# Mapping Tropical Forest Cover and Deforestation with Planet NICFI Satellite
Images and Deep Learning in Mato Grosso State (Brazil) from 2015 to 2021

Fabien H. Wagner
Institute of the Environment and Sustainability,
University of California, Los Angeles, CA 90095 USA;
and NASA-Jet Propulsion Laboratory
California Institute of Technology, Pasadena, CA 91109, USA;
and CTREES.org, Pasadena, United States;
wagner.h.fabien<EMAIL_ADDRESS>
Ricardo Dalagnol
Institute of the Environment and Sustainability,
University of California, Los Angeles, CA 90095 USA;
and NASA-Jet Propulsion Laboratory
California Institute of Technology, Pasadena, CA 91109, USA ;
and CTREES.org, Pasadena, United States;
<EMAIL_ADDRESS>
Celso HL Silva-Junior
Institute of the Environment and Sustainability,
University of California, Los Angeles, CA 90095 USA;
and NASA-Jet Propulsion Laboratory
California Institute of Technology, Pasadena, CA 91109, USA ;
and CTREES.org, Pasadena, United States;
<EMAIL_ADDRESS>
Griffin Carter
Institute of the Environment and Sustainability,
University of California, Los Angeles, CA 90095 USA;
and CTREES.org, Pasadena, United States;
<EMAIL_ADDRESS>
Alison L Ritz
Virginia Polytechnic Institute and State University,
Interdisciplinary Graduate Education
Program in Remote Sensing,
Blacksburg, VA 24061, United States;
<EMAIL_ADDRESS>
Mayumi CM Hirye
Quapá Lab
Faculty of Architecture and Urbanism
University of São Paulo—USP
São Paulo, SP, Brazil
<EMAIL_ADDRESS>
Jean PHB Ometto
Earth Observation and Geoinformatics Division
National Institute for Space Research – INPE
São José dos Campos, SP, Brazil
<EMAIL_ADDRESS>
Sassan Saatchi
Institute of the Environment and Sustainability,
University of California, Los Angeles, CA 90095 USA;
and NASA-Jet Propulsion Laboratory
California Institute of Technology, Pasadena, CA 91109, USA
<EMAIL_ADDRESS>
and CTREES.org, Pasadena, United States

###### Abstract

Monitoring changes in tree cover for rapid assessment of deforestation is
considered the critical component of any climate mitigation policy for
reducing carbon. Here, we map tropical tree cover and deforestation between
2015 and 2022 using 5 m spatial resolution Planet NICFI satellite images over
the state of Mato Grosso (MT) in Brazil and a U-net deep learning model. The
tree cover for the state was 556510.8 km2 in 2015 (58.1 % of the MT State) and
was reduced to 141598.5 km2 (14.8 % of total area) at the end of 2021. After
reaching a minimum deforested area in December 2016 with 6632.05 km2, the bi-
annual deforestation area only showed a slight increase between December 2016
and December 2019. A year after, the areas of deforestation almost doubled
from 9944.5 km2 in December 2019 to 19817.8 km2 in December 2021. The high-
resolution data product showed relatively consistent agreement with the
official deforestation map from Brazil (67.2%) but deviated significantly from
year of forest cover loss estimates from the Global Forest change (GFC)
product, mainly due to large area of fire degradation observed in the GFC
data. High-resolution imagery from Planet NICFI associated with deep learning
technics can significantly improve mapping deforestation extent in tropics.

_Keywords_ Tropical forests $\cdot$ Semantic segmentation $\cdot$ U-net
$\cdot$ TensorFlow 2 $\cdot$ Land-Cover and Land-Use

##  1 Introduction

Deforestation of tropical forests caused mainly by expanding forestry and
agriculture is currently the second largest source of anthropogenic carbon
emission, with 5.3$\pm$ 2.4 GtCO$2$ emitted per year on the period 2001-2019,
and still increasing Harris et al. (2021); Friedlingstein et al. (2022); Feng
et al. (2022); Shukla et al. (2022); Pendrill et al. (2019). In the Brazilian
Amazon, the deforestation rate of 2020 was the greatest of the decade so
farSilva Junior et al. (2021), and was recently overpass by the 2021
deforestation rate, with 13.032 km2 de Almeida et al. (2021). Apart from
carbon loss, $\sim$40 % of this deforestation happens in old-growth tropical
forests, causing also an increase in fragmentation Montibeller et al. (2020),
and constitutes a direct threat to biodiversity Harris et al. (2021).
Monitoring changes in tree cover for rapid assessment of deforestation is
considered the key component of any climate mitigation policies for reducing
emissions from deforestation and restoration of forests for biodiversity and
carbon sequestration Valeriano et al. (2004); Mitchard (2018).

However, maps of tree cover and deforestation are still challenging datasets
to produce. One of the most renown and accurate products of deforestation
monitoring is the PRODES from the National Institute of Space Research of
Brazil National Institute for Space Research (1988-2021) (INPE). Since 1988,
the PRODES monitor clear-cut deforestation in the Amazon and produced annual
deforestation rates for the Brazilian government. PRODES defines deforestation
as the suppression of areas of old-growth forested physiognomies by
anthropogenic actions, from intact forest cover to other lanf use de Almeida
et al. (2021). It is produced based on the visual interpretation of satellite
imagery and manual mapping of deforestation by trained specialists over the
course of the PRODES-year, which starts at August 1st and ends on July 31st de
Almeida et al. (2021). PRODES map is based on satellite dataset at 20 to 30 m
of spatial resolution and 5 to 16 days of revisit rate (currently Landsat-8,
SENTINEL-2 and CBERS-4/4A), and some pixels can have no observation due high
cloud coverage during the course of one year. One of the limits of the product
is that the manual sampling constrains the time of production of the map and
limits the scale of the areas that can be considered as deforestation. For
example PRODES does not map deforestation below 6.25 ha de Almeida et al.
(2021). Another major product to monitor deforestation is the forest loss year
dataset, a part of the Global Forest Change products (GFC), the first global
map at 30 m of forest cover freely available globally from the University of
Maryland Hansen et al. (2013). This product looks at the annual change in tree
cover in relation to the tree cover of the year 2000 at 30 m spatial
resolution globally. Forest loss is defined as the stand-replacement
disturbance or the complete removal of tree cover canopy at the Landsat pixel
scale. These two products, PRODES and GFC tree loss year product, are
reference datasets to monitor deforestation; however, they are delivered
annually and at 30 m of spatial resolution. Furthermore, Prodes is limited to
deforested areas above 6.25 ha and GFC tree loss year include forest fires and
forest degradation in the tree loss product, not only deforestation.

Recently, Planet satellite images over the tropics have been made available by
the Norway’s International Climate and Forest Initiative (NICFI,
https://www.nicfi.no/), to help save the world’s tropical forests while
improving the livelihoods of those who live off, in, and near the forests
Planet Team (2017). The Planet NICFI images are multispectral satellite images
containing red, green, blue, and near infrared bands at 4.78 m of spatial
resolution for the Normalized Analytic Basemaps. The temporal resolution of
the NICFI images is currently one month, and they are a mosaic composite of
the best daily acquisitions during the month. Consequently, Planet NICFI
images are mostly cloud free, thus providing the best freely available
multispectral dataset to monitor Land Use and Land Cover (LULC) changes in
tropical regions.

However, while Planet NICFI provide the best spatial and temporal resolution
to map tropical forest cover with optical data, the variation of reflectance
values between the different Planet satellite sensors, between date and
sometime within the same image is highly challenging. As state by the NICFI
documentation, the absolute radiometric accuracy is not guarantee for the
normalized surface reflectance basemaps Pooja Pandey and Curdoglo. (2021). For
the non-deep learning remote sensing analysis, where high quality reflectance
values are needed, this is almost a no-go, unless several preprocessing steps
are made, such as indices creations and bands normalisation Cheng et al.
(2020); Francini et al. (2020a). However, for deep learning methods, the high
quality or fidelity of reflectance values are not necessary. For example, to
recognize cars or plastic balloons in images, CNN models are even trained with
images where the hue is artificially changed (during data augmentation) to
impede the model to give too much importance to the color, which could be
considered as overfitting in this case Chollet and Allaire (2018).
Furthermore, colors (or values in the colour channels red, green, and blue)
can be seen as one feature when we currently know that several features and
multiple levels of abstraction are needed to reach state-of-the-art accuracy
of classification LeCun et al. (2015) as demonstrated by the success of CNNs
in computer vision tasks. CNNs correspond exactly to what is needed here, to
extract the maximum of information relevant to forest mapping using pixel
context and not only pixel-based colors values with minimal pre-processing
from the unequally calibrated images. Furthermore, deep learning has already
been applied with success for tasks such as classification of cloud, shadows
and land cover scenes in PlanetScope Shendryk et al. (2019) and to map
deforestation in Brazil with Landsat images Matosak et al. (2022); Maretto et
al. (2020).

