# Transfer learning to improve streamflow forecasts in data sparse regions

Roland Oruche, Lisa Egede, Tracy Baker, Fearghal O’Donncha
Department of Electrical Engineering & Computer Science, University of
Missouri–Columbia, USA
Human Computer Interaction Institute, Carnegie Mellon University, USA
The Nature Conservancy, New York, NY, USA
IBM Research Europe, IE
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Effective water resource management requires information on water availability
– both in terms of quality and quantity - spatially and temporally. In this
paper, we study the methodology behind Transfer Learning (TL) through fine-
tuning and parameter transferring for better generalization performance of
streamflow prediction in data-sparse regions. We propose a standard recurrent
neural network in the form of Long Short-Term Memory (LSTM) to fit on a
sufficiently large source domain dataset and repurpose the learned weights to
a significantly smaller, yet similar target domain datasets. We present a
methodology to implement transfer learning approaches for spatiotemporal
applications by separating the spatial and temporal components of the model
and training the model to generalize based on categorical datasets
representing spatial variability. The framework is developed on a rich
benchmark dataset from the US and evaluated on a smaller dataset collected by
The Nature Conservancy in Kenya. The LSTM model exhibits generalization
performance through our TL technique. Results from this current experiment
demonstrate the effective predictive skill of forecasting streamflow responses
when knowledge transferring and static descriptors are used to improve
hydrologic model generalization in data-sparse regions.

## Introduction

Machine learning has had limited impact in hydrological applications due
primarily to data sparsity, the large spatial and temporal extents involved,
and the challenges of generalising to conditions not observed in the training
data (particularly critical in the face of changing climate). Instead,
practitioners rely on complex hydrological modeling tools that are cumbersome
to set up, and generally require an expert both to configure and interpret
results. While some studies have explored LSTM networks to forecast
hydrological conditions, these have been applied on benchmark datasets from
regions with well developed, long-term environmental monitoring programs (?).
In this paper, we propose a framework based on transfer learning and
categorical descriptors of pertinent hydrological and geographical features to
forecast streamflow discharge in the Upper Tana region in Kenya.

Figure 1: LSTM static descriptors architecture

The 2018 edition of the United Nations World Water Development Report stated
that nearly 6 billion people will suffer from clean water scarcity by 2050
with some suggestions that even this number is an underestimate (?). Climate
change, population growth, and pollution are depleting water sources across
the globe, threatening the livelihoods of millions of people. The Nature
Conservancy (TNC) is a globally active conservation organization whose mission
is to conserve the water and lands on which life depends and to support
disparate water restoration projects. A key tool in the TNC toolbox is Water
Funds, which aims to support water conservation and restoration activities by
investing in land management to improve water quality and quantity and
generate long-term benefits for people and nature.

TNC has created 35 Water Funds in 12 countries and they project that by 2025,
water funds can be capable of providing 70 million people with water security,
improving livelihoods for 150,000 rural community members, and protecting 2
million hectares of freshwater in Latin America, Asia, and Africa (?). A
fundamental challenge to water fund success is choosing the most effective
soil and water conservation and land management strategies while also
remaining cognisant of the local environment and human activities. Another
critical challenge is that there is often little historical data available for
accurately representing the rainfall-runoff relationship and it is often
infeasible to collect sufficient (long-term) observation data to characterise
water quality and quantity over large regions. For example, the Upper Tana
Water Fund in Kenya covers an area approximately
$17.000\text{\,}\mathrm{k}\mathrm{m}^{2}$), and data is not well distributed
temporally and spatially. It is common practice to then fit physically based
models to the extent possible to describe the system’s potential response to
changes.

The most widely used hydrological models to use in data scarce regions is the
Soil and Water Assessment Tool (SWAT), which has seen widespread usage in
Africa (?) and globally. SWAT is a continuous‐time model that can operate on
daily, monthly, and annual time steps and was designed to predict the
potential impact of land management practices on water quality and quantity in
large ungauged agricultural watersheds. The model is both physically and
empirically based, computationally efficient, and capable of continuous
simulation over long periods. SWAT was developed by the USDA Agricultural
Research Services with development beginning in 1980 with many additional
modules implemented in the intervening years (?). Major model components
include weather, hydrology, soil temperature and properties, plant growth,
nutrients, pesticides, bacteria and pathogens, and land management. The model
has been applied numerous times in the Tana River Basin (?; ?; ?; ?).

The use of machine learning in hydrology has shown promising results for tasks
such as flood forecasting and run-off prediction (?; ?; ?; ?). However, these
successes typically occur in countries with well developed environmental
monitoring programs and relatively plentiful observation data to train on. No
previous studies have demonstrated the ability to effectively generalize to
regions where data are scarce and inconsistent due to the lack of temporal and
geospatial resources (geographical constraints, lack of population, funding
sources).

In this paper, we applied Transfer Learning (TL), in which the knowledge
manifested in one task is applied (or transferred) to similar objective tasks.
Specifically, we trained a Long Short-Term Memory (LSTM) on a source domain,
and fine-tuned the resulting model towards a target domain. We refer to the
source as the collection of inputs used to train the LSTM model, while the
target is the significantly smaller dataset used to fit and evaluate the
model’s performance. We employed a transfer learning technique that optimizes
the weights during training and fine-tunes the parameters on a selected basin
during validation. These parameters were then transferred to a replicated
multi-layered LSTM network to generalize the streamflow response over
different regions. We trained and optimized our model on our source domain
using the US Catchment Attributes and Meteorology for Large-Sample Studies
(CAMELS-US) dataset, a benchmark dataset that covers 531 basins across the US
region (?; ?). We then performed TL over our target domain to generalize the
performance using a dataset we curated with The Nature Conservancy (TNC) which
covers the Upper Tana River Basin in Kenya. Previous studies have demonstrated
that combining categorical geospatial features (e.g., land cover, climate
metrics, soil properties) in addition to corresponding dynamic inputs
(precipitation, air temperature) helps improve the performance of streamflow
prediction for ML-based hydrological models (?). Influenced by this, the
structure of the Kenya dataset generated the same static basin feature as
additional input to our LSTM model as illustrated in Figure 1. We validated
the utility of this approach by conducting extensive experiments to
demonstrate effective predictive skill of a standard LSTM model with transfer
learning and static descriptors.

The contributions of this paper are as follows:

  * •

We present a novel dataset of hydrological and categorical catchment
attributes collected from the Upper Tana River Basin in Kenya, which follows
the same structure as the benchmark CAMELS dataset. We highlight examples of
scarcity in the temporal and geospatial data collected.

  * •

We propose a transfer learning approach that optimizes the models performance
over the CAMEL-US source domain dataset to address the issues of data sparsity
in smaller datasets. We use a fine-tuning method that randomly selects a basin
that’s performance score was less than 50% the median distribution. The
model’s parameters are then transferred over the TNC dataset to fit and
predict the streamflow response.

