time diagram overlaps OC40S; the other
mooring locations are within 100 $m$ of the transect. For each of the four
tides spanning the time period shown in the space-time diagram, the lead bore
detected in the mooring observations (McSweeney et al., 2020a) is back plotted
onto the space-time diagram. It is evident that the mooring-detected arrivals
well-align with the bright waves in the space-time diagram, with some
deviations to be discussed herein.

Figure 7: Space-time diagram spanning four internal tides, with the onshore
component of wind speed plotted above. The teal dashed lines are temporal
delineations between the tides. Grey dots are bore arrivals in the mooring
data.

Mooring data between 50 $m$ and 9 $m$ water depth for these three tides is
shown in Figure 8, with OC40S thru STR3B moorings spatially overlapping the
transect used for the space-time diagram. Eastward velocity is plotted for
moorings OC50, OC40S, and STR3B, with positive values towards the East
(roughly onshore); velocity was not recorded for OC32S, OC25SB, or OC17S $m$.
Temperature is contoured for each mooring at 1∘ intervals with the 15∘ contour
plotted in bold. Tides A-C are delineated using cyan lines, which correspond
to the times and locations of the cyan lines in Figure 7. The arrival of the
lead bores for each tide in the mooring data are also plotted as grey dots in
Figure 8.

Figure 8: Mooring observations at OC50, OC40S, OC32S, OC25SB, OC17S, and STR3B
(refer to Figure 1 for locations). Eastward velocity is plotted in blue and
Westward velocity is plotted in red. Temperature is contoured at 1∘ intervals
with the 15∘ contour plotted in bold.

Figure 7 indicates that Tides A-C are visibly different from one another. Tide
A is characterized by a bright and prominent lead bore, with some leading
waves closer to shore but no trailing waves. Using the radar data alone, it
would be relatively straightforward to identify this bore. Tides B and C,
however, contain many waves within the packet. Initial comparisons between the
tides indicate that the packet of waves in Tide C is both more tightly spaced,
as well as faster (noticeably steeper slope) than Tide B. Within all three
packets, it is evident that the space-time trajectory of each wave changes
with decreasing distance to shore. While Tides A and C contain consistently
imaged features from offshore to onshore, Tide B is visibly more complicated,
with many of the linear features broken into shorter segments throughout the
cross-shore. In the proceeding sections, we will discuss inter-packet speed
variability, spatial complexity, and polarity reversal using a close analysis
of these three tides.

###  4.3 Event-based characteristics of internal tides

Tides A and C demonstrate vastly different intra-tidal, or intra-packet wave
characteristics. In Tide A, a very large bore behaves nearly independently,
with some short-lived trailing and leading waves. Tide C, on the other hand,
holds a consistent shape of rank-ordered waves with the lead bore the largest
followed by a series of waves of decreasing amplitude. Here we examine each
tide in more detail through coupled radar and mooring analysis.

As seen in the space-time radar imagery in Figure 10, Tide A contains a
prominent bright bore that propagates to shore. The lead bore was easily
traced using the edge detection routine from 7000m ($>$40m depth) to shore,
indicated by a red line. Two trailing waves were additionally identified in
the space-time diagram, indicated by blue and green dots at the locations they
cross the 32m and 25m moorings. However, approximately 100m after crossing the
25m mooring location, the signatures of these waves appear to slow (indicated
by their shallowing slopes) and disappear. The space-time diagram also
indicates many higher frequency waves before the main bore, which are
traveling slower than the main bore, appear to merge into one another, and are
eventually overtaken by the main bore of Tide A. If we turn to the mooring
observations, we can see that Tide A contains a large, rarefied lead bore. The
mooring-identified bore is shown as a grey dot, and the time-stack identified
bore is shown as a red arrow, indicating good agreement between the tracking
in each method. The two trailing waves identified in the space-time diagram
are plotted on the mooring data as blue and green dots. These two waves are
seen at the 32m mooring as waves of elevation trailing the lead bore of
depression, and have dramatically shrunk in size by the 25m mooring to the
point of barely being detectable in the temperature contours. The leading
waves seen in the space-time diagram are also very small amplitude in the
mooring data, and are not distinguishable between moorings. Therefore, the
behavior of the lead bore overtaking the slower-traveling but leading higher
frequency waves can only be detected in the space-time diagram.