To segment evergreen forest tree cover in Planet images with deep learning,
the U-net model was chosen Ronneberger et al. (2015). This is a state-of-the-
art model for the segmentation tasks, that returns the probability of the
object to map per pixel and has shown excellent performance on tropical forest
cover mapping in very high-resolution images Wagner et al. (2019). Here, the
tree cover model returns a binary value for each pixel, 0 or 1 for non-forest
or forest, respectively, that constitute the tree cover mask. Then the
deforestation date is deducted from the forest masks at each date. The model
was tested in the Mato Grosso States in Brazil with the complete the Planet
dataset from December 2015 to March 2022, corresponding to a total of 85056
images. The region located in the south of the Amazon Forest was chosen
because it has overcome large amounts of deforestation the past. Between the
years 2000 and 2013 there has been estimated 76,360 km2 of deforestation in
Mato Grosso which corresponded to 32% of the whole Brazilian Amazon estimates
Tyukavina et al. (2017). Furthermore, Mato Grosso state also contains some of
the main current deforestation fronts of the region know as the arc of
deforestation of the Brazilian Amazon forest.

In this work is presented (i) the product of evergreen tree cover and
deforestation obtained with Planet images and deep learning, (ii) the
validation of the product with independent tree cover mask based on Planet
data and also from an independent LiDAR dataset, and (iii) the comparison with
PRODES deforestation and GFC tree loss year loss products.

##  2 Materials and Methods

Figure 1: Planet NICFI mosaic of September 2021 constituted of the 2658 Planet
image tiles covering the Mato Grosso State - Brazil.

###  2.1 Planet satellite images of Mato Grosso - Brazil

There were 2658 Planet tiles of $\sim$ 20 $\times$ 20 km at $\sim$ 4.78 m
spatial resolution covering the entire Mato Grosso State of Brazil, Fig 1.
This study region cover represents $\sim$ 1 millions of km2. Images were
downloaded trough the Planet API https://api.planet.com/basemaps/v1/mosaics
and with the PlanetNICFI R package Mouselimis (2022) for all the 32 available
dates Planet Team (2017) at the time of the study. The complete dataset
contains 85056 planet images for the following dates, bi-annually: 2015-12-01,
2016-06-01, 2016-12-01, 2017-06-01, 2017-12-01, 2018-06-01, 2018-12-01,
2019-06-01, 2019-12-01, 2020-06-01; and, monthly from 2020-09-01 to
2022-06-01. All bands in raw image digital numbers (12 bits),Red (0.650-0.682
$\mu$m), Green (0.547-0.585 $\mu$m), Blue (0.464 -0.517 $\mu$m) and the NIR
bands (0.846-0.888 $\mu$m) Planet (2021), were, first, truncated to the range
0–2540 for the RGB bands and scaled between 0 and 2540 for the NIR bands (i.e.
divided by 3.937). Second, the 4 bands were scaled to 0–255 (8 bits) by
dividing by 10 and then the Red-Green-Blue-NIR (RGBNIR) composite was built.
The forest reflectance values are low in the RBG bands ($<$ 500) and the
specific scaling of these bands was made optimize the range of values of the
forest reflectance in 8 bits. The scaling of NIR is only a min-max (0-10000)
scaling as forest reflectance values are not low in this band. No atmospheric
correction was performed. A second image was generated from the composite
adding a mirroring border of 128 pixels on each side for the deep learning
prediction in order to remove border effects.

###  2.2 LiDAR tree cover dataset

To validate our tree cover map, we compared our results to the LiDAR height
data that were acquired in the scope of the Amazon Biomass Estimate project
(EBA/INPE) which covers mainly Amazonian forests. The complete dataset is
constituted of discrete-return LiDAR data with 610 flightlines of 15 km long
by 0.5 km wide distributed randomly over the amazon region ($\sim$ 787 ha
each; total area of 480,000 ha). Among this dataset, we selected 141
flightlines that intersected entirely with the Planet images over Mato Grosso.
The LiDAR dataset was acquired during 2016 using the Trimble HARRIER 68i laser
scanning system onboard an airplane at an average flight altitude of 600 m.
Multiple LiDAR returns were recorded with a minimum point density of 4 points
per m2. The horizontal and vertical accuracy ranged from 0.035 m to 0.185 m
and from 0.07 m to 0.33 m, respectively. It is currently the largest available
non-continuous dataset of high-resolution airborne LiDAR over Amazon forests
The LiDAR point clouds were processed into digital terrain models (DTM) and
canopy height models (CHM) with 1 $\times$ 1 m cell size, as described by
Dalagnol et al. (2019, 2021) and the median of the data was aggregated at the
resolution of the planet data.

###  2.3 Deforestation history datasets

To test if our tree cover and deforestation products were consistent with
independent datasets over the region, we compared our results to two existing
products. The first product is PRODES from INPE - the Brazilian National
Institute for Space Research National Institute for Space Research (1988-2021)
(INPE). The PRODES project monitors clear-cut deforestation in the Brazilian
Legal Amazon based on satellite images of $\sim$20 to 30 m of spatial
resolution (Landsat-8, CBERS-4 and similar imagery) and delivers official
annual rates of forest loss for Brazil (1988-2021). The publicly available
annual maps are all corrected/edited manually by experts and contain polygons
with the following labels: forest, non-forest, deforestation of the year,
previous deforestation, clouds, and water. The minimum area considered for
mapping deforestation is 6.25 ha. The second product is the forest loss year
component of the Global Forest Change map of the University of Maryland at 30
m spatial resolution based on Landsat data Hansen et al. (2013),
https://developers.google.com/earth-
engine/datasets/catalog/UMD_hansen_global_forest_change_2021_v1_9. Forest
Cover Loss is defined as a stand-replacement disturbance, or a change from a
forest to non-forest state, during the period from 2000 to 2021. Both datasets
were compared to our product (i) by computing the total area intersecting with
our product at each time period, and (ii) by computing the percentage of
intersections of GFC tree loss year or PRODES with our product by tiles
weighted by the number of observations in GFC or PRODES.

###  2.4 Neural Network Architecture

Figure 2: Architecture of the U-net model used for evergreen tropical forest
mapping in Planet data.

The segmentation of evergreen forest cover in the Planet images of Mato Grosso
was made with a classical U-net model Ronneberger et al. (2015), Fig. 2. This
model provides state of the art performance for segmentation of remote sensing
images and have been already used to mapping forest cover in the tropics Huang
et al. (2018); Wagner et al. (2019, 2020). More specifically the U-net model
return the probability of presence of evergreen tropical forest in each pixel
of a given input image. In our case, the model inputs are 4 bands RGB-NIR
images made up of 256 $\times$ 256 pixels at $\sim$4.77 m of spatial
resolution and the output a mask of one band and 256 $\times$ 256 pixels
containing 1 (forest cover, pixel probability $>=$0.5) or 0 (non-forest,
probability $>=$0.5). The model has a total of 34,614,657 parameters, of which
34,600,641 are trainable. The model was coded in R language R Core Team (2016)
with RStudio interface to Keras and TensorFlow 2.8 Chollet et al. (2015);
Allaire and Chollet (2016); Allaire and Tang (2020); Abadi et al. (2015).

###  2.5 Training

To produce the training sample of evergreen forest cover for the U-net model,
10 Planet images where randomly selected over Mato Grosso for each monthly
Planet mosaic dates from September 2020 to September 2021 included, to
constitute a dataset of 130 images. Among this dataset, 75 image were then
selected after removing images with a high cloud cover or large variations in
illumination. Then for the selected images, we ran the k-textures model Wagner
et al. (2022) to perform image self-segmentation with 8 classes. We kept only
23 images where the forest segmentation was deemed accurate according to the
visual interpretation of the forest and non-forest areas. Small errors of the
forest mask were manually corrected. At this step, the masks had only two
classes: forest (1), and non-forest (0) which contained agriculture, urban
area, water surface and bare ground. From this dataset, we trained a first
U-net model Ronneberger et al. (2015) and applied predict tree cover over the
original 130 images. Then, we removed the 23 images used in the first training
sample, and selected a second set of 21 images with atmospheric conditions
such as cloud and haze. The forest masks were selected or manually corrected
to keep forests with no haze or very thin transparent cloud, while removing
all clouds and heavy haze. Here, we want the model to classify all cloud and
haze as non-forest when forests are difficult to recognize even by the human
eye, keeping only the forests that a human can easily say it is forest cover.
From the first and the second samples, we constitute a second training sample
of 6663 images patches of 256 x 256 pixels and their associated labelled masks
for the final U-net model. 6187 were containing forest and background and 476
only background. 80% (5331) were used for training and 20% (1332) for
validation.

Each image patch goes through a data augmentation process that consists in
random vertical and horizontal flips. No additional data augmentation was
necessary because of the natural data augmentation provided by different
atmospheric conditions and illumination due the different dates of the
sampling images Wagner (2021). After data augmentation, the images were then
fed to the U-net model.