  * •

We present experimental results that compare models using transfer learning
and static descriptors to its counterparts. These extensive experiments show
the model’s effective predictive skill of forecasting streamflow response
where there lacks sufficient data needed for effective generalization.

  * •

Finally, we discuss further research opportunities to apply machine learning
to improve water management and conservation practises.

## Related Work

Recent advancements in machine learning has led to widespread interest amongst
hydrologists and environmental scientists as a solution to address the
challenges that persist with streamflow and run-off forecasting. While
previous works have approached performance levels of state-of-the-art physics-
based methods (?; ?; ?), the challenge remains whether it can generalize in
different domains, and specifically if it can perform in regions with limited
training data. Hence, we review the existing literature on transfer learning
techniques and its applicability for use cases around hydrology.

Previous works such as (?) categorize the types of transfer learning as
inductive, transductive, and unsupervised, which has built foundation for many
techniques used in machine learning, more specifically, time-series
forecasting (?; ?; ?). For instance, Lin et al. (?) forecasts traffic
conditions in poorly monitored urban areas through a transfer learning
approach that extracts useful spatio-temporal features from data-rich traffic
areas. McCarthy et al. (?) has shown that using a unified vector
representation approach between the source and target data to transfer
knowledge in heterogeneous feature spaces improves flight delay forecasting.
Fawaz et al. (?) uses convolutional neural networks (CNNs) to invoke transfer
learning by fine-tuning the weights of a pre-trained CNN model over the UCR
time-series benchmark dataset for classification tasks. In cases relevant to
hydrology, Kimura et al. (?) implements a feature-representation transfer
approach via a CNN from time-series data for predicting floods. The CNN takes
advantage of low-level feature extraction to transfer knowledge in a target
domain based on a pre-processing conversion mechanism between the spatial
imagery and time-dependent data inputs. Gang et al. (?) addresses the data-
sparsity issue in small and inconsistent target datasets via transfer learning
to improve the performance of flood forecasting in both data-rich and data-
sparse urban areas. The success of transfer learning, however, has mainly been
applied in computer vision for low to mid-level feature mapping or
implementing a freezing mechanism on CNNs to reduce computational overhead.
There presents a lack of literature on transfer learning used for LSTM
networks and other variations of RNNs. In addition, data sparsity and
inconsistency remains a critical issue behind the performance degradation in
generalization.

Physics- or empirical-based hydrological models are well established in the
literature with research in the space receiving significant impetus with the
US Clean Water Act of 1977. In these systems, the hydrological processes of
water movement are represented by finite difference equations that are
resolved over defined spatial and temporal ranges. Data inputs to resolve
streamflow processes include meteorological forcing and a large number of
parameters describing the physical characteristics of the catchment (soil
moisture content, initial water depth, topography, topology, dimensions of
river network, etc.) (?). Popular modelling systems include SWAT (?), MIKE SHE
(?), and the VIC framework (?). On the SWAT model alone, there are over 4,500
peer-reviewed journal articles describing its application to different
hydrology studies (?).

More recently, extensive research efforts have focused on the potential of
deep learning (DL) for hydrology studies (?; ?). In particular, research has
focused on the potential of recurrent networks and LSTM to resolve the
complex, nonlinear, spatiotemporal relationship between meteorological
forcing, soil moisture and streamflow (?). In a provocative recent paper, (?)
argued that there is significantly more information in large-scale
hydrological data sets than hydrologists have been able to translate into
theory or models. This argument for increased scientific insight and
performance from machine learning rests on the assumption that large-scale
data sets are available globally (over sufficient historical periods) to
condition and inform on hydrological response. While significant progress on
hydrology dataset curation has been achieved in the US (?; ?), and Europe (?)
this is not implemented for many other regions. Inspired by the existing
research contributions, we investigate and apply a transfer learning approach
to address the issue of data sparsity when forecasting the streamflow response
of catchment areas.

## Data Collection

The Tana River supplies 95% of Nairobi’s water and half of the country’s
energy. The entire Tana River Basin covers
$\sim$$95.000\text{\,}\mathrm{k}\mathrm{m}^{2}$ and is home to millions of
wildlife species as well local communities (?). Since the 1970s, the number of
small subsistence farms in the upper basin has skyrocketed (?). The dual
effects of climate change and rapid population growth are leading to increased
sediment erosion in the Upper Tana River Basin, reducing the capacity of
reservoirs, and increasing water treatment costs. Water scarcity is predicted
to disproportionately affect communities that have least contributed to
climate change, such as the millions of farmers and fishermen who rely on the
Tana River and its tributaries for survival and economic prosperity. Taking
action now to prevent further damage and help these communities preserve their
water resources is critical to ensure the region’s water security for the
future. To this end, The Nature Conservancy created a water fund, an
organization that brings together public and private stakeholders to address
threats to water security at their source through targeted, long-term
investments in watershed conservation and management activities (?). In 2015,
The Nature Conservancy established the Upper Tana-Nairobi Water Fund as the
first of its kind in Africa. The Water Fund aims to improve water quality and
quantity for all stakeholders, and is founded on the principle that it is less
expensive to prevent water problems at the source than it is to address them
further downstream.

The primary focus of the Upper Tana-Nairobi Water Fund (UTNWF) is to improve
farming practices in the watershed. Water Fund partners work with almost
50,000 farmers to provide the skills, training and resources they need to
conserve water, reduce soil runoff, and improve productivity. Effective
management of water conservation activities require quantitative metrics on
differences in water availability in different areas of the watershed, subject
to different water conservation strategies (and control watersheds with no
intervention). Previously, TNC have relied on a combination of SWAT model
simulations together with data measurements to inform Water Fund operations
(?). However, this comes with the challenges outlined earlier related to
computational cost, complexity, and need for high level of user expertise.

UTNWF collects a relatively rich hydrologic dataset that provides
opportunities for DL-based forecasting. Data were collected from February 2015
to January 2020 at 26 stream locations as described in Table 1 in (?). Data on
water level and temperature were collected at 30-minute intervals using Onset
HOBO water level data logger. HOBO loggers are a robust, widely-used tool to
continuously monitor water level and temperature with specified device
accuracy of 0.1%. A standard challenge in hydrology is that water level data
are easy to measure with low cost equipment while flow or discharge are
difficult to measure requiring sophisticated equipment and experienced
personnel. On the other hand, water level communicate little meaningful
information with streamflow or discharge necessary to quantify water
availability and hydrological processes.