To gain a better picture of the leading and trailing waves surrounding the
large bore of depression that characterizes Tide A, we can turn to three radar
images that show the bore as it moves to shore. Figure 10 shows that the bore
is uniform in the alongshore for its complete propagation to shore. However,
the leading and trailing waves are not. 10b shows the lead bore as it crosses
the 25m mooring, the leading waves are scalloped in shape. Additionally, these
leading waves have variable orientations in the alongshore, and appear to have
a much stronger southward orientation on the northern portion of the bore
(around OC25N). Interestingly, the trailing waves are much more prominent
toward the southern half of the region, and consistent with the space-time
diagram and mooring analysis, are only visible between the 32m and 25m
moorings. Both the scalloped structures preceding the lead bore, and the
minimal presence of trailing waves in this particular packet, are unique
characteristics of Tide A.

Figure 9: Mooring observations (left) and space-time diagram (right) plotted
for the duration of Tide A.

Figure 10: Three frames showing the propagation of Tide A in the radar
imagery. The colored symbols are the locations of moorings in the radar
footprint (refer to Figure 1). The dashed grey line is the transect from which
the space-time diagram is extracted. The location of the bore is indicated. In
frame a) the bore is between OC40S and OC32S. In frame b) the bore is arriving
at OC25SB. In frame c) the bore is between OC17S and STR3B.

Tide C, on the other hand, is characterized by a lead bore followed by a clean
packet of trailing waves, which are visible in both the space-time diagram and
mooring observations (Figure 11). The mooring observations during Tide C, as
seen in Figure 11b, reveal a packet of solitary waves of depression at OC50
that maintains its shape as it transits to OC17S. The stratification before
the arrival of the bore remains relatively consistent, with the pycnocline
above the mid water depth. Similarly, in the space-time diagram (Figure 11b),
the packet is well-imaged and shows consistent linear features from 40 $m$
water depth to shore. In addition to the lead bore, the next three waves of
the packet have been traced. The intercepts of these traces with the mooring
locations are plotted using the same colored triangles on the mooring data in
Figure 11a. By looking at Tide C in the radar imagery in Figure 12, it is
apparent that the waves are less spatially uniform in the alongshore
direction, distinctly different from Tide A. In Figure 12a, it appears that
there are potentially two packets of internal waves that merge just south of
the of the transect used for the space-time diagram (dashed grey line).

Figure 11: Mooring observations (left) and space-time diagram (right) plotted
for the duration of Tide C.

Figure 12: Three frames showing the propagation of Tide C in the radar
imagery. The cyan dots are the locations of moorings in the footprint (40-9
$m$ depth). The dashed grey line is the transect from which the space-time
diagram is extracted. In frame a the bore is between OC40S and OC32S. In frame
b the bore is arriving at OC25SB. In frame c the bore is about to arrive at
STR3B.

The continuous cross-shore wave imaging also allows us to estimate the speeds
of the waves within a wave packet - something that would be much more
difficult using the mooring data alone. Figure 13 shows wave speed as a
function of cross-shore distance. The color of the speed profiles in Figure 13
corresponds to the traced waves in Figure 11b. The black line represents the
linear speed estimated from the sorted stratification for this time period. It
is important to note that the sorted stratification represents a background, 6
hour moving average product, and thus cannot reveal linear speed estimations
on a wave-by-wave basis. This is because the sorted product represents the
wave guide that the packet of waves enters into, and not an instantaneous
speed prediction for each wave. In order to use the linear speed estimation on
a wave-by-wave basis, one would need to solve Equation 2 at instantaneous
density profiles immediately before each wave passes through that
stratification. Given that Equation 2 yields a maximum speed value when the
pycnocline is in the middle of the water column, the passing of the first wave
will depress the water column to closer to mid water depth, which would solve
for a faster second wave than first wave.