During network training, we used a standard stochastic gradient descent
optimization. The loss function was designed as a sum of two terms: binary
cross-entropy and Dice coefficient-related loss of the three predicted masks
Dice (1945); Allaire and Chollet (2016); Chollet et al. (2015) and finally the
optimizer Adam Kingma and Ba (2014) with a learning rate of 0.0001 was used.
We used the accuracy (i.e. the frequency with which the prediction matches the
observed value) as the metrics to assess the model performance. The network
was trained for 20000 epochs with a batch size of 256 images and the model
with the best weighted accuracy was kept for prediction (epoch 17058 and
training/validation accuracy of 99.40 and 97.70% respectively and
training/validation loss of 0.0088 and 0.0574 respectively,). The training of
the models took approximately 12.5 hours using a Nvidia RTX2080 Graphics
Processing Unit (GPU) with an 8 GB memory.

####  2.5.1 Prediction

For prediction, the Planet tiles of 4096 $\times$ 4096 pixels a border of 64
neighboring pixels were added on each side with mirroring image. This border
method was used to avoid border artefacts during prediction, a known problem
for the U-net algorithm Ronneberger et al. (2015). Then the prediction was
made on the entire image. To belong to the evergreen forest class, the pixel
prediction value must be greater than or equal to 0.5. The prediction for one
Planet tile took approximately 7 seconds using a Nvidia RTX3090 GPU.

####  2.5.2 Cloud temporal filter

Even with the high temporal resolution of Planet satellite image acquisition
(daily), the monthly mosaic product can still contains some clouds. To filter
for these clouds, we designed a simple temporal filter: to be classified as an
evergreen forested pixel, a pixel at a given date must have been detected as
evergreen forest cover at least half of the observations on the period defined
by the closest 3 dates before and after its date. For the borders of the time
series, we used the all the available dates in the $\pm$ 3 months around the
date to filter.

####  2.5.3 Tree cover and deforestation masks

At this step, we considered that the mask images filtered by cloud are almost
cloud free and we give only a maximum of two chances to have a forest that
have been misclassified as non-forest on the 2015-2021 period due to cloud.
For example, if on the period of the cloud filter ($\pm$ 3 months), the
forested pixel has never been observed due to clouds.

Then, we process the time series from the present to the past to reconstruct
tree cover map per pixel. we keep only biannual time series as this point. For
each pixel at each date, we keep or correct the value of the pixel with rules
described in the following text and we attribute a value of trust of the
classification, trust = 1 for "confirmed", the pixel is tree cover or
deforestation, trust = 2 for "very likely", when a pixel is tree cover on all
the data, non-tree cover on only 1 or 2 dates and with this dates being not
the first and the last, note that the non-forested pixel values are all set to
tree cover in this case. And finally trust = 3 for "unconfirmed". This value
is only for the deforestation of the last available date, to be confirmed with
next available date. Deforestation is defined by a pixel classified as non
forest at the date N. The data are processed starting from the last date to
the first date and the rules are applied at each date. Deforestation is also
confirmed during the process, when a pixel at N-1 is classified as deforested
and still deforested at N, is confirmed as deforestation and its trust value
become 1. The result of this steps are corrected tree cover and deforestation
masks for all the dates and their associated values of classification
confidence.

##  3 Results

###  3.1 Tree cover validation with independent validation sample

The tree cover algorithm presented a high level of segmentation accuracy, with
an overall accuracy of 98.11% and an F1-Score of 0.982 (precision = 0.975,
recall = 0.988) on the 1332 images patches (256x256 pixels) of the validation
sample.

###  3.2 Tree cover validation with airborne LiDAR data

The validation of our tree cover product was made by comparing the median
height measured by LiDAR at 1 m spatial resolution inside the planet pixels
overlapping entirely with the 141 EBA LiDAR footprints above Mato Grosso. it
represented 47267739 pixels for tree cover and 1912151 pixels for non-tree
cover. In the non-tree cover pixels classified by the U-net, very few pixels
have a median high above 2.5 meters, Fig. 3a.

We found that 94 % of the non-treecover pixels had a value below 0.55 m, 1 %
in the values ranging between 0.55 m and 2.07 m, 3 % in the range between 2.07
m and 4.45 m and only 2% above 4.45 m. For the tree cover pixels, the height
was a distributed in a Gaussian shape around the median height of 16.9 m, Fig.
3b, and 5 % of the tree cover pixels has median height below 5.02 m. overall,
based on the LiDAR data, it is observed that our tree cover model segment
trees with a very high accuracy, that is, 98% non forest pixels have height
below 5 m, and around 95 % of the tree cover pixels have height above 5 m.

Figure 3: Distribution of median height in the Planet pixels that intersects
(with full overlap) with 141 EBA LiDAR datasets randomly distributed over Mato
Grosso, for planet pixels classified by the U-net as non-tree cover (a) and
tree cover (b), respectively.

###  3.3 Tree cover and deforestation of Mato Grosso on the period 2015-12-01
to 2021-12-01

For the Mato Grosso State, a tree cover of 556510.8 km2 was found for the year
2015, which represent 58.1 % of the State area, Table 1. Between December 2015
and December 2021, this area was diminished of $\sim$ 141598.5 km2 (14.8 % of
total area). The biannual deforestation data did not show a clear seasonal
preference for deforestation. After reaching the minimum value of
deforestation area in December 2016 with 6632.05 km2, the biannual
deforestation area showed a tendency to increase between December 2016 and
December 2019 and a sharp increase was observed thereafter, with the area of
December 2021 (19817.8 km2) been almost the double of the area of December
2019 (9944.5 km2). Note that new forests growing after 2015 are not accounted
in these numbers. Furthermore, even with the 6-months cloud filter some clouds
can remains in the last date, and results for 2021-12-01 should be taken with
caution.

classes | Area (km²) | Area (%)
---|---|---
2015 total tree cover | 556510.76 | 58.08
2015 non-tree cover | 401689.23 | 41.92
2015 remaining tree cover | 414912.25 | 43.30
deforestation 2016-06 | 18763.20 | 1.96
deforestation 2016-12 | 6632.05 | 0.69
deforestation 2017-06 | 8511.05 | 0.89
deforestation 2017-12 | 8037.00 | 0.84
deforestation 2018-06 | 8066.52 | 0.84
deforestation 2018-12 | 9016.48 | 0.94
deforestation 2019-06 | 8812.72 | 0.92
deforestation 2019-12 | 9944.50 | 1.04
deforestation 2020-06 | 9998.04 | 1.04
deforestation 2020-12 | 15297.82 | 1.60
deforestation 2021-06 | 18701.35 | 1.95
deforestation 2021-12 | 19817.78 | 2.07
Table 1: Tree cover and its change due to deforestation on the period from
2015-12-01 to 2021-12-01 obtained with Planet data and the U-net tree cover
model. Figure 4: Tree cover and tree cover loss per year for the 2658 Planet
tiles covering the Mato Grosso State - Brazil, with area represented with the
real size found in each tiles. Remaining tree cover of 2015 is in green, and
the yellow to red color scale represent the deforestation with the red being
the more recent.

Only 52 (1.96 %) of the 2658 Planet tiles of Mato Grosso States have less than
1 km2 of tree cover change on the period between 2015 and 2021, Fig. 4. The
mean deforested area was 57.2 km2 per tiles (tile area is 382.9019 km2). 455
tiles (17.1 %) shown deforested area above 100 km2. Some areas with current
intensive tree cover change are clustered such as areas located in the south
of Mato Grosso, and near the largest forest remnants.

Figure 5: Example of our deforestation product for five Planet quad on the
period 2015-12-01 to 2021-12-01.

Large scale deforestation still appeared in 2021, for example, in the Xingu
region, Fig. 5a. Note that roads built previously for logging appeared in the
largest patch. Depending on the size of the rivers inside forest, it can take
some times to appear clearly in the images due to satellite view angle, and
small rivers can be mapped as deforestation in the first years, such as the
river in the bottom right. Gradually advancing smaller scale deforestation can
also be observed in the Xingu region, Fig. 5b, accompanied by new roads inside
forests. Large scale deforestation is visible on the recent deforestation
front between the Mato Grosso and Amazonas states, Fig. 5c. Some isolated dots
are logging decks or roads and indicate where the deforestation will likely
appear in a near future. An unusual circular pattern of deforestation for Mato
Grosso shows still active deforestation in 2021, Fig. 5d. It is located in the
deforestation front between the Mato Grosso and Amazonas states. Some pattern
of tree cover change can be natural or not, for example, in the South Central
region of Mato Grosso, Fig. 5e. Natural changes in river border were mapped as
deforestation and this region also present new roads, and large areas of tree
cover change. For this region, the large deforestation could be linked to the
change from biannual to monthly time series and inclusion as forest when they
are deciduous and now observed as deforestation. Note that not all this
deforestation is necessary illegal. However, for most of these images small
new roads or logging decks located inside the forest can be observed,
indicated that logging is active and planned to continue.

###  3.4 Similarities and discrepancies of our product with Prodes
deforestation and GFC tree loss year data in the Amazon region of Mato Grosso

In this section, we detailed which deforested pixels detected by Prodes or GFC
tree loss year are found or not by our product for the Legal Amazon region of
the Mato Grosso State.

|  | Deforestation |
---|---|---|---
| our product (km2) | Prodes (km2) | GFC tree loss year (km2)
2016 | 7762.19 | 1370.87 | 5192.88
2017 | 5453.75 | 1327.95 | 7638.69
2018 | 5959.05 | 1407.31 | 3950.39
2019 | 6271.88 | 1848.54 | 3666.78
2020 | 7638.07 | 1817.41 | 5636.31
2021 | 13930.53 | 1860.23 | 3987.56
Table 2: Annual tree cover change in the Amazonian region of Mato Grosso in
km2 on the period from 2016 to 2021-12-01 obtained with our product, Prodes
data and GFC tree loss year data.