Water level data were converted to streamflow using established heuristic
relationship between water level and discharge (termed stage–discharge rating
curve analysis) (?). This required the collection of streamflow data,
development of water level-streamflow rating curve or relationships, and using
the generated relationship to convert water level measurements to streamflow.
Flow measurements were collected using an Acoustic Doppler Current Profiler
(ADCP) instrument. An ADCP is a type of sonar that measures and records water
current velocities over a range of depths using the Doppler effect of sound
waves scattered back from particles within the water column. Point
measurements were collected at multiple locations across the river channel and
these used to compute the streamflow discharge at a point in time. Collecting
these measurements under multiple streamflow conditions allows one to create
the water level discharge relationship. In our case, flow measurements were
collected at each station between 50–200 times over the study period. Where
sufficient flow data were not available robust streamflow relationships could
not be generated. The HOBO water level measurements were converted to
streamflow measurements using the above process (19 of the 26 stations had
suitable flow measurements) and data were resampled from 30 minute intervals
to daily values. Figure 2 presents a heatmap illustrating range of observed
values and also highlighting challenges related to data sparsity. For most
stations, there are periods when data was not recorded, extending for up to
50% of the time at some stations. While this is a challenge for DL, it also
highlights the possibility of using DL models to address data gaps and inform
hydrology processes.

Figure 2: Visualisation of range and availability of streamflow measurements
($\mathrm{m^{3}}/s$) at each of the 19 stations. The colorbar indicates values
of streamflow while whitespace denotes periods where no data were collected.
Figure 3: Main architecture of the transfer learning approach in which the
knowledge learned from training over a fairly large source domain dataset (US
CAMELS) is applied to a smaller and sparse target dataset (Kenya) for
generalization performance of streamflow prediction.

## Proposed Approach

In this section, we first introduce a standard recurrent neural network,
specifically Long Short-Term Memory (LSTM), that uses a loss function commonly
used in hydrology. We then describe in detail the method behind our transfer
learning approach.

### Model Architecture

We propose a multi-layer LSTM network as the base model to perform transfer
learning through two separate domains, as shown in Figure 3. We set up our
deep learning architecture from the open-source work of (?; ?), where the
inputs are dynamic features from water catchment processes such as: min/max
air temperature, vapor pressure, and precipitation. The aim of the
hydrological LSTM model is to learn the prediction of the continuous
streamflow response on a target dataset (Kenya) from the long-term dependent
features given the knowledge learned from a large source domain dataset (US
CAMELS).

To calculate the model’s performance of streamflow prediction, we present the
use of Nash-Sutcliffe efficiency (NSE) (?), which is a statistical estimator
that measures the predictive skill of a hydrological model. NSE is defined as:

$NSE=1-\frac{\sum_{t=1}^{T}(Q_{m}^{t}-Q_{o}^{t})^{2}}{\sum_{t=1}^{T}(Q_{o}^{t}-\bar{Q}_{o})^{2}},$
(1)

where $Q_{m}^{t}$ is the modeled discharge (streamflow) at time $t$ , and
$Q_{o}^{t}$ is the observed discharge at time $t$. The value of the NSE varies
on the interval $(-\infty,1]$, where a set of conditions are used as criteria
for understanding model performance. $NSE=0$ denotes equivalent predictive
skill of the model to the mean of the time-series (i.e., sum of squared
errors). In a perfect model situation, $NSE=1$ is when the estimation error
variance equal to zero. If $NSE\leq 0$, the estimation error variance of the
model is larger than the variance of observation. Hence, if the observed mean
is a better predictor than the model, then the model has little predictive
skill.

Similar models such as mean squared error (MSE) are widely adopted for
calibration and model evaluation with observed data (?). Unlike MSE, NSE does
not overweight catchments with larger streamflow values, which could have
influence on the models performance (?) (Figure 2 indicates the large data
ranges involved). Hence, this study focuses solely on the performance of the
hydrological model through the NSE coefficient.

### Transfer Learning Approach

Given a LSTM model $f$ that fits over a domain dataset $D={(X,Y)}$ where $X$
is the set of inputs $(x_{1},x_{2},...,x_{n})$ and $Y$ is the set of labels
$(y_{1},y_{2},...,y_{n})$, we define a source source domain dataset to be
$D_{s}={\\{(x_{s}^{i},y_{s}^{i})\\}}_{i=1}^{m_{s}}$ and a target domain
dataset to be $D_{t}={\\{(x_{t}^{i},y_{t}^{i})\\}}_{i=1}^{m_{t}}$. We also
define a domain task $T={(Y,f(X))}$ where we denote the source task domain as
$T_{s}=\\{{(y_{s}^{i},f(x_{s}^{i}))}\\}$ and the target task domain as
$T_{t}=\\{{(y_{t}^{i},f(x_{t}^{i}))}\\}$. The objective transfer function
learning aims to learn the target conditional probability distribution
$P=(Y_{t}|X_{t})$ in $D_{t}$ with the knowledge gained from both $D_{s}$ and
$T_{s}$, where $D_{s}\neq D_{t}$ or $T_{s}\neq T_{t}$ (?).

The source and target domains share similar feature input space, however
labels $Y_{s}$ and $Y_{t}$ are heterogeneous. Hence, the LSTM model $f$ cannot
be fine-tuned directly on the target domain dataset $D_{t}$ due to the
discrepancy of task objectives. A common technique to transfer knowledge and
fine-tune is to split $f$ between an LSTM model representation
$F_{\bar{\theta}}$ (parametrized by $\bar{\theta}$) and the task-oriented
head–which is the topmost layer used for regression–denoted as
$G_{\theta_{s}}$ (parametrized by $\theta_{s}$). The head of the LSTM model
can be replaced with $H_{\theta_{t}}$ (parametrized to $\theta_{t}$), where a
random initialization is used to fit the distribution of target domain label
$Y_{t}$. We can then perform a standard fine-tuning method over a loss
function in Equation 2 for parameter optimization:

$\operatorname*{arg\,min}_{\bar{\theta},\theta_{t}}\frac{1}{|D_{t}|}\sum_{i=1}^{m_{t}}L(H_{\theta_{t}}(F_{\bar{\theta}}(x_{t}^{i},y_{t}^{i}))).$
(2)

Using Equation 1, we apply the replacement and initialization of the LSTM
regression layer to refine the NSE loss function as:

$\operatorname*{arg\,min}_{\bar{\theta},\theta_{t}}~{}1-\frac{\sum_{t=1}^{m_{t}}(H_{\theta_{t}}(F_{\bar{\theta}}(x_{t}^{i}))-y_{t}^{i})^{2}}{\sum_{t=1}^{m_{t}}(y_{t}^{i}-\bar{y}_{t}^{i})^{2}}.$
(3)

The loss function in Equation 3 enables the LSTM model to use its learned
weights as a starting point for training as opposed to traditional random
initialization methods such as Gaussian distribution. Hence, this yields an
opportunity at improving the generalization performance of predicting
streamflow response.