However, nonlinear theory indicates speed should be amplitude dependent
(Equation 3), and thus a rank-ordered packet should have decreasing wave
speeds on a wave-by-wave basis. In Liu et al. (2014), large amplitude waves
O(100m) are observed in SAR satellite data, and nonlinear speed values are
estimated using a correction term is applied to linear theory (Equation 2)
resulting in Equation 3. In Liu et al. (2014), subsurface observations were
not available therefore amplitude ($\zeta$) and $\alpha$ values were guessed
($\zeta$ ranging from 120 m to 20m and $\alpha$ of 0.02). The nonlinear
correction term indicates that nonlinear speed values will scale with wave
amplitude. For Tide C, the observed lead bore speeds are shown in Figure 13 as
solid lines, with colors corresponding to the identified waves in Figure 11).
Indeed, the lead bore is the fastest wave among the packet, with the trailing,
smaller waves slightly slower. Here we estimate a nonlinear speed by adding a
first-order KdV correction to the linear speed:

$c=c_{0}+\frac{\alpha\zeta}{3}$ (3)

where $\zeta$ is the internal wave amplitude and $\alpha$ is the nonlinearity
coefficient of the KdV equation:

$\alpha=\frac{3c_{0}}{2}\frac{\int_{-H}^{0}(\frac{d\Phi}{dz})^{3}dz}{\int_{-H}^{0}(\frac{d\Phi}{dz})^{2}dz}$
(4)

The theoretical nonlinear speeds are computed in the cross-shore for the first
four waves of the internal wave packet by using $\alpha$ computed at OC40S,
OC32S, OC25SB, and OC17S. We manually estimated the amplitude of these waves
by looking at each wave in the mooring salinity data. Amplitude values at the
40m mooring decreased from 18m down to 7m from first wave to last wave in the
packet. The overall amplitudes of the packets decreased as the packet traveled
to shore, resulting in an amplitude range of 7m to 5m at the 17m mooring. An
$\alpha$ value of 0.015 $s^{-2}$ was computed using the sorted stratification
product, and was found to be roughly consistent for each of the mooring
locations. The nonlinear speeds are plotted in Figure 13 as dashed lines,
where the estimated nonlinear corrections are added to the linear speed (black
line). The magnitude of the nonlinear speeds is more comparable to the
magnitude of the radar-estimated speeds than the linear speeds, which
indicates nonlinearity is a better representation in this case than linear
theory. Amplitude-dependent speed decrease between the first wave and last
wave is observed in the theoretical nonlinear speeds. While the lead bore is
observed to be the fastest wave, the observed speed variation between waves
among the wave packet is less distinguishable, and such variation likely falls
within the margin of error for observed speed estimation.

Figure 13: Observed, linear, and nonlinear speeds during Tide C. Solid colored
lines represent cross-shore speeds estimated during Tide C from the radar
space-time diagram, for the first four waves of the packet. Black solid line
represents the linear theory estimated using a 6-hour sorted stratification
product. Dashed lines represent the linear speed plus a KdV nonlinear
correction term using the amplitude of the first four waves of the packet.

