Trend analysis

Packages

import ee
import geemap
import pandas as pd
import numpy as np
import geopy_ee_ts_fun as geopy

try:
    ee.Initialize(project="ee-pflugmacher")
except Exception as e:
    ee.Authenticate()
    ee.Initialize(project="ee-pflugmacher")

Set up map window

In this exercise, we are going to map trends. First, we define a region, a coordinate reference system crs, and a scale(pixel size) or affine transformation crsTransform.

region = ee.Geometry.Polygon(
        [[[12.901495385988614, 53.14672196059798],
          [12.901495385988614, 52.850010520689125],
          [13.426092553957364, 52.850010520689125],
          [13.426092553957364, 53.14672196059798]]])
scale = 20
crs = 'EPSG:3035'

Map = geemap.Map()
Map.centerObject(region, zoom=11)
Map

Visualization parameters

# Sentinel-2 surface reflectance
s2Vis = {"bands": ["B4", "B3", "B2"], "min":0, "max": 1200}

# NDVI color palette
ndviVis = {"palette": ['white', 'green'], "min":0.2, "max": 1}

Sentinel-2 data

The code block below filters the Sentinel-2 collection, masks clouds and calculats NDVI. The cloud masking approach uses the AI-based Cloud Score+ available in the Cloud Score+ Harmonized-S2 dataset. The Sentinel-2 harmonized dataset fixes different scaling factors in the original Sentinel-2 dataset.

dateFilter = ee.Filter.And(ee.Filter.dayOfYear(200, 365), 
                           ee.Filter.date(ee.Date.fromYMD(2018, 1, 1), 
                                          ee.Date.fromYMD(2021, 12, 31)))

# Harmonized Sentinel-2 Level 2A collection.
s2 = ee.ImageCollection('COPERNICUS/S2_SR_HARMONIZED')

# Cloud Score+ image collection. Note Cloud Score+ is produced from Sentinel-2
# Level 1C data and can be applied to either L1C or L2A collections.
csPlus = ee.ImageCollection('GOOGLE/CLOUD_SCORE_PLUS/V1/S2_HARMONIZED')

# Use 'cs' or 'cs_cdf', depending on your use case; see docs for guidance.
QA_BAND = 'cs_cdf'

# The threshold for masking; values between 0.50 and 0.65 generally work well.
# Higher values will remove thin clouds, haze & cirrus shadows.
CLEAR_THRESHOLD = 0.60

def mask_s2(image):
    return image.updateMask(image.select(QA_BAND).gte(CLEAR_THRESHOLD))

s2m = s2.filterBounds(region) \
        .filter(dateFilter) \
        .linkCollection(csPlus, [QA_BAND]) \
        .map(mask_s2)

# Create NDVI Collection from masked Sentinel-2 data
ndviS2 = s2m.map(geopy.calcNdviS2)

ndviS2

• ImageCollection COPERNICUS/S2_SR_HARMONIZED (1052 elements)
  • type:ImageCollection
  • id:COPERNICUS/S2_SR_HARMONIZED
  • version:1737911578969955
  • bands: []
  • properties: Object (21 properties)
    • date_range: [1490659200000, 1647907200000]
      • 0:1490659200000
      • 1:1647907200000
    • description:

      Sentinel-2 is a wide-swath, high-resolution, multi-spectral imaging mission supporting Copernicus Land Monitoring studies, including the monitoring of vegetation, soil and water cover, as well as observation of inland waterways and coastal areas.

      The Sentinel-2 L2 data are downloaded from scihub. They were computed by running sen2cor. WARNING: ESA did not produce L2 data for all L1 assets, and earlier L2 coverage is not global.

      The assets contain 12 UINT16 spectral bands representing SR scaled by 10000 (unlike in L1 data, there is no B10). There are also several more L2-specific bands (see band list for details). See the Sentinel-2 User Handbook for details. In addition, three QA bands are present where one (QA60) is a bitmask band with cloud mask information. For more details, see the full explanation of how cloud masks are computed.

      EE asset ids for Sentinel-2 L2 assets have the following format: COPERNICUS/S2_SR/20151128T002653_20151128T102149_T56MNN. Here the first numeric part represents the sensing date and time, the second numeric part represents the product generation date and time, and the final 6-character string is a unique granule identifier indicating its UTM grid reference (see MGRS).

      Clouds can be removed by using COPERNICUS/S2_CLOUD_PROBABILITY. See this tutorial explaining how to apply the cloud mask.

      For more details on Sentinel-2 radiometric resolution, see this page.