TNC Dataset in Kenya Region
---
Model | $\mathrm{Median_{NSE}}$ | $\mathrm{Mean_{NSE}}$ | $\mathrm{Max_{NSE}}$ | $\mathrm{Min_{NSE}}$ | $\mathrm{STD}$ | $\mathrm{NSE>0}$
$\mathrm{LSTM}$ | -0.85 | -30.70 | 0.58 | -1259.73 | 2.08 | 2
$\mathrm{LSTM_{SCA}}$ | -0.93 | -19.70 | 0.62 | -423.45 | 1.29 | 1
$\mathrm{LSTM_{TL}}$ | -0.49 | -14.33 | 0.53 | -314.46 | 0.46 | 3
$\mathrm{LSTM_{TL+SCA}}$ | -0.51 | -17.48 | 0.67 | -577.23 | 1.02 | 3
Table 1: Model comparison of the proposed methods using the NSE model
estimation. The statistics were averaged across 30 runs using a different seed
each time.

## Experiments

### Experimental Setup

Datasets We collected data from two primary sources: (i) US CAMELS and (ii)
TNC. The US CAMELS (?) is a large sample benchmark dataset that has covered
temporal and geospatial information related hydrology across 531 basins from
October 1st, 1980 through September 30th, 2014. All of the LSTM model
calibration – which is utilized for transfer learning – was performed from
October 1st, 1999 through September 30th, 2008, while the evaluations were
performed from October 1st, 1989 through September 30th, 1999. The total time
period used for model simulation over the geospatial input data amounts to
10,227 days (28 years).

The Kenya dataset contained hydrological data from 2015-2019, in which 26
river stations were gauged. However, only 12 had sufficient hydrological data
(i.e. observations were available $\mathrm{>50\%}$ of time) to use for this
study. We combined with geospatial static catchment attributes to characterise
the hydrological profile of the area similar to the US CAMELS data. These
categorical static attributes included: catchment area, elevation, and slope,
mean precipitation, high and low precipitation frequency and duration, mean
evapotranspiration, and aridity (ratio of precipitation and
evapotranspiration), and vegetation indices such as leaf area index and NDVI.
Suitably conditioned data were available from 01/09/2015–16/10/2019 which was
split into training between period 01/09/2015–22/02/2018 and testing from
22/02/2018–16/10/2019. The total time period used for this experiment is 1,507
days ( $>$ 4 years). In both datasets, we leverage the same subset of
geospatial features which include land cover, climate, soils, as well as
dynamic meteorological descriptors. The target variable is a continuous value
of the daily observed streamflow (or discharge) at each station.

Model Comparison We present the following models used for this experiment:

  * •

Standard LSTM: We employ a standard multi-layered LSTM network based on the
existing work on predicting streamflow (?; ?). The output layer uses a
regression head to simulate the discharge over unseen data.

  * •

LSTM with static catchment attributes(SCA) ($\mathrm{LSTM_{SCA}}$): We
incorporate static basin descriptors related to land cover, soil, climate,
vegetation, and topography that has shown to augment ability for the LSTM to
generalize (?).

  * •

LSTM with transfer learning ($\mathrm{LSTM_{TL}}$): We present the existing
transfer learning solution using an LSTM over the US CAMELS dataset. The
weights are transferred to generalize over the Kenya dataset.

  * •

LSTM with TL + SCA ($\mathrm{LSTM_{TL+SCA}}$): We combine the ability of
additional catchment attributes to augment the model performance along with
our proposed transfer learning approach.

Hyperparameters and Evaluation Metrics The hyperparameters of the baseline
models were employed uniformly. The LSTM network structure consisted of 128
hidden units and 1 output regression layer used for forecasting daily
streamflow over 30 epochs. We initialize the learning rate $\eta$ to 0.001
during the first epoch and adjust to 0.0005 over the remaining epochs. To
penalize inconsistent predictions, we implement an ADAM optimizer as our
regularization term.

Our primary evaluation metric for this study is the Nash-Sutcliffe model
Efficiency estimation (NSE) (?), which estimation of predicting discharge
compared to similar metrics such as MSE and RMSE. We leveraged six statistical
metrics from our generated NSE values to calculate the performance of the
propose models which include: median ($\mathrm{Median_{NSE}}$), mean
($\mathrm{Mean_{NSE}}$), maximum ($\mathrm{Mean_{NSE}}$), min
($\mathrm{Min_{NSE}}$), standard deviation ($\mathrm{STD}$), and number of
basins where the NSE value is greater than 0 ($\mathrm{NSE>0}$).

Figure 4: A colormap visualization that demonstrates the performance of the
fine-tuned LSTMTL model and its response to generalize on sparse data inputs
over 30 runs.

### Comparing Proposed Methods

We evaluated the performance of each proposed model over the TNC Kenya dataset
and compared their results based on our aforementioned metrics. From Table 1,
the $\mathrm{LSTM_{TL}}$ displayed the best overall performance, notably
achieving a higher $\mathrm{Median_{NSE}}$ and $\mathrm{Mean_{NSE}}$ score.
The optimized weights served as a better starting point for the target LSTM
model than random initialization. Augmenting the standard LSTM model with
static catchment attributes (i.e., $\mathrm{LSTM_{SCA}}$,
$\mathrm{LSTM_{TL+SCA}}$) showed a lack of improvement in performance relative
to their counterparts. While some modest improvement of mean NSE and standard
deviation were reported by addition of SCA to baseline LSTM, this did not
translate to the model with transfer learning. This likely results from the
different scale of geospatial datasets used in the CAMELS and Kenya dataset
which may require some statistical conditioning to resolve more closely. As an
example climate indicators for Kenya were extracted from the ECMWF ERA5 global
reanalysis product at horizontal resolution of
$\sim$$28\text{\,}\mathrm{k}\mathrm{m}$ while the US CAMELS dataset used the
US National N15 Daymet dataset at resolution of
$\sim$$1\text{\,}\mathrm{k}\mathrm{m}$. The difference in the standard
deviation (Std) between the standard LSTM model and $\mathrm{LSTM_{TL}}$ (2.08
vs. 0.46) was largely due to the lack of temporal streamflow observations that
cause large gaps during model training and validation. Employing transfer
learning from a model train on a significantly richer source domain enables
the model to reduce deviation for producing more effective results on unseen
streamflow.

Figure 4 reports the performance of 30 $\mathrm{LSTM_{TL}}$ model instances
trained with random initialization. It indicates that 4 of the 12 stations
reported $\mathrm{NSE<<0}$ while 3 reported $\mathrm{NSE>0}$ and further 3 had
values $\sim 0$. While these values return moderate predictive skill, they
present a promising demonstration of the potential of transfer learning to
address the combined challenge of data sparsity, data uncertainty, and complex
physical and empirical relationships.

Figure 5: Comparisons on the prediction of streamflow at a given basin between
the baseline LSTM and LSTMTL model.

In Figure 5, we compare the performance of the proposed transfer learning
model ($\mathrm{LSTM_{TL}}$) with a baseline model to predict the streamflow
response at a given station. Herein, demonstrates the ability of the transfer
learning model to address the limitations of data collection in areas where
geospatial information is sparse and inconsistent. Compared to the standard
LSTM model, ($\mathrm{LSTM_{TL}}$) provides a more realistic prediction over
the TNC Kenya dataset by leveraging the learned weights from a data-rich
source domain dataset (US CAMELS). Thus, the simulated streamflow from the
($\mathrm{LSTM_{TL}}$) model can be of more effective use in informing
stakeholders in developing areas (e.g., farmers, Kenyan water funds) about the
water availability within the Kenyan region for better water resource
management.