###  4.4 Polarity reversal

Figure 14 shows six snapshots of the propagation of Tide B to shore. Through
use of the combined radar and in situ dataset, we have identified an instance
of potential polarity reversal during this tide. Tide B begins at OC50 as a
packet of internal waves of depression which can be seen in Figure 15. This is
to be expected, as the pycnocline has restored to its position higher in the
water column (shallower than the critical depth) following tide A. While the
radar imaging does not reveal NLIWs at OC50 due to lower signal at far ranges
during this time period, it is apparent in the moorings that the NLIWs in the
packet have relatively constant temporal spacing throughout the packet (Figure
15a). Between OC50 and OC40S, however, the temporal gap between the lead bore
and second wave widens in the mooring data. In the OC40S mooring data, the gap
between the first (red) and second (orange) wave is larger than the gap
between the first and second waves in OC50, and also wider than the rest of
the waves in the packet at OC40S. This is additionally apparent in the radar
imagery, as seen in Figure 14a. At this time stamp, the packet of waves are
arriving at OC40S. There is considerably more spacing between the lead bore
and the rest of the packet of NLIWs. Additionally, the offshore radar
signature of the wave packet is a series of wide, dark bands with the bright
band between the waves the same intensity as the background intensity. The
packet is imaged almost entirely via dark signatures evident of divergence,
with little to no evidence in the radar imagery of increased backscatter above
the background backscatter. This imaging mechanism is unique to this tide when
compared to Tides A and C.

Figure 14: Six frames showing the propagation of Tide B in the radar imagery.
The colored symbols are the locations of moorings in the radar footprint
(refer to Figure 1). The dashed grey line is the transect from which the
space-time diagram is extracted. The location of the bore is indicated.

At OC32s, the structure of the first two waves (red and orange) is comparable
to OC40S, although the gap between them has shortened. This is evident in both
the mooring observations, as well as in the space-time diagram (Figure 15) as
the red and orange traces veer closer together near the 32 $m$ depth contour.
In looking at the mooring data, the rest of the packet looks somewhat
different. At OC40S the packet of waves of depression each restored to a
relatively consistent pycnocline depth ($\sim$30 $m$). However, at OC32S, the
waves trailing the second (orange) wave do not restore to a comparable depth
as the red and orange waves, rather they only restore to $\sim$20 $m$. This is
a potential hinting of polarity reversal. The OC25S mooring observations
reveal a potential initiation of polarity reversal among the red and orange
waves. The pycnocline leading up to the red wave has now depressed to $\sim$15
$m$. The amplitude of the red wave is now quite small, roughly 5 $m$ compared
to 10-15 $m$ at OC50 thru OC32S. Between the red and orange wave, there is an
evident wave of elevation emerging from the depressed pycnocline. This wave of
elevation between the red and orange waves becomes even more apparent at
OC17S.

Figure 15: Mooring observations (left) and space-time diagram (right) plotted
for the duration of Tide B.

Tide B additionally illustrates the strong benefits of combining remote
sensing observations with in situ observations for tracking of internal bores.
In the radar images (Figure 14) and space-time diagram (Figure 15), the lead
bore is straightforward to identify from 40 to 32 $m$ depths, as it is the
leading wave and the brightest feature. However, between 32 $m$ and 25 $m$
(Figure 14 panel c) the signature behind the lead bore becomes the brightest
feature in the radar. This is actually the restored pycnocline between the
first (red) and second (orange) waves, which we hypothesize becomes a wave of
evolution. In the radar imagery alone (Figure 14) the lead bore would not be
correctly identified as the imaging changes as the waves propagates to shore.
This has implications in the ability to estimate speeds from the radar imagery
alone, because the identification would jump between waves causing an
artificial jump in speed. In the space-time diagram (Figure 15), the lead and
second bores are more visible as distinct features than in the radar images
(Figure 14), due to the intensity normalization and the ability to track a
single feature over time. However, without comparison to the mooring
observations, the identified bore would likely be the second wave of the
packet, as this is a brighter feature than the first wave.