For the tree cover change in the Amazon for region of Mato Grosso, Table 2, we
found more tree cover changes that the Prodes and GFC tree loss year product.
We found between 3.39 to 7.49 times more deforestation than Prodes depending
on the date and the largest discrepancies were found on the last date of our
time series. We found between 0.71 and 3.49 times the deforestation from GFC
tree loss year product, with the last date having the largest differences.
Note that the spatial resolution of Planet is 4.77 m while the spatial
resolution of Prodes data and GFC tree loss is 30 m, and that even with the
6-months cloud filter some clouds can remains for the last date of our
product, so results for 2021-12-01 should be taken with caution.

Because of the $\pm$ 3 dates aggregation for cloud filtering, our
deforestation data start to be consistent in time with PRODES only when
monthly Planet data are available. On the period 2020-06 to 2021-12, our model
intersects with 72.5% of the Prodes data for the year 2021, Table 3. On the
period 2019-06 to 2021-06, our model intersects with 64.4% of the Prodes data
for the year 2020. On the period 2018-06 to 2020-06, our model intersects with
60.7% of the Prodes data for the year 2019. The largest class outside the
intersection of dates is always forest cover, meaning that we have classified
as tree cover a pixel that was classified as deforested by Prodes. This can be
errors from our model or small patches of forests that are present in the 30 m
pixels classified as deforestation by Prodes.

|  |  |  | PRODES |  |  |  |
---|---|---|---|---|---|---|---|---
| non-forest | forest | deforestation | deforestation | deforestation | deforestation | deforestation | deforestation
|  |  | 2016-12 | 2017-12 | 2018-12 | 2019-12 | 2020-12 | 2021-12
OURS |  |  |  |  |  |  |  |
non-forest | 67.30 | 1.30 | 52.60 | 22.20 | 9.30 | 6.00 | 5.00 | 3.50
forest | 19.50 | 95.30 | 8.60 | 9.90 | 13.60 | 11.60 | 17.90 | 17.20
deforestation |  |  |  |  |  |  |  |
2016-06 | 2.10 | 0.20 | 8.00 | 15.50 | 3.00 | 1.40 | 1.10 | 0.60
2016-12 | 0.70 | 0.10 | 1.90 | 1.70 | 0.50 | 0.40 | 0.30 | 0.20
2017-06 | 0.90 | 0.10 | 4.40 | 13.60 | 4.20 | 1.40 | 0.80 | 0.50
2017-12 | 0.80 | 0.10 | 3.30 | 9.80 | 13.40 | 2.40 | 1.00 | 0.50
2018-06 | 0.90 | 0.10 | 3.10 | 5.30 | 16.80 | 4.20 | 1.50 | 0.60
2018-12 | 0.90 | 0.10 | 3.80 | 4.60 | 11.40 | 18.70 | 2.70 | 0.90
2019-06 | 0.80 | 0.10 | 3.20 | 3.20 | 7.70 | 18.60 | 7.20 | 1.40
2019-12 | 0.90 | 0.20 | 2.70 | 3.30 | 5.00 | 12.50 | 19.80 | 2.10
2020-06 | 0.80 | 0.20 | 1.90 | 2.70 | 3.20 | 6.70 | 16.30 | 3.60
2020-12 | 1.20 | 0.40 | 2.70 | 3.40 | 5.50 | 7.40 | 11.80 | 20.20
2021-06 | 1.60 | 0.50 | 2.10 | 2.70 | 3.90 | 5.50 | 9.30 | 34.00
2021-12 | 1.50 | 1.40 | 1.70 | 2.10 | 2.70 | 3.30 | 5.50 | 14.70

Table 3: Intersection in percentage of annual Prodes data and our bi-annual
deforestation product.

Regarding the GFC tree loss year data, on the period 2020-06 to 2021-12, our
model intersects with 57.9% of the GFC tree loss year data, Table 3; on the
period 2019-06 to 2021-06 with 33.9% of the GFC tree loss data for the year;
and, on the period 2018-06 to 2020-06 with 36.7% of the GFC tree loss data.
The largest class is always forest cover, meaning that we have classified as
tree cover a pixel that was classified as tree loss in GFC data. These are
errors from our model and are also expected because GFC tree loss year
consider a change inside the Landsat pixels, but this does not always
represent a clear cut of the forest in the pixel, only a significant change in
tree cover.

|  |  |  | GFC tree loss year |  |  |  |
---|---|---|---|---|---|---|---|---
| non forest | forest 2000 | deforestation | deforestation | deforestation | deforestation | deforestation | deforestation
|  |  | 2016-12 | 2017-12 | 2018-12 | 2019-12 | 2020-12 | 2021-12
OURS |  |  |  |  |  |  |  |
non forest | 26.90 | 72.90 | 26.30 | 7.80 | 5.40 | 4.90 | 3.90 | 5.30
forest | 67.40 | 13.10 | 42.30 | 50.30 | 38.70 | 36.50 | 56.40 | 31.60
deforestation |  |  |  |  |  |  |  |
2016-06 | 0.80 | 2.70 | 5.40 | 5.30 | 2.20 | 1.10 | 0.60 | 0.90
2016-12 | 0.30 | 0.70 | 1.20 | 0.70 | 0.30 | 0.60 | 0.30 | 0.40
2017-06 | 0.30 | 1.20 | 2.70 | 4.60 | 2.80 | 0.90 | 0.40 | 0.50
2017-12 | 0.30 | 1.00 | 2.20 | 5.00 | 5.30 | 1.40 | 0.40 | 0.60
2018-06 | 0.30 | 1.00 | 1.90 | 3.10 | 9.00 | 1.90 | 0.50 | 0.70
2018-12 | 0.30 | 1.20 | 2.50 | 3.10 | 9.20 | 8.90 | 0.90 | 0.60
2019-06 | 0.30 | 1.00 | 2.20 | 2.60 | 5.90 | 11.40 | 1.50 | 0.60
2019-12 | 0.30 | 1.00 | 2.30 | 2.50 | 4.50 | 9.40 | 6.00 | 0.80
2020-06 | 0.40 | 0.70 | 1.70 | 2.20 | 2.90 | 5.10 | 7.00 | 1.00
2020-12 | 0.50 | 1.10 | 2.80 | 4.30 | 5.10 | 6.00 | 13.50 | 6.20
2021-06 | 0.80 | 1.30 | 2.60 | 3.90 | 4.20 | 6.00 | 5.90 | 21.80
2021-12 | 1.20 | 1.20 | 3.70 | 4.50 | 4.40 | 6.00 | 2.70 | 28.90

Table 4: Intersection in percentage of annual GFC tree loss data and our bi-
annual deforestation product.

For the year 2020, the histogram of the percent of intersection per quad for
our product and the Prodes deforestation data shown that our product share
consistent information with the Prodes data, Fig. 6a. The median intersection
of Prodes with our product was 67.2%, and 75% of the pixels had more than
50.9% of intersection and 25% had more than 81.7%. Note that the Prodes data
are manually made with data at 20 to 30 m spatial resolution and do not
consider deforested area below 6.25 ha. Our product show less agreement with
the GFC tree loss year data, Fig. 6b. In 2020, only 25% of our data had an
intersection above 54%. An important proportion of the GFC tree loss data of
2020 show almost no intersection with our product as shown by the median of
intersection of 27.5%. Still, for some quad the intersections can be above
80%, Fig. 6b.

Figure 6: Histogram of intersection per Planet tiles of our product of
deforestation with the Prodes data (a) and GFC tree cover loss (b) per quad
for the year 2020. Each intersection value was weighed by the number of pixels
with intersection observed in the tile, of Prodes or of GFC tree loss year
data, respectively.

Figure 7: Example of good agreement, i.e. $>$ 90% of intersection, of our
product of deforestation with the Prodes data (Planet quad 0711-0955) and of
our product and GFC tree loss year data (Planet quad 0727-0962) for the year
2020. Planet NICFI quad is $\sim$ 19.5 $\times$ 19.5 km.

For the tiles with similar results with our deforestation map, that is with
more than 90% of intersections between our deforestation mapping and the
Prodes or GFC tree loss year data, Fig. 7, several observations can be made.
Between Prodes and our deforestation data, Fig. 7 a-b for the images and e-f
for the deforestation masks, it can be observed that Prodes (as expected by
the Prodes methodology) included non-forest pixels that are captured by our
algorithm with Planet image at 5 m, for example the logging decks. It can also
be observed that our algorithm detected deforested pixels on the borders of
the forest patches and roads. It indicates that the pixel attribution has
changed, however, it remains difficult to conclude if this is because of real
change in tree cover or because of imprecise georeferencing of Planet images
between the two dates. Between GFC tree loss year data and our product, Fig. 7
c-d for the images and g-h for the deforestation masks, it can be observed
that GFC tree loss year data produced large patches and also some roads are
detected, while our product detected a lot more changes in pixels. Inside
deforestation patches, sometimes some tree remains, and our algorithm still
detect them as forests. Overall Prodes and GFC tree loss year data give more
homogeneous surfaces, and our product, with its finer spatial resolution gives
a tree cover at the scale of the tree and small gap in the canopy can be
marked as non-forest or few remaining trees inside a deforested patch can be
still recognized as tree cover. Outside the forest the change of attribution
of our product is difficult to interpret, as it can be due to deforestation or
to imprecise Planet images georeferencing.