## Discussion

While employing transfer learning has many implications of success, as
demonstrated by the improved performance of the standard LSTM model, there
still presents many concerns regarding its applicability for time series
tasks. Successful transfer learning theories and application has largely been
applied to computer vision tasks (?; ?). Transfer learning has also seen
success in multi-lingual machine translation as it has the ability to create
word representation (embeddings) (?; ?). However, this success has not been
extended to time-series related tasks due to the lack of existing literature
in both theory and application. Time-series models may be elusive, as it is
harder to find a useful hierarchy or a set of intermediate representations
that could generalize to different domains. In addition, negative transfer
learning is a common challenge as it can negatively affect performance when
performing knowledge transferring from a source to a target domain (?).

Furthermore, the initiatives to improve hydrological modeling have equally, if
not more so, relied on sufficient data collection processes as it does on the
model’s underlying algorithm. The sparsity and inconsistency of the Upper Tana
dataset caused performance issues related to the predictive streamflow skill.
This can be due to the lack of equipment used to track meteorological data
such as precipitation and air temperature in developing countries. The US and
many areas across Europe, on the other hand, have had the means of curating
large-sample datasets through the accessible resources of weather tracking at
their disposal. The insufficient data of tracking meteorological information
in countries from Africa presents an issue of informing stakeholders (e.g.,
farmers, water funds) about the water quality and availability for proper
water resource management. Programs such as The trans-African hydro-
meteorological observatory (TAHMO) (?) which aims for continuous monitoring of
precipitation data across Sub-Saharan Africa are key initiatives to allow for
more effective, applied DL to hydrology. In addition, the resolution provided
by different geoscientific monitoring and modelling tools such as NASA and
European Centre for Medium Weather Forecasting (ECMWF) at continental and
global scales could be reconciled towards higher resolution in places such as
Kenya (to provide resolution comparable to that currently provided for US and
Europe by the respective organisations). Approaches for statistical
downscaling of these global datasets (e.g. (?)) could improve the performance
of transfer learning across different geographies.

## Conclusion

The science of hydrology has largely derived from using known physical and
empirical relationships – guided by user expertise and intuition – to resolve
complex rainfall – discharge relationships. Huge potential exists to use deep
learning to augment these inference and intuition from large hydrological
datasets. A fundamental challenge currently is data sparsity.

In this paper, we address the limitations of streamflow predictions in data
sparse regions using transfer learning. We first address the underlying
problem by collecting, pre-processing, and curating a dataset, (TNC Kenya),
similar to that of US CAMELS, which has the same dynamic meteorological input
features and the same set of static features describing each catchment area.
Our approach attempts to generalize a model in small target domain datasets
given knowledge learned from a significantly larger source domain dataset by
knowledge transfer. We employ a multi-layer LSTM model to forecast streamflow
response using the dynamic meteorological inputs as well as static catchment
descriptors. We experimented with the predictive skill of forecasting
streamflow under four conditions concerning transfer learning, static
catchment attributes, both, or neither. Our results demonstrate the improved
performance and potential of transfer learning to give insights on streamflow
prediction in areas where data is sparse due to the lack of temporal and
geospatial resources. Our future work aims to look at other drivers of
predictive streamflow forecasting using spatial-temporal methods such as graph
neural networks and convolutional neural networks.

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Electron Impact Excitation of O III: An Assessment

Kanti M Aggarwal

Astrophysics Research Centre, School of Mathematics and Physics, Queen’s
University Belfast,

Belfast BT7 1NN, Northern Ireland, UK

e-mail<EMAIL_ADDRESS>
Keywords: electon impact excitation of O III, collision strengths, effective
collision strengths, accuracy assessments

###### Abstract

Tayal and Zatsarinny [Astrophys. J. 2017, _8_ 50, 147] have reported results
for energy levels, radiative rates (A-values), lifetimes, and effective
collision strengths ($\Upsilon$) for transitions among 202 levels of C-like O
III. For the calculations they have adopted the multi-configuration Hartree-
Fock (MCHF) code for the energy levels and A-values, and B-spline $R$-matrix
(BSR) code for $\Upsilon$. Their reported results cover a (much) larger range
of levels/transitions than generally available in the literature, and appear
to be accurate for energy levels and A-values. However, the magnitude and
behaviour of $\Upsilon$ do not appear to be correct for several transitions.
We demonstrate this through our independent calculations by adopting the
flexible atomic code (FAC) and recommend a fresh calculation for this
important ion.

##  1 Introduction

C-like O III is an ion of astrophysical importance, because many of its
emission lines are frequently observed in a variety of plasmas, such as
planetary nebulae, H II regions and solar atmospheres, and have been very
useful for both temperature and density diagnostics – see for example the
paper by Tayal and Zatsarinny [1] and references therein. For this reason,
there have been several calculations for the generation of its atomic data,
mainly for energy levels, radiative rates (A-values), collision strengths
($\Omega$), and effective collision strengths ($\Upsilon$). Measurements for
some energy levels have been performed by a few workers and these have been
compiled and assessed by the NIST (National Institute of Standards and
Technology) team, and their recommended values are freely available at their
website: http://www.nist.gov/pml/data/asd.cfm. However, for the remaining
parameters one has to depend on theoretical results.

During the past four decades there have been several calculations for the
determination of A-values, $\Omega$ and $\Upsilon$, for which a variety of
methods have been adopted. Particularly for the calculations of $\Omega$ and
$\Upsilon$, required for diagnostics and modelling of plasmas, the two most
common methods are distorted wave (DW) and $R$-matrix. The main difference
between the two is in the exclusion/inclusion of closed-channel (Feshbach)
resonances, which have significant effect on the values of $\Upsilon$,
particularly for the forbidden transitions, and at almost all temperatures of
interest. For this reason, the $R$-matrix calculations are preferred for
(mainly) $\Upsilon$.

The first $R$-matrix calculations for O III were initiated by Baluja et al.
[2]. After that many other have followed, but the most important and widely
used one are those of Aggarwal and Keenan [3], who reported results for
transitions among 46 levels belonging to the 2s22p2, 2s2p3, 2p4, and
2s22p3$\ell$ configurations. This range of levels/transitions was considerably
extended by Storey et al. [4], who considered 146 levels. Unfortunately, their
results have (very) limited application because their focus was only on 10
transitions belonging to the 5 levels of the 2s22p2 ground configuration
alone. Therefore, Tayal and Zatsarinny [1] have performed yet another
calculation for this important ion, and have considered 202 levels of the
2s22p2, 2s2p3, 2p4, 2s22p3$\ell$, 2s22p4$\ell$, 2s2p23$\ell$, and 2s22p5s
configurations, 14 in total.