      Provider: European Union/ESA/Copernicus
      

      Revisit Interval
      5 days

      Bands

      Name	Description
      B1	Aerosols
      B2	Blue
      B3	Green
      B4	Red
      B5	Red Edge 1
      B6	Red Edge 2
      B7	Red Edge 3
      B8	NIR
      B8A	Red Edge 4
      B9	Water vapor
      B11	SWIR 1
      B12	SWIR 2
      AOT	Aerosol Optical Thickness
      WVP	Water Vapor Pressure. The height the water would occupy if the vapor were condensed into liquid and spread evenly across the column.
      SCL	Scene Classification Map (The "No Data" value of 0 is masked out)
      TCI_R	True Color Image, Red channel
      TCI_G	True Color Image, Green channel
      TCI_B	True Color Image, Blue channel
      MSK_CLDPRB	Cloud Probability Map (missing in some products)
      MSK_SNWPRB	Snow Probability Map (missing in some products)
      QA10	Always empty
      QA20	Always empty
      QA60	Cloud mask
      Bitmask for QA60 Bit 10: Opaque clouds 0: No opaque clouds 1: Opaque clouds present Bit 11: Cirrus clouds 0: No cirrus clouds 1: Cirrus clouds present

      Image Properties

      Name	Type	Description
      AOT_RETRIEVAL_ACCURACY	DOUBLE	Accuracy of Aerosol Optical thickness model
      CLOUDY_PIXEL_PERCENTAGE	DOUBLE	Granule-specific cloudy pixel percentage taken from the original metadata
      CLOUD_COVERAGE_ASSESSMENT	DOUBLE	Cloudy pixel percentage for the whole archive that contains this granule. Taken from the original metadata
      CLOUDY_SHADOW_PERCENTAGE	DOUBLE	Percentage of pixels classified as cloud shadow
      DARK_FEATURES_PERCENTAGE	DOUBLE	Percentage of pixels classified as dark features or shadows
      DATASTRIP_ID	STRING	Unique identifier of the datastrip Product Data Item (PDI)
      DATATAKE_IDENTIFIER	STRING	Uniquely identifies a given Datatake. The ID contains the Sentinel-2 satellite, start date and time, absolute orbit number, and processing baseline.
      DATATAKE_TYPE	STRING	MSI operation mode
      DEGRADED_MSI_DATA_PERCENTAGE	DOUBLE	Percentage of degraded MSI and ancillary data
      FORMAT_CORRECTNESS	STRING	Synthesis of the On-Line Quality Control (OLQC) checks performed at granule (Product_Syntax) and datastrip (Product Syntax and DS_Consistency) levels
      GENERAL_QUALITY	STRING	Synthesis of the OLQC checks performed at the datastrip level (Relative_Orbit_Number)
      GENERATION_TIME	DOUBLE	Product generation time
      GEOMETRIC_QUALITY	STRING	Synthesis of the OLQC checks performed at the datastrip level (Attitude_Quality_Indicator)
      GRANULE_ID	STRING	Unique identifier of the granule PDI (PDI_ID)
      HIGH_PROBA_CLOUDS_PERCENTAGE	DOUBLE	Percentage of pixels classified as high probability clouds
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B1	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B1 and for all detectors
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B2	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B2 and for all detectors
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B3	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B3 and for all detectors
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B4	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B4 and for all detectors
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B5	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B5 and for all detectors
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B6	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B6 and for all detectors
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B7	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B7 and for all detectors
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B8	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B8 and for all detectors
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B8A	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B8a and for all detectors
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B9	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B9 and for all detectors
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B10	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B10 and for all detectors
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B11	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B11 and for all detectors