Figure 8: Example of poor agreement, i.e. $<$ 10% of intersection, of our
deforestation product and the Prodes data (Planet quad 0717-0961) and GFC tree
loss year data (Planet quad 0701-09948) for the year 2020. Planet NICFI quad
is $\sim$ 19.5 $\times$ 19.5 km.

For the tiles with bad agreement between our deforestation mapping and the
Prodes or GFC tree loss year data, that is with less than 10% of intersection,
Fig. 8, fire mapped as deforestation in Prodes and in GFC tree loss year data
seems to be the culprit. While our algorithm mapped almost no change for the
quad 0717-0961 in 2020, Fig. 8 a-b and e-f, a large area of forest that have
burnt is delineated as deforestation in Prodes, while it is still forest. The
same observation is made for the GFC data for the region of the quad 0701-0948
in 2020, Fig. 8 c-d and g-h. In this tile, large fires have been mapped as
deforestation, while the forest is still here. These fires included in the
mapping of deforestation in Prodes or GFC tree loss year data leads to major
discrepancy with our product.

Figure 9: Worst cases of non - agreement, between Prodes data (Planet quad
0716 - 0965), Hansen tree cover loss (Planet quad 0728 - 0957) and our
product, i.e. $<$ 10% of intersection, for the year 2020. Planet NICFI quad is
$\sim$ 19.5 $\times$ 19.5 km.

For the tiles that shows the most extreme difference, that is our product find
a lot more pixels than Prodes or GFC tree loss year data ($>$100000 pixels),
Fig 9, it can be observed that the changes are mainly isolated pixels. It is
also common around rivers and on the border of forests, where our algorithm
can pick up really small changes as the spatial resolution is 5 m. However, it
remains difficult to say if it is an anthropic or natural change or even a
referencing problem. This also seems to happen for short forest stand that are
not highly packed.

Figure 10: Comparison of our product of deforestation with the Prodes data (a)
and GFC year of tree loss for the Planet quad 0681-0967 for the period from
2015-12-01 to 2021-12-01. Planet NICFI quad is $\sim$ 19.5 $\times$ 19.5 km.

Comparing the deforestation time series between our product, GFC tree loss
year data and Prodes product, it can be observed that our product tends to
delay the timing of deforestation for the first years of the time series, even
if it detected well these areas spatially. This is due to the cloud filter
that consider three dates around each date to filter for clouds. For example,
for the year 2015, the three next data are 2016-06-01, 2016-12-01 and
2017-06-01 for the cloud filter. This issue is resolved after 2020-12-01, as
the first date of the monthly time series is 2020-09-01 and will not appear
for the future dates. For the last periods, our product is well aligned with
Prodes and GFC tree loss year data tree loss year dataset. Our product appears
noisier, because of the temporal resolution (6 months) and also the spatial
resolution (5 m) while Prodes and GFC tree loss year data produce more
homogeneous area. With our product is is possible to see the deforestation
each 6 months. As our product does not use expert interpretation, the results
are fast to produce, it just depends on the release of the image and the
deforestation of the end of 2021 is included in our dataset, as for GFC tree
loss year data but not in Prodes data (Prodes year end in August, our study
was made in May). Our product missed some small regions, this is mainly due to
remaining trees, as if some tree are remaining, our model still consider that
a forest is present, while in 30 m resolution the change of signal between the
two dates must be very high and easy to detect. Overall, our results seem to
indicate that the time series of deforestation can be obtained with the Planet
data at least when the Planet time series is monthly.

##  4 Discussion

###  4.1 Mapping dense tropical tree cover

In the Mato Grosso state - Brazil, the U-net network identified dense tropical
forests with an overall accuracy and a F1-score above 98% on the validation
dataset, showing again the high capacity of deep learning to support
vegetation mapping Kattenborn et al. (2021) even in tropical environment
Wagner et al. (2019, 2020). The 2015 tree cover represented 58.1 % of the
state. We found that 43.3 % of Mato Grosso in 2021 is still forested with
forest that was present in 2015. Note that our model does not account for
regrowing forest. The high performance of the segmentation could be explained
by the unique spectral values and textural information of the dense forests
when considering the context, for example Fig. 7-10. The k-textures model,
that is used to make the training sample, was previously shown to segment
these forests by textures without human supervision, which further confirm
that they contain unique feature characteristics Wagner et al. (2022). Here,
the accuracy shown the robustness and generalization of the U-net model to
such segmentation task. No heavy pre-processing was used, only a scaling to
have 8 bits images and a border added with mirroring image to avoid border
effect during the prediction. This minimal pre-processing eased the processing
of large image database, such as the database of 85056 images used here.
Planet images have high spectral variability due to lack of inter-sensor
calibration between satellites, as well as the intra-annual variation of
sunlight and atmospheric conditions present in the daily imagery that are
combined to make the monthly or bi-annual composites Planet (2021). As only a
light data augmentation was used, the generalisation of the model seems to
confirm that natural variation of reflectance present in the satellite images
is sufficient to improve the accuracy of the model and to help the model to
generalize Wagner (2021). Our model was very robust to estimate non-forest as
shown by the independent validation with LiDAR data, Fig. 3a and made only few
false positive errors ($\sim$5%) on the forest cover, Fig. 3b. We are less
preoccupied by false positives than false negatives, that is, we don’t want to
miss any forested pixels. False positives can be easily corrected later with
the temporal filter. For example, with a simple rule, i.e., the forest
attribution cannot change from forest to non-forest and to forest again.
Relatively few other observed errors were mainly sparse trees and with dark
green background. As our algorithm look at the context, if there are still
some trees, it can be classified as a forest, and as it does not look at the
change between date, the model is unaware of the diminution of the number of
trees and can miss the detection. This is a rare case and is more related to
degradation that to deforestation. Predicting the tree cover at each date
independently of the other dates is a major difference with models like GFC
map Hansen et al. (2013) where the authors looked at the difference between
date to identify change. In future work, the deep learning model with Planet
NICFI images could be adapted to work with the times series, such as works
made to detect deforestation with Landsat Matosak et al. (2022); Maretto et
al. (2020), and to account for more subtle change in tree cover. As the tree
cover is initially self-segmented by the k-textures algorithm, an expert only
define which classes found by the k-textures algorithm are representing tree
cover, it is still a challenge to relate our tree cover meaning to the
official forest definitions such as the FAO definition: ’Land spanning more
than 0.5 hectares with trees higher than 5 meters and a canopy cover of more
than 10 percent, or trees able to reach these thresholds in situ. It does not
include land that is predominantly under agricultural or urban land use’ FAO
(2020). The U-net model needs a lot of data to train, and we still do not
dispose of sufficient LiDAR dataset to try to get closer to the official
definition. In the future, this could be done, by training a U-net including
the height from LiDAR, for example.

###  4.2 Mapping of deforestation

The tree cover maps at very high accuracy for all Planet dates enabled to
reconstruct the history of deforestation for the region on the period 2015 -
2021 at a spatial resolution of $\sim$ 5 m and temporal resolution of 6
months. We observed that deforestation is still widespread in Mato Grosso and
the deforestation front are actives, Fig . 4. Our numbers show that
deforestation is still alarmingly increasing is this state, with the tree
cover change for the second semester of 2021 representing $\sim$2% (19817.78
km2) of the size of the state, Table 2. Several types of tree cover changes
were observed, from small scale deforestation to large scale deforestation,
roads, and also natural changes such as changes in river paths, Fig 5. One of
the major differences with GFC tree cover loss year and Prodes product Hansen
et al. (2013); National Institute for Space Research (1988-2021) (INPE), is
that the entire tree cover is mapped at each date and not only the change in
relation to a fixed previous year (2000 in the case of the GFC map) or
previous dataset (deforested areas for Prodes) so we can compute the actual
deforestation number. Our product is also bi-annual when other products are
annual and in the next version, we will integrate a cloud mask specific to
Planet NICFI image to have a monthly product. Large areas mapped by Prodes or
GFC tree cover loss year that where not found by our product were mainly
linked to forest fire or degradation due to fire mapped as deforestation, Fig.
8. We considered as tree cover pixelx that were classified as tree cover on
half of the dates on a moving windows of $\pm$ three months. Consequently, it
is possible that vegetation that is leafless during drought months start to be
detected only with the monthly dataset (since 2020-09-01) and be classified as
deforestation, which could increase the detection for the first years of the
monthly dataset. This artifact should be resolved with a longer monthly time
series. Furthermore, rivers roads and clearcut boundaries are also mapped with
increased accuracy during the time series as it can take some time to see them
clearly due to minor errors in the co-registration and the satellite view
angles of images at different dates Francini et al. (2020b). For example, a
river can appear as deforestation for the first dates of the series, until the
larger path of the river is mapped. This artifact should also be also resolved
with more monthly data. The monthly time series is still really new, and some
more months would be needed to properly calibrate the deforestation map.
Furthermore, our map also accounts for natural changes in tree cover, and not
all changes that we observe are human made tree cover changes. This can
explain, in addition to the spatial resolution of 5 m, why our model maps up
to 7.49 times more change in tree cover than the 30 m products, Table 2. In
future work, we will also include regrowing forest, as there is a significant
amount of regrowing forests and they are very interesting for carbon
sequestration, with the potential of contributing 5.5% to Brazil’s 2030 net
emissions reduction target Silva Junior et al. (2020); Heinrich et al. (2021);
Rosan et al. (2019). As for other tropical forest monitoring systems based on
optical imagery, our model can be still limited on areas with high cloud cover
and for fast detection of deforestation, for early warning systems, radar
monitoring could be preferred as they are now efficient and used such as
DETER-SAR in Brazil Doblas et al. (2022). In future work, we will adapt the
methodology to radar images to circumvent the cloud cover issues. Besides
supporting forest monitoring and carbon assessment, our 5 m resolution tree
cover map could also support forest fragmentation analysis for conservation as
forest fragmentation is currently increasing in the Amazon Montibeller et al.
(2020); Vogt et al. (2009); Strassburg et al. (2016). Finally, our work
demonstrates the potential of deep learning for mapping deforestation with
Planet NICFI image.