For the generation of wavefunctions, i.e. for the calculations of energy
levels and A-values, Tayal and Zatsarinny [1] have adopted the multi-
configuration Hartree-Fock (MCHF) code (in combination with the B-spline
expansion), and for the collisional data, i.e. $\Omega$ and $\Upsilon$, the
B-spline $R$-matrix (BSR) code. Since they have used flexible term dependent
orbital sets to represent the target states, their calculated energies and
A-values are (apparently) accurate, as shown in their tables 1 and 2.
Regarding the other parameters, namely $\Omega$ and $\Upsilon$, they have
presented results for only the latter, but for all transitions (20 301) among
the considered 202 levels. Since corresponding data for 1035 transitions among
the lowest 46 levels are already available [3], they have assessed the
accuracy of their results based on comparisons for these, but have made no
attempt to assess similar accuracy for the remaining 19 266 (95%) transitions.
Therefore, our main focus is on these transitions among the higher lying
levels to see if these results are equally accurate.

In the absence of corresponding $\Omega$ data it is (comparatively) more
difficult to make accuracy assessments, particularly when the oscillator
strengths (f-values) or A-values are also not given. Nevertheless, we make an
attempt by other means, mainly by performing our own calculations with the
easily available and generally reliable Flexible Atomic Code (FAC) of Gu [5],
currently hosted at the website: https://www-amdis.iaea.org/FAC/. This code is
based on the DW method, is fully relativistic, is highly efficient to run, and
more importantly, produces results for background values of collision
strengths ($\Omega_{B}$) which are comparable with other methods, such as the
$R$-matrix. This has been demonstrated in many of our earlier papers for a
wide range of ions – see for example, figure 2 of Aggarwal and Keenan [6] for
Mg V. However, resonances are (generally) not resolved in this code, but
results for $\Omega_{B}$ will be helpful to draw conclusions.

Before we discuss the results of Tayal and Zatsarinny [1] in detail, we will
like to note that in a more recent paper Mao et. al. [7] have reported
$\Upsilon$ data for several C-like ions (N II to Kr XXXI), including O III,
and for even a larger number of levels, i.e. 590 which arise from 24
configurations of 2$\ell{{}^{3}}{n}\ell^{\prime}$ with $n$ = 2 to 4, $\ell$ =
0 and 1, and $\ell^{\prime}$ = 0 to 3, plus three additional configurations of
2s22p5$\ell$ with $\ell$ = 0, 1 and 2. For the calculations they have used the
AutoStructure (AS) code to generate the wavefunctions, i.e. to calculate
energy levels and the A-values, and for the values of $\Omega$ and
subsequently $\Upsilon$ their intermediate coupling frame transformation
(ICFT) method alongwith the standard $R$-matrix code. Although these
calculations are the largest performed so far, and are the most recent one,
yet their accuracy and the subsequent use for applications, particularly for
the analysis of the nebular plasmas, has recently been questioned by Morisset
et al. [8], who have assessed their data to be less accurate, especially at
low temperatures, and have actually warned against their use. Since these
calculations are the largest, as already stated, no direct comparisons are
possible with other existing results, and therefore a detailed assessment of
the accuracy of their data, for a larger number of transitions and over a
wider range of temperature, may be the subject of a separate paper.
Furthermore, the deficiency of the ICFT calculations has already been
discussed in the literature for several other ions [9], although in the
concerned calculations the authors have made some improvements, particularly
by extending the energy range of the $\Omega$ data up to three times the
ionization potential, before extrapolating the range, for the determination of
$\Upsilon$ up to very high temperatures, required for fusion plasmas.

Figure 1: Our calculated values of $\Omega$ with FAC for the 44–202
(triangles: 2s22p(2Po)3d 1F${}^{o}_{3}$–2s2p2(2P)3d 1D2), 45–202 (diamonds:
2s22p(2Po)3d 1P${}^{o}_{1}$–2s2p2(2P)3d 1D2), and 164–202 (stars: 2s2p2(2P)3s
1P1–2s2p2(2P)3d 1D2) transitions of O III.  Figure 2: Our calculated values of
$\Upsilon$ with FAC for the 44–202 (triangles: 2s22p(2Po)3d
1F${}^{o}_{3}$–2s2p2(2P)3d 1D2), 45–202 (diamonds: 2s22p(2Po)3d
1P${}^{o}_{1}$–2s2p2(2P)3d 1D2), and 164–202 (stars: 2s2p2(2P)3s
1P1–2s2p2(2P)3d 1D2) transitions of O III.  Figure 3: The calculated values of
$\Upsilon$ by Tayal and Zatsarinny [1] with BSR for the 44–202 (triangles:
2s22p(2Po)3d 1F${}^{o}_{3}$–2s2p2(2P)3d 1D2), 45–202 (diamonds: 2s22p(2Po)3d
1P${}^{o}_{1}$–2s2p2(2P)3d 1D2), and 164–202 (stars: 2s2p2(2P)3s
1P1–2s2p2(2P)3d 1D2) transitions of O III.

##  2 Collision strengths and effective collision strengths

For our calculations we have considered the same 202 levels among the 14
configurations (given in section 1) as by Tayal and Zatsarinny [1], and
exclusively listed in their table 1. For easy understanding we will use the
same level indices, while referring to a transition, as by them. In Figure 1
we show our results for $\Omega$ for three transitions, namely 44–202
(2s22p(2Po)3d 1F${}^{o}_{3}$–2s2p2(2P)3d 1D2, f = 0.2798), 45–202
(2s22p(2Po)3d 1P${}^{o}_{1}$–2s2p2(2P)3d 1D2, f = 0.3960), and 164–202
(2s2p2(2P)3s 1P1–2s2p2(2P)3d 1D2), out of which the first two are allowed and
have significant f-values, calculated with FAC, whereas the third one is
parity forbidden, and all energies shown in this (and other) figures are
incident. Since all these three (and many more) transitions involve the
highest level of the calculations, their values of $\Omega$ (in any
calculation) vary smoothly, as expected, although no comparisons can be made
with the corresponding (non available) data of Tayal and Zatsarinny [1]. Our
calculated values of $\Upsilon$ are shown in Figure 2 in the same temperature
range as by them, i.e. between 100 and 100 000 K, and the values (expectedly)
vary smoothly with increasing Te. The corresponding results of Tayal and
Zatsarinny are shown in Figure 3, the magnitudes of which are not only lower,
by up to a factor of four, especially for the allowed transitions, but their
behaviour are also anomalous, particularly towards the higher end of the
temperature range. Some differences in $\Upsilon$ values may be due to the
corresponding differences in the f-values, but since both 44–202 and 45–202
are strong transitions, their f-values are not expected to differ by more than
20% – see for example, similar calculations for S III in table 1 of Aggarwal
[10]. Before we discuss the (possible) reasons for the discrepancies we
consider a few more transitions to show that the above three are not the
isolated examples.