      MEAN_INCIDENCE_AZIMUTH_ANGLE_B12	DOUBLE	Mean value containing viewing incidence azimuth angle average for band B12 and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B1	DOUBLE	Mean value containing viewing incidence zenith angle average for band B1 and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B2	DOUBLE	Mean value containing viewing incidence zenith angle average for band B2 and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B3	DOUBLE	Mean value containing viewing incidence zenith angle average for band B3 and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B4	DOUBLE	Mean value containing viewing incidence zenith angle average for band B4 and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B5	DOUBLE	Mean value containing viewing incidence zenith angle average for band B5 and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B6	DOUBLE	Mean value containing viewing incidence zenith angle average for band B6 and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B7	DOUBLE	Mean value containing viewing incidence zenith angle average for band B7 and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B8	DOUBLE	Mean value containing viewing incidence zenith angle average for band B8 and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B8A	DOUBLE	Mean value containing viewing incidence zenith angle average for band B8a and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B9	DOUBLE	Mean value containing viewing incidence zenith angle average for band B9 and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B10	DOUBLE	Mean value containing viewing incidence zenith angle average for band B10 and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B11	DOUBLE	Mean value containing viewing incidence zenith angle average for band B11 and for all detectors
      MEAN_INCIDENCE_ZENITH_ANGLE_B12	DOUBLE	Mean value containing viewing incidence zenith angle average for band B12 and for all detectors
      MEAN_SOLAR_AZIMUTH_ANGLE	DOUBLE	Mean value containing sun azimuth angle average for all bands and detectors
      MEAN_SOLAR_ZENITH_ANGLE	DOUBLE	Mean value containing sun zenith angle average for all bands and detectors
      MEDIUM_PROBA_CLOUDS_PERCENTAGE	DOUBLE	Percentage of pixels classified as medium probability clouds
      MGRS_TILE	STRING	US-Military Grid Reference System (MGRS) tile
      NODATA_PIXEL_PERCENTAGE	DOUBLE	Percentage of No Data pixels
      NOT_VEGETATED_PERCENTAGE	DOUBLE	Percentage of pixels classified as non-vegetated
      PROCESSING_BASELINE	STRING	Configuration baseline used at the time of the product generation in terms of processor software version and major Ground Image Processing Parameters (GIPP) version
      PRODUCT_ID	STRING	The full id of the original Sentinel-2 product
      RADIATIVE_TRANSFER_ACCURACY	DOUBLE	Accuracy of radiative transfer model
      RADIOMETRIC_QUALITY	STRING	Based on the OLQC reports contained in the Datastrips/QI_DATA with RADIOMETRIC_QUALITY checklist name
      REFLECTANCE_CONVERSION_CORRECTION	DOUBLE	Earth-Sun distance correction factor
      SATURATED_DEFECTIVE_PIXEL_PERCENTAGE	DOUBLE	Percentage of saturated or defective pixels
      SENSING_ORBIT_DIRECTION	STRING	Imaging orbit direction
      SENSING_ORBIT_NUMBER	DOUBLE	Imaging orbit number
      SENSOR_QUALITY	STRING	Synthesis of the OLQC checks performed at granule (Missing_Lines, Corrupted_ISP, and Sensing_Time) and datastrip (Degraded_SAD and Datation_Model) levels
      SOLAR_IRRADIANCE_B1	DOUBLE	Mean solar exoatmospheric irradiance for band B1
      SOLAR_IRRADIANCE_B2	DOUBLE	Mean solar exoatmospheric irradiance for band B2
      SOLAR_IRRADIANCE_B3	DOUBLE	Mean solar exoatmospheric irradiance for band B3
      SOLAR_IRRADIANCE_B4	DOUBLE	Mean solar exoatmospheric irradiance for band B4
      SOLAR_IRRADIANCE_B5	DOUBLE	Mean solar exoatmospheric irradiance for band B5
      SOLAR_IRRADIANCE_B6	DOUBLE	Mean solar exoatmospheric irradiance for band B6
      SOLAR_IRRADIANCE_B7	DOUBLE	Mean solar exoatmospheric irradiance for band B7
      SOLAR_IRRADIANCE_B8	DOUBLE	Mean solar exoatmospheric irradiance for band B8
      SOLAR_IRRADIANCE_B8A	DOUBLE	Mean solar exoatmospheric irradiance for band B8a
      SOLAR_IRRADIANCE_B9	DOUBLE	Mean solar exoatmospheric irradiance for band B9
      SOLAR_IRRADIANCE_B10	DOUBLE	Mean solar exoatmospheric irradiance for band B10
      SOLAR_IRRADIANCE_B11	DOUBLE	Mean solar exoatmospheric irradiance for band B11
      SOLAR_IRRADIANCE_B12	DOUBLE	Mean solar exoatmospheric irradiance for band B12
      SNOW_ICE_PERCENTAGE	DOUBLE	Percentage of pixels classified as snow or ice
      SPACECRAFT_NAME	STRING	Sentinel-2 spacecraft name: Sentinel-2A, Sentinel-2B
      THIN_CIRRUS_PERCENTAGE	DOUBLE	Percentage of pixels classified as thin cirrus clouds
      UNCLASSIFIED_PERCENTAGE	DOUBLE	Percentage of unclassified pixels
      VEGETATION_PERCENTAGE	DOUBLE	Percentage of pixels classified as vegetation
      WATER_PERCENTAGE	DOUBLE	Percentage of pixels classified as water
      WATER_VAPOUR_RETRIEVAL_ACCURACY	DOUBLE	Declared accuracy of the Water Vapor model

      Terms of Use
      

      The use of Sentinel data is governed by the Copernicus Sentinel Data Terms and Conditions.