###  4.3 Advances in training sample production for land cover

Here, we used an alternative to produce the large training sample needed to
parameterize the U-net model, which is usually made by hand. The production of
our training sample was made in a large part automatically with the use of
self-segmentation by the k-textures algorithm Wagner et al. (2022). The
k-textures algorithm provides self-supervised segmentation of an image in a k
number of texture classes and is fully deep learning based. The manual steps
in this process consisted of the choice of number of classes returned by the
k-textures model, to identify which classes represented forest, to perform
small corrections on the mapped results, and to find additional samples of
atmospheric conditions, such as forests below thin clouds which does not need
segmentation (only adding in the sample 256$\times$256 images with thin cloud
cover over forest). The training sample was therefore produced with minimal
human supervision, that is, without the need of manually mapping samples of
pixels that belonged to the forest class. This enables to make the training
sample very quickly, in a couple of days, and which could explain the high
accuracy of segmentation. Furthermore, the k-textures already use several
features and multiple levels of abstraction, to separate each class, and a
human cannot be as consistent and fast Brodrick et al. (2019). Drawing
polygons manually cannot be as accurate as the model, as the manual
delineation of pixels cannot follow the pixels in the borders of forests
accurately, for example. As a result, the production of the sample was faster
and the training sample more accurate. More generally, this approach could
ease the production of training sample for classes that have a different
texture from other land-covers.

##  5 Conclusions

In this work, we showed the great potential of the popular CNN semantic
segmentation architecture U-net Ronneberger et al. (2015) to map tropical tree
cover and deforestation with Planet NICFI images. The U-net accurately
segmented dense tree cover (F1-score $>$ 0.98) with minimal preprocessing and
even with the high reflectance variability of the Planet NICFI images. The
mapping of tree cover at a 6-month scale enables to map of deforestation
further and brings information on forest ecosystem change at an unprecedented
time scale. In addition, our method detects small patches of deforestation,
which allows us to locate areas in an early stage, helping institutions
responsible for fiscalization and law enforcement to avoid illegal
deforestation. The model will be further improved to work with images from in
different tropical forest regions, more solar/view angles and atmospheric
conditions by increasing the training sample and scaled on the cloud to
provide global biannual tree cover and deforestation data for the entire
Planet NICFI dataset.

##  6 Authors contributions

Conceptualization, F.H.W. and S.S.; methodology, F.H.W., R.D., C.H.L.S.J,
G.C., A.L.R., M.C.M.H.; software, F.H.W. and R.D.; validation, F.H.W., R.D.
and J.P.H.B.O; formal analysis, F.H.W.; investigation, F.H.W. R.D. S.S.;
resources, S.S.; data curation, F.H.W.; writing—original draft preparation,
F.H.W., R.D., C.H.L.S.J, and S.S.; writing—review and editing, F.H.W., R.D.,
C.H.L.S.J and S.S; visualization, F.H.W.; supervision, S.S.; project
administration, S.S.; funding acquisition, S.S. All authors have read and
agreed to the published version of the manuscript.

##  7 funding

This research received no external funding

##  8 Data availability

The TensorFlow 2 U-net model with trained weights is available on Zenodo
https://doi.org/10.5281/zenodo.7324753. Planet NICFI data are freely available
trough the Planet Mosaics API
https://developers.planet.com/docs/basemaps/reference/##tag/Basemaps-and-
Mosaics.

##  9 acknowledgments

We thank the team from the EBA project for supporting the use of the airborne
LiDAR datasets. Part of this work was carried out at the Jet Propulsion
Laboratory, California Institute of Technology, under a contract with the
National Aeronautics and Space Administration (NASA).

##  10 Conflicts of interest

The authors declare no conflict of interest. The funders had no role in the
design of the study; in the collection, analyses, or interpretation of data;
in the writing of the manuscript; or in the decision to publish the results.

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# Measuring topological entanglement entropy using Maxwell relations

Sarath Sankar  School of Physics and Astronomy, Tel Aviv University, Tel Aviv
6997801, Israel  Eran Sela  School of Physics and Astronomy, Tel Aviv
University, Tel Aviv 6997801, Israel  Cheolhee Han  School of Physics and
Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel

###### Abstract

Topological entanglement entropy (TEE) is a key diagnostic of topological
order, allowing to detect the presence of Abelian or non-Abelian anyons.
However, there are currently no protocols to measure TEE in condensed matter
systems. Here, we propose a scheme to measure the TEE, based on a nontrivial
connection with the thermodynamic entropy change occurring in a quantum point
contact (QPC) as it pinches off the topological liquid into two. We show how
this entropy change can be extracted using Maxwell relations from charge
detection of a nearby quantum dot. We demonstrate this explicitly for the
Abelian Laughlin states, using an exact solution of the sine-Gordon model
describing the universal crossover in the QPC. Our approach might open a new
thermodynamic detection scheme of topological states also with non-Abelian
statistics.

## Introduction.

Entanglement describes non-local correlations between quantum objects and is
essential in understanding quantum many-body systems. It is also at the heart
of quantum computation and information sciences and plays a pivotal role in
models such as topological quantum computation Nayak et al. (2008) and
measurement based quantum computation Briegel et al. (2009). Topological
entanglement entropy (TEE) Kitaev and Preskill (2006); Levin and Wen (2006);
Jiang et al. (2012) is the ultimate diagnostic of topological order as defined
by fractional quasiparticles carrying anyonic statistics. In a
(2+1)-dimensional topological phase where the system size is larger than the
correlation length, the entanglement entropy in the ground state
$|\Psi\rangle$ between subsystems $A$ and $B$,
$S_{EE}=-\text{Tr}[\rho_{A}\log\rho_{A}]$ where
$\rho_{A}=\text{Tr}_{B}[|\Psi\rangle\langle\Psi|]$, takes the general form
Kitaev and Preskill (2006); Levin and Wen (2006),

$S_{EE}=\alpha L-\gamma+...~{}.$ (1)

Here $L$ is the length of the entanglement cut, $\alpha$ is a non-universal
constant describing short range entanglement, and the second sub-leading term
is the TEE. The TEE $\gamma=\log\mathcal{D}$ is uniquely related to the total
quantum dimension of the topological phase,
$\mathcal{D}=\sqrt{\sum_{a}d_{a}^{2}}$, with $d_{a}$ being the quantum
dimensions of each individual anyon type labeled by $a$. In the presence of
$N$ anyons of type $a$, the gapped topological liquid has a degeneracy that
scales as $d_{a}^{N}$, so that $d_{a}>1$ ($d_{a}=1$) refers to a non-Abelian
(Abelian) anyon. For instance, there are $m$ Abelian anyons in the fractional
quantum Hall (FQH) Laughlin state with filling fraction $\nu=1/m$, with
$d_{a}=1$ $(a=0,1,\dots,m-1)$ and $\mathcal{D}=\sqrt{m}$; The Moore-Read state
at $\nu=5/2$ has four abelian anyons and two non-Abelian anyons having
$d_{a}=\sqrt{2}$, with $\mathcal{D}=2\sqrt{2}$.