In Figure 4 we show our $\Omega$ for three other transitions, namely 185–202
(2s2p2(2P)3p 1D${}^{o}_{2}$–2s2p2(2P)3d 1D2, f = 0.0896), 186–202 (2s2p2(2P)3p
1P${}^{o}_{1}$–2s2p2(2P)3d 1D2, f = 0.2899), and 201–202 (2s2p2(2P)3d
1F3–2s2p2(2P)3d 1D2). Again, the first two are allowed and the third one is
forbidden, but the magnitudes of all these three transitions are significantly
larger than of those shown in Figure 1. This is because these transitions have
very small energies ($<$ 4 eV) and hence converge slowly. Our results of
$\Upsilon$ and those of Tayal and Zatsarinny [1] are shown in Figures 5-6.
Unfortunately the trends (and hence the conclusions) are the same as for those
shown in Figure 1, i.e. their magnitudes are lower than those of ours by up to
a factor of four and the behaviours are incorrect.

The lower magnitudes of $\Upsilon$ of Tayal and Zatsarinny [1] are (probably)
due to the insufficient number of partial waves included by them, i.e. with
angular momentum $J\leq$ 25.5. Although they have included the contribution of
higher neglected partial waves through a top-up procedure, it still
underestimates the magnitudes if implemented at an early stage, and an example
of this can be seen in figure 2 of Aggarwal and Keenan [11] for a transition
of Ni XI. For the incorrect behaviours seen in Figures 3 and 6, there are
three possibilities. First, there may be pseudo-resonances, at energies above
thresholds, which abnormally affect the determinations of $\Upsilon$ results.
Second, their adopted energy mesh of 0.2 Ryd at energies above thresholds, is
too coarse, in comparison to the very fine 0.000 04 Ryd in the thresholds
region, i.e. up to about 4.5 Ryd. However, the possibility of pseudo-
resonances is less likely because of the nature of the wavefunctions they
adopted. Otherwise also, their presence (if any) will increase the magnitude
of $\Upsilon$, rather than decreasing its value. Moreover, they have also
mentioned a smooth variation of $\Omega$ at higher energies, and have
demonstrated that in their figures 1–3. Therefore, (one of) the definite
reason for the anomalous behaviour of their $\Upsilon$ results is the “coarse”
energy mesh adopted at higher energies. This particularly affects transitions
involving the higher levels, as already noted in Figures 3 and 6, and also
discussed earlier for transitions of Mg V [6] and S III [10]. Finally, the
third possibility is the extrapolation they performed for determining $\Omega$
values beyond 30 Ryd, the energy up to which they did actual calculations.
Often the formulae adopted for the extrapolations (for various types of
transitions) are not very robust and many examples of this have been discussed
in our earlier work [9]. As a result of this sometimes the extrapolations do
more harm than good and it is clear from the behaviour of $\Upsilon$ shown in
Figures 3 and 6. More importantly, it is beyond our understanding why they had
to do the extrapolation in the first place, because the highest threshold is
below 5 Ryd (see their table 1) and therefore they have a wide range of over
25 Ryd to calculate $\Upsilon$ up to 105 K, which is only 0.63 Ryd. Hence, the
anomalous behaviours of their $\Upsilon$ results are (probably) due to a
combination of the coarse energy mesh as well as the (unnecessary)
extrapolations.

Figure 4: Our calculated values of $\Omega$ with FAC for the 185–202
(triangles: 2s2p2(2P)3p 1D${}^{o}_{2}$–2s2p2(2P)3d 1D2), 186–202 (diamonds:
2s2p2(2P)3p 1P${}^{o}_{1}$–2s2p2(2P)3d 1D2), and 201–202 (stars: 2s2p2(2P)3d
1F3–2s2p2(2P)3d 1D2) transitions of O III.  Figure 5: Our calculated values of
$\Upsilon$ with FAC for the 185–202 (triangles: 2s2p2(2P)3p
1D${}^{o}_{2}$–2s2p2(2P)3d 1D2), 186–202 (diamonds: 2s2p2(2P)3p
1P${}^{o}_{1}$–2s2p2(2P)3d 1D2), and 201–202 (stars: 2s2p2(2P)3d
1F3–2s2p2(2P)3d 1D2) transitions of O III.  Figure 6: The calculated values of
$\Upsilon$ by Tayal and Zatsarinny [1] with BSR for the 185–202 (triangles:
2s2p2(2P)3p 1D${}^{o}_{2}$–2s2p2(2P)3d 1D2), 186–202 (diamonds: 2s2p2(2P)3p
1P${}^{o}_{1}$–2s2p2(2P)3d 1D2), and 201–202 (stars: 2s2p2(2P)3d
1F3–2s2p2(2P)3d 1D2) transitions of O III.  Figure 7: Our calculated values of
$\Omega$ with FAC for the 83–98 (triangles: 2s22p(2Po)4d
3F${}^{o}_{4}$–2s22p(2Po)4f 3G5), 89–93 (diamonds: 2s22p(2Po)4d
3D${}^{o}_{3}$–2s22p(2Po)4f 3F4), and 94–101 (stars: 2s22p(2Po)4d
1F${}^{o}_{3}$–2s22p(2Po)4f 1G4) transitions of O III.  Figure 8: Our
calculated values of $\Upsilon$ with FAC for the 83–98 (triangles:
2s22p(2Po)4d 3F${}^{o}_{4}$–2s22p(2Po)4f 3G5), 89–93 (diamonds: 2s22p(2Po)4d
3D${}^{o}_{3}$–2s22p(2Po)4f 3F4), and 94–101 (stars: 2s22p(2Po)4d
1F${}^{o}_{3}$–2s22p(2Po)4f 1G4) transitions of O III. Continuous curves:
present results with FAC and broken curves: earlier results of Tayal and
Zatsarinny [1] with BSR.

We now focus our attention on some other transitions which do not involve the
higher lying levels. In Figure 7 we show our values of $\Omega$ for three
transitions, namely 83–98 (2s22p(2Po)4d 3F${}^{o}_{4}$–2s22p(2Po)4f 3G5),
89–93 (2s22p(2Po)4d 3D${}^{o}_{3}$–2s22p(2Po)4f 3F4), and 94–101 (2s22p(2Po)4d
1F${}^{o}_{3}$–2s22p(2Po)4f 1G4), all of which are allowed and have large
magnitudes. The corresponding results of $\Upsilon$ from our calculations and
those of Tayal and Zatsarinny [1] are shown in Figure 8. As earlier, their
$\Upsilon$ values are underestimated by over a factor of three at most
temperatures, although the differences decrease with increasing Te. The reason
for these large differences, particularly at the lower end of Te, is rather
simple. The thresholds region is often dominated by resonances (although
mostly for forbidden transitions) and therefore a top-up for higher neglected
partial waves is performed only at energies above thresholds. Subsequently,
values of $\Omega$ in the thresholds region remain uncorrected if sufficient
number of partial waves are not included, as is the case with their
calculations. This underestimation can be confirmed by an example shown in
figure 4 of Aggarwal et al. [12] for the 3s3p 1P${}^{o}_{1}$ – 3p2 1D2
transition of Fe XV in which the whole $\Omega_{B}$ is lifted upwards with the
inclusion of larger ranges of $J$. The threshold region for these transitions
is of over 1.2 Ryd, about twice the range of Te (0.63 Ryd) considered by them,
and therefore values of $\Upsilon$ are underestimated over the entire Te
range. Since these (and several other) transitions have comparatively larger
magnitudes their effect on modelling applications are larger than of the
weaker ones.