###  4.5 Wave reflection from shore

Throughout the two month experiment, space-time diagrams were made every 6
hours intended to capture each internal tide. Through visualization of these
space-time diagrams we noticed several tides that contained an offshore-
propagating bright feature, indicative of potential internal wave reflection.
Two representative space-time diagrams are shown in Figure 16. In the left
diagram, a bright linear feature propagates to shore with a sharp turnaround
at 3500m offshore (32m depth). In this diagram it is quite evident that the
off-shore propagating wave is connected to the incoming bore, indicating
likely imaging of a reflected internal wave. On the right panel, however,
there is an offshore-propagating feature that begins farther inshore at 1000m
meters from shore, or 12m water depth. While this offshore propagating feature
appears to occur at a time period when a packet of waves is coming to shore,
it is not possible to link this feature to potential reflection of a specific
incoming internal wave. A similar front is observed inshore at 1000m in the
left diagram time as well. While these potential reflected internal waves are
not investigated further in this study, we have included them as an indication
of the capabilities of this sensing method, as well as motivation for future
work.

Figure 16: Space-time diagrams from two different time periods showing
examples of offshore-propagating features indicative of reflected internal
waves.

##  5 Discussion

The presented speed estimation technique using X-Band radar proves comparable
to speed estimation from moorings, while providing higher cross-shore
resolution of the shoaling speed profile. Overall cross-shore trends in
internal tide bore speed are consistent with the findings of McSweeney et al.
(2020a), thereby demonstrating the utility of an X-Band radar as a tool for
long-term bore speed estimation. However, interestingly, the lead bore as seen
via in situ observations is not consistently the brightest wave in the X-band
radar observations. Specifically, around 40 $m$ water depth in Tide B, the in
situ observed lead wave is evident as the brightest and leading wave in the
X-band radar observations. However, around 32 $m$ depth, the second wave
becomes significantly brighter in the X-band radar. It can be seen in the in
situ observations that the first wave, between 32 and 25 $m$ depth has
transformed from a wave of depression to a wave of elevation, and in the
process become a smaller amplitude wave. We hypothesize that this amplitude
reduction during the polarity reversal process is leading to the reduction in
brightness of the first wave in the radar. With the radar alone, the brighter,
second wave is likely to be identified as the leading edge of the wave packet
in Tide B, thus missing the first wave observed in the in situ observations.
This highlights the importance of the cross-sensor comparison.

Tides A and C contain solitary waves of depression. While Tide A contains,
mostly, a single bore of depression, Tide C is a packet of waves of
depression. We show that the first four waves are rank ordered in amplitude,
with the lead bore exhibiting the fastest speed. Using a KdV nonlinear
correction to the theoretical linear speeds, the nonlinear speed reasonably
well predicts the speed of this wave packet. Additionally, while Tide B does
not show rank ordered waves in terms of speed, it is evident in the space-time
diagram and mooring observations that the wavelength of the first waves is
longer than the rest of the packet. Several studies have looked at inter-
packet dynamics. In Fu and Holt (1982), Seasat SAR images show that there is
variability in wavelength within a group of waves in deep water. This study
hypothesized that the waves were rank ordered in amplitude, but was not able
to confirm without mooring observations. Additionally, Sutherland (2001)
studies wave packet dispersion. Using fully nonlinear finite amplitude
simulations, they predict that for small-amplitude waves, the amplitude within
a wave packet decreases over time. This is consistent with the findings in
Tide C. It is of note that the phase speed of the waves in Tide C are the
fastest that were detected using the radar throughout the experiment; however,
at this point it is unclear why. For packets with larger amplitude waves,
Sutherland (2001) predict that the amplitude will increase initially, but then
decreases as the packet subdivide, which is a possible explanation for the
increased spacing that occurs during Tide B.

Tide B additionally demonstrates a potential polarity reversal from waves of
depression to one or multiple waves of elevation. To corroborate the
likelihood of this happening, we compute the parameter $\alpha$ (Equation 4)
over depth for each of the three tides. To do this, density measurements at
OC50, OC40S, OC32S, OC25S, OC17S, and STR3B are averaged 1 hour before the
arrival of the bore at each mooring. A 1 hour window was chosen as the
representative wave guide into which the packet of NLIWs was entering. The
values of $\alpha$ for Tides A and C fall between 0.01 and 0.02 $s^{-2}$,
while values of $\alpha$ are 0.01 $s^{-2}$ at the 40m mooring but dip to 0
$s^{-2}$ at the 30m mooring and -0.01 $s^{-2}$ by the 17m mooring. Positive
values of alpha indicate the pycnocline is above mid water-column, and NLIWs
should be waves of depression. A sign change in $\alpha$ to negative indicates
the pycnocline has dropped below mid water-column, and waves of elevation are
likely. In both Tides A and C, values of alpha remain positive through all
depths, and, only waves of depression are observed. In Tide B, however, the
decrease in $\alpha$ to a negative value supports the observations that
initial waves of depression offshore undergo a polarity reversal into waves of
elevation beginning around the 25 $m$ depth contour.