Measuring entanglement entropy in many-body systems is a daunting task since
it requires full state tomography. Variants of $S_{EE}$ such as the Rényi
entanglement entropy can be accessed in controllable many-body quantum
simulators such as cold atoms or trapped ions using many-copy methods Daley et
al. (2012); Pichler et al. (2013); Islam et al. (2015); Azses et al. (2020) or
randomized measurement techniques van Enk and Beenakker (2012); Elben et al.
(2018); Brydges et al. (2019). The latter, remarkably, was employed recently
to measure the Rényi TEE of Kitaev’s toric code Kitaev (2003) prepared on a
quantum processor Satzinger et al. (2021). Unambiguously extracting the
subleading TEE term $\gamma$ requires dividing the system into three
subsystems and forming appropriate combinations of the different partitions
such as to cancel the leading term Kitaev and Preskill (2006); Levin and Wen
(2006). This was successfully implemented Satzinger et al. (2021) thanks to
the zero correlation length of the toric code eigenstates. However, measuring
TEE in condensed matter systems, such as in the realm of the FQH effect in two
dimensional electron systems, remains elusive.

Interestingly, there exists an intimate relation between TEE and a
thermodynamic entropy loss associated with a quantum point contact (QPC)
Fendley et al. (2007). The QPC allows tunneling between two points on the edge
(see parameter $\lambda$ in Fig. 2 below). For an edge carrying fractional
quasiparticles this is a relevant perturbation that introduces an energy scale
$T_{B}$ Kane and Fisher (1992). For $T\gg T_{B}$ the system is described by
the ultraviolet (UV) fixed point that is unaffected by tunneling. As
temperature decreases below $T_{B}$, tunneling processes proliferate and the
system flows to the infrared (IR) fixed point where the droplet is
disconnected into two droplets $A$ and $B$. Their anyonic charge $a$ can no
longer fluctuate since each droplet can not support an overall fractional
anyonic charge, leading to an entropy reduction. Using the bulk-edge
correspondence, it was shown for a general (2+1)-dimensional chiral
topological phase that the thermodynamic entropy change of the QPC coincides
with a bulk property being precisely the TEE Fendley et al. (2007),

$S_{UV}-S_{IR}=\log\mathcal{D}.$ (2)

This entropy difference can be read out from the temperature dependence of the
entropy as shown in Fig. 1 below. However, can this thermodynamic entropy
change be measured experimentally?

Recently there has been a surge of interest in measuring entropy in mesoscopic
systems Yang and Halperin (2009); Schmidt et al. (2017); Hou et al. (2012);
Chickering et al. (2013); Pyurbeeva et al. (2021); Kleeorin et al. (2019);
Pyurbeeva et al. (2021); Rozen et al. (2021); Saito et al. (2021). Among
those, a general entropy detection approach is based on charge measurements of
quantum dots combined with Maxwell relations. While its promising applications
were identified theoretically early on Cooper and Stern (2009); Yang and
Halperin (2009); Viola et al. (2012); Ben-Shach et al. (2013, 2015); Smirnov
(2015); Sela et al. (2019); Hou et al. (2012); Han et al. (2022), it was
actually demonstrated very recently Kuntsevich et al. (2015); Hartman et al.
(2018); Child et al. (2022a, b).

In this paper, we rely on the general relation Eq. (2) in the context of
charge-based entropy detection experiments Hartman et al. (2018); Child et al.
(2022a, b), to formulate a protocol to measure the TEE in FQH systems.

## Model.

The chiral edges of a FQH system described by an Abelian Laughlin state with
filling factor $\nu=1/m$ are described by the bosonized Hamiltonian Wen
(1990); SM

$H_{0}=\frac{1}{4\pi\nu}\int
dx(\partial_{x}\varphi_{L})^{2}+(\partial_{x}\varphi_{R})^{2},$ (3)

where $L/R$ denotes left/right movers. Here we have set the edge velocity to
unity. The QPC at $x=0$ induces tunneling of quasiparticles between the edge
states, which is described by Wen (1991); Kane and Fisher (1992)

$H_{B}=\lambda\cos(\varphi_{L}(x=0)-\varphi_{R}(x=0)),$ (4)

where $\lambda$ is the tunneling strength. For $m>1$ it leads to the energy
scale $T_{B}=C\lambda^{\frac{1}{1-\nu}}$ across which the crossover from UV to
IR limits happen, with $C$ being a non-universal constant of appropriate
dimensions Kane and Fisher (1992). The integer quantum Hall case $m=1$ is also
described by the same Hamiltonian, but in this case the tunneling of electrons
is marginal in the renormalization group sense and there is no crossover.
While for experiments in electronic systems only odd values of $m$ are
relevant, corresponding to fermions, we also consider theoretically the case
of even $m$, corresponding to a Laughlin state of bosons.

Figure 1:  Exact TBA results for the thermodynamic entropy change in a QPC at
$\nu=1,1/2,1/3$ with a net entropy difference of $0$, $\log\sqrt{2}$, and
$\log\sqrt{3}$, respectively. The results for $\nu=1/2$ are also obtained
using refermionization SM .

In general for $\nu=1/m$, the model can be mapped into the boundary sine-
Gordon model which is integrable Ghoshal and Zamolodchikov (1994). It can be
solved using thermodynamic Bethe ansatz (TBA) Fendley et al. (1994); Fendley
et al. (1995a, b); SM , which allows to obtain the free energy $F[T,T_{B}]$,
and from it the boundary entropy $S=-\partial_{T}F$.

Applying the TBA we compute the entropy along the full crossover from the UV
to IR fixed points SM , see Fig. 1, which gives
$\mathcal{S}_{UV}-\mathcal{S}_{IR}=\log\sqrt{m}$. This yields a finite entropy
change only in the fractional case, $m>1$. We included also the case $\nu=1/2$
which can be solved exactly using refermionization, and which corresponds to
an effective Majorana fermion Guinea (1985); Kane and Fisher (1992); Matveev
(1995); Smirnov (2015); SM . Similar methods Schiller et al. (2020); col  can
be applied to extract the entropy of parafermion modes.

Figure 2: A QPC in a chiral topological phase facilitates tunneling between
the edge states. The tunneling strength $\lambda$ is controlled by the charge
on a nearby QD, $\lambda=\lambda(N)$, that itself depends on the chemical
potential $\mu$ of a nearby weakly tunnel-coupled reservoir. An adjacent
charge detector measures the charge of the QD $\langle N\rangle$. Sufficiently
charging the QD drives the QPC between UV to IR limits.

Next, we present two schemes to realize the crossover in the QPC between the
UV and IR limits, and measure the resulting thermodynamic entropy change.

## Scheme 1: Side coupled quantum dot.

We now illustrate the TEE measurement protocol for the $\nu=1/m$ Laughlin
states using a side coupled quantum dot (QD) following the approach of Ref.
Sela et al., 2019. As shown in Fig. 2, we attach to the QPC a QD in the
Coulomb blockade regime described by a classical energy function
$E(N,\mu)=E_{c}N^{2}-\mu N$. Here $E_{c}$ is the charging energy, $N$ is the
number of electrons in the QD, and $\mu$ is a local chemical potential of the
QD, controlled by a gate voltage. The QD interacts electrostatically with the
QPC, as described by a dependence of the tunneling amplitude on the number
operator of the QD, $\lambda=\lambda(N)$. Thus, the crossover energy scale
$T_{B}$ is controlled by $N$, $T_{B}=T_{B}(N)$. We now show how, from such a
dependence, one can extract $S_{UV}-S_{IR}$.

Under these conditions the partition function of the combined system is

$Z_{\text{tot}}=\sum_{N}e^{-\frac{1}{T}[F(T,T_{B}(N))+E(N,\mu)]},$ (5)

where $F(T,T_{B}(N))$ is the free energy corresponding to the Hamiltonian in
Eqs. (3) and (4) with $\lambda\to\lambda(N)$. By attaching a charge detector
to the QD, using the Maxwell relation $\frac{d\langle
N\rangle}{dT}=\frac{dS}{d\mu}$, one can extract the entropy change produced by
a change of $\mu$,

$\Delta S_{\mu_{1}\to\mu_{2}}=\int_{\mu_{1}}^{\mu_{2}}\frac{d\langle
N\rangle}{dT}d\mu.$ (6)

When $T\ll E_{c}$, upon increasing $\mu$, there are several quantized charge
steps in $N$, see inset of Fig. 3(a), and the QPC gets closer to pinch-off,
corresponding to an increase in $T_{B}$. The desired entropy change will occur
if, by charging the QD by $\Delta N$ electrons, the QPC transitions from the
UV limit $(T_{B}\ll T)$ to the IR limit $(T_{B}\gg T)$.

Figure 3:  Illustration of the proposed TEE measurement. (a) The conductance
between the ohmic contacts for filling factor $\nu=1/2,1/3$ changes upon
charging a nearby QD by varying the chemical potential $\mu$ of the QD. The
charging curve is shown in the inset. Here $T/E_{c}=0.1$, $G_{0}=e^{2}/h$, and
$T_{B}(N)/E_{c}=e^{N/2}-1$ Sela et al. (2019). (b) Temperature derivative of
the QD charge as a function of $\mu$. For $\nu=1$ (dashed) the curve is
exactly anti-symmetric, while for $\nu=1/3$ (solid) it is not. (c) The
resulting entropy change (QPC+QD), obtained by integrating $d\langle
N\rangle/dT$ using the Maxwell relation Eq. (6), for $\nu=1$ (dashed blue) and
$\nu=1/2,1/3$ (solid curves). The peaks are associated with the $\log 2$
entropy of the dot at its charge steps. The plateaus of the entropy decrease
for $\nu=1/2,1/3$ as a function of $\mu$, and converge to the desired TEE once
the UV-IR crossover is accomplished.