Figure 9: Our calculated values of $\Omega$ with FAC for the 2–51 (triangles:
2p2 3P1–2s2p2(4P)3s 3P1), 2–72 (diamonds: 2p2 3P1–2s22p(2Po)4p 3P1), and 3–23
(stars: 2p2 3P2–2s22p(2Po)3p 1P1) transitions of O III.  Figure 10: Our
calculated values of $\Upsilon$ with FAC for the 2–51 (triangles: 2p2
3P1–2s2p2(4P)3s 3P1), 2–72 (diamonds: 2p2 3P1–2s22p(2Po)4p 3P1), and 3–23
(stars: 2p2 3P2–2s22p(2Po)3p 1P1) transitions of O III.  Figure 11: The
calculated values of $\Upsilon$ by Tayal and Zatsarinny [1] with BSR for the
2–51 (triangles: 2p2 3P1–2s2p2(4P)3s 3P1), 2–72 (diamonds: 2p2
3P1–2s22p(2Po)4p 3P1), and 3–23 (stars: 2p2 3P2–2s22p(2Po)3p 1P1) transitions
of O III.

Some differences in the $\Omega$, and subsequently $\Upsilon$ values, shown in
Figure 8 can be due to the differences in the corresponding f-values, because
our wavefunctions are rather simple whereas those of Tayal and Zatsarinny [1]
are more sophisticated with the inclusion of larger CI (configuration
interaction). Therefore, as a test we have performed additional calculations
by including further CI with the (2p3) 3$\ell$ and 4$\ell$ configurations.
These seven configurations give rise to additional 192 levels (i.e. 394 in
total) but all (except one) have energies higher than the 202 considered
earlier. Effectively there is no close interaction between these additional
levels/configurations and those considered earlier. For this reason the
f-value for the 83–98 (2s22p(2Po)4d 3F${}^{o}_{4}$–2s22p(2Po)4f 3G5)
transition is invariant ($\sim$0.14) and hence should not affect the
calculations of $\Omega$ and $\Upsilon$. The other two transitions, namely
89–93 (2s22p(2Po)4d 3D${}^{o}_{3}$–2s22p(2Po)4f 3F4) and 94–101 (2s22p(2Po)4d
1F${}^{o}_{3}$–2s22p(2Po)4f 1G4), are comparatively weaker, and therefore for
the former it changes from 0.030 to 0.022 with the inclusion of larger CI but
not for the latter, which remains $\sim$0.016. This conclusion has been
confirmed with the similar calculations performed with the GRASP (general-
purpose relativistic atomic structure package) code.

Before concluding we consider three more transitions, namely 2–51 (2p2
3P1–2s2p2(4P)3s 3P1), 2–72 (2p2 3P1–2s22p(2Po)4p 3P1), and 3–23 (2p2
3P2–2s22p(2Po)3p 1P1), all of which are forbidden. Our calculated values with
FAC for $\Omega$ and $\Upsilon$ are shown in Figures 9 and 10, and the results
of Tayal and Zatsarinny [1] in Figure 11. Our values for both parameters vary
smoothly, mainly because we have not considered the contribution of
resonances, but Tayal and Zatsarinny have. However, not only the magnitudes of
their $\Upsilon$ results are lower, the behaviours also appear to be abnormal,
particularly of the 3–23 transition – note the sharp fall in magnitude at
temperatures above 103.7 K.

##  3 Conclusions

In this paper we have made an assessment of the $\Upsilon$ results reported by
Tayal and Zatsarinny [1] for transitions among 202 levels of O III. It is a
pity that even after performing sophisticated calculations with elaborate
wavefunctions (which involved up to 800 configurations for each level) and
using large computational resources (which involved up to 256 processors
handling Hamiltonian of the sizes as large as 70 000 for a partial wave), the
reported results are found to be anomalous, deficient and inaccurate for
several transitions of all types, such as: allowed, forbidden,
intercombination, and semiforbidden, and at all temperatures. We have arrived
at this conclusion based on comparisons with our own independent (non-
resonant) calculations with FAC and our long experience with similar work on a
wide range of ions. For the large discrepancies noted, in both magnitudes and
behaviours, we have identified three possible sources of errors, which are:
(i) inclusion of insufficient number of partial waves, (ii) a coarse energy
mesh at energies above thresholds, and (iii) unrequired extrapolation of
$\Omega$ to energies higher than actually calculated.

Excitation rates for the transitions of O III are required for up to (at
least) 1.62$\times$105 K, the temperature derived by Doschek [14] and Keenan
and Aggarwal [15] from the measured line intensities in the solar transition
regions. For this reason in our earlier work [3] we reported values of
$\Upsilon$ up to Te = 2.0$\times$105 K. Therefore, it is not understandable
why Tayal and Zatsarinny [1] restricted their results up to Te = 105 K, in
spite of calculating $\Omega$ up to an energy of 30 Ryd, in comparison to only
13 Ryd by us [3]. Perhaps they were aware of the inaccuracies at higher
temperatures! Unfortunately, their listed results cannot be extrapolated to
higher temperatures, because this will lead to much larger inaccuracies than
noted already. Furthermore, for the diagnostics and modelling of plasmas, a
complete set of atomic data for energy levels, A-values and $\Upsilon$ (and
preferably for $\Omega$ too because these are very useful for accuracy
assessments) are required, but Tayal and Zatsarinny have reported A-values for
only a few transitions, among the lowest 15 levels alone.

We will like to stress here that the discrepancies in $\Upsilon$ values noted
in this paper for several transitions of O III, and earlier for other ions
such as Mg V [6] and S III [10], are not because of the code (BSR) adopted by
them, but its implementation. If the code is used judiciously with due care
for the requirements than the results obtained should be in reasonable
agreement with those from others, such as the standard $R$-matrix code, Dirac
atomic $R$-matrix code (DARC), or even FAC, particularly for the allowed
transitions. Therefore, in our opinion fresh calculations (with any version of
the $R$-matrix code) should be performed for O III, an ion of great
astrophysical importance. And this is in spite of the availability of more
recent and larger calculations by Mao et al. [7], which have already been
assessed to be inaccurate by Morisset et al. [8]. Finally, our results with
FAC can either be easily generated or obtained from the author on request.

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