Contrary to the theory of weakly nonlinear amplitude dispersion, which
predicts a rank-ordered decrease in wave speed throughout the packet, linear
theory predicts an increased wave speed throughout a wave packet due to the
individual waves’ influence on the stratification. In other words, the waves
themselves influence the wave guide for the proceeding waves. According to
Equation 3, the stratification that produces the fastest wave speed will be
when the pycnocline is at the critical wave depth, or perfectly mid-depth.
Typically, the stratification is elevated above the critical depth, or on
rarer occasions, depressed below the critical depth, before the arrival of the
wave. Therefore, the depression or elevation of the pycnocline due to the
presence of the waves to the critical depth will increase the theoretical
linear wave speed.

Last, we discuss the influences that internal tides may have on each other. It
is of particular note that at OC50, OC40S, and OC32S, the pycnocline between
Tides A and B restores too a depth above mid-water column. However, at OC25S
and shoreward, the pycnocline remains depressed below mid water-column between
Tides A and B. This is a potential cause of the complicated dynamics of this
tide, and highlights the importance of tide-to-tide influences on the
stratification.

##  6 Conclusions

We have analyzed observations of internal waves in a unique dataset collected
during the Inner Shelf Dynamics Experiment, September-October 2017. The
observations consist of X-Band radar imagery gathered over a 10 $km$ radius of
the nearshore inner shelf, as well as stationary mooring observations. Not
only is this a novel dataset due to its high spatial and temporal resolution
and long deployment duration, it is additionally a novel study due to the
relatively shallow water in which the waves are observed. Thus, we have
provided an overview of observed transformation dynamics as the waves shoal,
beginning in depths $<$40 $m$ and eventually to shore dissipation. The
developed detection routine using the X-Band radar uses space-time diagrams
along cross-shore transects to detail internal wave speed and angle with
resolution of O(5 $m$) and O(2 $sec$). We analyze 7 leading bores (largest
wave in internal tide) for an overall cross-shore speed trend, which indicates
that the internal bores propagate with steady speed that is slower than linear
theory until $\sim$32 $m$ depth, surpasses linear theory, and then decreases
at a rate faster than linear theory would predict. In addition to analysis of
lead bores, the remote sensing technique allows for in depth analysis of
multiple waves within a packet, which is traditionally difficult to analyze
over a transformation region from mooring data alone. We have detailed three
nearly-consecutive tides using both X-Band radar and mooring observations to
reveal evidence of polarity reversal and inter-packet dispersion dynamics.
Wave-to-wave variability among packets indicates that the waves are rank
ordered, with the lead bore being the fastest wave. Additionally, by looking
in detail at three packets, we have identified likely tide-to-tide influences
on the stratification that influence consecutive packet behavior. In the
instance of observed polarity reversal dynamics, the second waves appears to
transform from a wave of depression to a wave of elevation, and computed
values of the KdV nonlinearity coefficient $\alpha$ switches signs indicating
that a polarity reversal is likely. Using the X-band radar, the ability to
track single features consistently through space makes it possible to reveal
inter-packet dynamics which poses a challenge to the point observations of
moorings alone. However, the presence of in situ observations with spatial and
temporal overlap to the radar remains critical for understanding the
subsurface shape and behavior of the remotely sensed internal waves.

###### Acknowledgements.

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