In Fig. 3, we assume an $N$ dependence of $T_{B}(N)$ such that across a
charging of $\Delta N=3$ electrons one achieves $T_{B}(N)\ll T\ll
T_{B}(N+\Delta N)$. In Fig. 3(b) we compute $\frac{d\langle N\rangle}{dT}$,
where $\langle N\rangle$ is extracted from the total free energy
$F_{\text{tot}}=-T\ln Z_{\text{tot}}$ as $\langle N\rangle=-\partial
F_{\text{tot}}/\partial\mu$. The total entropy change $S_{\mu_{1}=0\to\mu}$
from a selected $\mu_{1}$ till a varying $\mu$ is shown in panel Fig. 3(c). We
can see that this entropy contains a series of $\log 2$ peaks corresponding to
charge degeneracies of the QD, riding on top of the slowly varying entropy
along the crossover for $\nu\neq 1$. For $T\ll E_{c}$ these two effects are
well separated and we can measure the TEE by taking the difference between the
corresponding entropy plateaus where $\langle N\rangle$ weakly fluctuates; see
dashed horizontal lines in Fig. 3(c).

Several experiments observed the crossover between UV and IR limits of a gate-
tuned QPC for several FQH states through the conductance and the shot noise
which crosses from quasiparticle tunneling to electron tunneling Chung et al.
(2003); Griffiths et al. (2000); Dolev et al. (2008); Cohen et al. (2022). Our
method outlined in Fig. 2, however, requires to drive this crossover as a
function of the chemical potential of the dot. The conductance for such a
crossover is plotted in Fig. 3(a).

In practice, however, a side-coupled QD as in Fig. 2 may have a limited
electrostatic affect on the QPC. To enhance the effect, we now discuss a
different setup with the QD embedded in the constriction.

Figure 4:  (a) Two QPCs in series creating a QD with Coulomb interaction
$E_{c}$ within the FQH system. By tuning the gate voltage $N_{g}$ at fixed
$\lambda_{1,2}$, this resonant double barrier-like system, yields an effective
backscattering strength in Eq. (8) which drops at resonance and vanishes for
symmetric barriers. The charge detector measures the charge of the QD. (b) The
average occupation of the QD for the case of symmetric barriers at $\nu=1/3$.
Increasing temperature shifts the curve to right/left for $N_{g}$ above/below
$1/2.$ (c) Entropy change inferred from Maxwell relation for several ratios of
$T/T_{B}^{0}$. As $T\to 0$, $N_{g}=0$ tends to the IR limit and the entropy
difference at $N_{g}=1/2$ (which is always the UV limit) tends to the TEE.

## Scheme 2: Coulomb blockade in the FQH regime.

One can realize a nearly complete change of transmission by replacing the QPC
by a double barrier consisting of two QPCs, see Fig. 4(a). For non-interacting
electrons it is well known Datta (1997) that the transmission has an abrupt
resonance peak which is maximized for symmetric barriers. Here we demonstrate
that this resonance effect also occurs in a double barrier system on a
fractional edge.

As shown in Fig. 4(a), we consider two QPCs in series separated by a distance
$2\ell$, having backscattering amplitudes $\lambda_{1,2}$, and defining a QD
characterized by a charging energy $E_{c}$. In this case the Hamiltonian is

$\displaystyle H_{QD}=$
$\displaystyle\lambda_{1}\cos(\varphi_{L}(-\ell)-\varphi_{R}(-\ell))+\lambda_{2}\cos(\varphi_{L}(\ell)-\varphi_{R}(\ell))$
$\displaystyle+\frac{E_{c}}{4\pi^{2}}\Big{(}\sum_{i=L,R}(\varphi_{i}(-\ell)-\varphi_{i}(\ell))-2\pi
N_{g}\Big{)}^{2}.$ (7)

In the limit of a large charging energy $E_{c}\ell\gg 1$ one obtains an
effective $N_{g}$ dependent boundary condition between the bosonic field Lee
et al. (2020); Morel et al. (2022); Lee et al. (2012); SM . As a result, the
two QPCs behave as one effective QPC with

$H_{B}^{\prime}=\lambda_{{\rm{eff}}}e^{i(\varphi_{L}(0)-\varphi_{R}(0))}+{\rm{h.c.}},$
(8)

where

$\lambda_{{\rm{eff}}}\propto\lambda_{1}e^{-i\pi N_{g}}+\lambda_{2}e^{i\pi
N_{g}}.$ (9)

This effective model holds for $E_{c}\gg T_{B1,2}=C\lambda_{1,2}^{1/(1-\nu)}$.
We assume a finite reflection at the QPCs such that $T_{B1,2}\gg T$. However,
using Eq. (9), we see that when the barriers are symmetric
$\lambda_{1}=\lambda_{2}\equiv\lambda_{0}$, we have
$\lambda_{{\rm{eff}}}\propto\lambda_{0}\cos(\pi N_{g})$, or equivalently
$T_{B}=T_{B}^{0}|\cos(\pi N_{g})|^{1/(1-\nu)}$. Hence at $N_{g}=1/2$ the
effective barrier vanishes and $T_{B}=0$.

Thus, the TEE can be extracted as the entropy reduction between the resonance
condition $N_{g}=1/2$ and the off-resonance limit $N_{g}=0,1$. This entropy
difference can be directly measured by attaching a charge detector to the QD
and using the Maxwell relations. Eq. (6) applies with $\mu\to 2E_{c}N_{g}$.

In Fig. 4(b,c) we plot $\langle N\rangle$ and the extracted entropy. Here
$\langle N\rangle=-\frac{1}{2E_{c}}\frac{\partial F}{\partial N_{g}}$ is
computed from the TBA free energy $F(T,T_{B})$ with
$T_{B}=T_{B}[\lambda_{{\rm{eff}}}(N_{g})]$ carrying the $N_{g}$-dependence via
Eq. (9). Different than the side coupled QD, by using the resonance effect,
the crossover is fully accomplished along the way from a Coulomb peak
($N_{g}=1/2$) and a nearby Coulomb valley ($N_{g}=0,1$). Also, different from
the side coupled QD which was only weakly coupled to the lead, and hence led
to clear $\log 2$ entropy peaks at its charge degeneracy points [Fig. 3(c)],
here the strong tunnel-coupled QD has a negligible contribution to the entropy
which scales as
$S_{QD}\sim\frac{T_{B}^{2-2\nu}}{T^{2-2\nu}}\frac{T^{2}}{E_{c}^{2}}$ SM .
However, in order to explore the full UV-IR crossover, the barriers should be
high enough such that $T_{B}^{0}\gg T$, see Fig. 4(c).

## Discussion.

In the above schemes, to isolate the subleading TEE term $\gamma$ in Eq. (1),
we assume that when the FQH droplet splits into two, the change in the length
of the edge, $L\to L+\Delta L$, leads only to a negligible entropy change as
compared to the order $\mathcal{O}(1)$ TEE. We can now justify this
assumption. The resulting entropy change of the 1D gapless edge modes with
velocity $v_{F}$ can be estimated as

$\Delta S_{L}\sim\frac{T}{\hbar v_{F}}\Delta L.$ (10)

Since the TEE part of the entropy change is $O(1)$, we can neglect $\Delta
S_{L}$ if $\Delta L\ll\hbar v_{F}/T$. For $T=100$ mK and
$v_{F}=10^{6}{\rm{m/s}}$, we find that the requirement $\Delta L\ll
10^{-4}{\rm{m}}$ is easily satisfied.

We also note that the TEE is a property of the ground state, which actually
vanishes above a certain temperature determined by the energy gap to bulk
excitations Lu et al. (2020). This restriction applies to our schemes to
measure TEE as well.

## Summary.

Despite its importance, the measurement of the topological entanglement
entropy is still elusive in condensed matter systems. Using a relation between
TEE and a thermodynamic entropy change in fractional QPCs, we proposed
realistic approaches to measure the TEE employing charge measurement and the
Maxwell relation. We illustrated our protocols for the experimentally simplest
and yet nontrivial case of Abelian fractional topological order.

Our proposed setups are also applicable to extract the entropy change between
UV and IR fixed points in the more general boundary sine-Gordon model
describing an impurity in a Luttinger liquid with any value of the interaction
parameter, within mesoscopic systems simulating this model such as in Ref.
Anthore et al., 2018.

Since the relation we applied is general, our method can be applied to more
exotic systems, such as complex hierarchical states Röösli et al. (2021),
other topological systems such as spin liquid systems Savary and Balents
(2016), and most interestingly FQH states with non-Abelian quasiparticles.

_Acknowledgements._ We gratefully acknowledge support from the European
Research Council (ERC) under the European Unions Horizon 2020 research and
innovation programme under grant agreement No. 951541. ARO (W911NF-20-1-0013)
and the Israel Science Foundation grant numbers 154/19. We thank Anne Anthore,
Christophe Mora, Yigal Meir, and particularly Frederic Pierre for illuminating
discussions.

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