    • keywords: List (7 elements)
      • 0:copernicus
      • 1:esa
      • 2:eu
      • 3:msi
      • 4:reflectance
      • 5:sentinel
      • 6:sr
    • period:0
    • product_tags: ['msi', 'sr', 'reflectance']
      • 0:msi
      • 1:sr
      • 2:reflectance
    • provider:European Union/ESA/Copernicus
    • provider_url:https://earth.esa.int/web/sentinel/user-guides/sentinel-2-msi/product-types/level-2a
    • sample:https://mw1.google.com/ges/dd/images/COPERNICUS_S2_SR_sample.png
    • source_tags: ['eu', 'esa', 'copernicus', 'sentinel']
      • 0:eu
      • 1:esa
      • 2:copernicus
      • 3:sentinel
    • system:visualization_0_bands:B4,B3,B2
    • system:visualization_0_max:3000.0
    • system:visualization_0_min:0.0
    • system:visualization_0_name:RGB
    • tags: List (7 elements)
      • 0:copernicus
      • 1:esa
      • 2:eu
      • 3:msi
      • 4:reflectance
      • 5:sentinel
      • 6:sr
    • thumb:https://mw1.google.com/ges/dd/images/COPERNICUS_S2_SR_thumb.png
    • title:Sentinel-2 MSI: MultiSpectral Instrument, Level-2A
    • type_name:ImageCollection
    • visualization_0_bands:B4,B3,B2
    • visualization_0_max:3000.0
    • visualization_0_min:0.0
    • visualization_0_name:RGB
  • features: List (1052 elements)
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      • type:Image
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    • 1051: Image (1 band)
      • type:Image
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        • 0: "ndvi", float, EPSG:32633, 5490x5490 px
          • id:ndvi
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          • crs_transform: [20, 0, 300000, 0, -20, 6000000]
            • 0:20
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        • system:time_start:1640772959976

Harmonic function

In this exercise, I fit a first-order harmonic model of the form:

\(\hat{y}(t) = \beta_0 + \beta_1 t + \beta_2 cos(2\pi \frac{t}{T}) + \beta_3 sin(2\pi \frac{t}{T})\)

The harmonic model is fitted to every pixel, i.e., we will estimate the parameters \(\beta_0\), \(\beta_1\), \(\beta_2\), and \(\beta_3\) for every pixel. The \(\beta\)´s are parameters for \(t\) and different transformations of \(t\), where \(t\) is time in days or (fractional) years and \(T\) is the period. The period \(T\) describes how many \(t\) it takes to complete a full harmonic cylce. For example, \(T\) is 1 if \(t\) is in years and 365 if \(t\) is in days. The model estimates \(\hat{y}\), which is the NDVI in our case.

The first step is to assemble the response (NDVI) and predictor variables. The function addPredictorVariables() adds four bands to an NDVI image. The four bands are the intercept and the predictors, i.e., \(t\) and its harmonic tranformations. The band constant has the pixel value 1 and is required to fit an intercept. The band time, associated with parameter \(\beta_1\), is time \(t\) expressed in years since January 1st 1970. It is a fractional value with decimal places. Now, the predictor variables do not vary for a specific NDVI image, i.e., \(t\) is constant for a specific image. However, recall that we fit time series to individual pixels. You also see that we can create an image with a scalar value simply by calling ee.Image(scalar). Earth engine will expand the scalar value in space as needed, i.e., defined by the output region.

def addPredictorVariables(image):
    
    date = ee.Date(image.get('system:time_start'))
    time = date.difference(ee.Date('1970-01-01'), 'year')
    sin = time.multiply(2 * np.pi ).sin()
    cos = time.multiply(2 * np.pi ).cos()
    
    predictors = ee.Image([sin, cos, time, 1]).float()

    return predictors.addBands(image.select(['ndvi'])) \
                     .rename(['sin', 'cos', 'time', 'constant', 'ndvi']) \
                     .set('system:time_start', image.get('system:time_start'))

trainingCollection = ndviS2.map(addPredictorVariables)

Fit model

Now, we can estimate the \(\beta\)´s using linear regression. The reducer is not aware of variable names and their order in the image. It assumes the last variable to be the response variable. So, we make sure that NDVI (responseName) is selected last.

The output of the regression reduction (trend) contains two bands: coefficients and residuals. However, trend is an array image and not a standard image. An array image can store an entire vector/array in each pixel not just a scalar value. For example, an array image can store a six-band image in a single band, where each pixel would contain a vector of length six. However, pixels in an array image do not have to store vectors of the same length. In our case, the coefficients band stores the coefficients in a 4x1 array in each pixel. To turn this into a four band image, we need to apply arrayProject (removes the second axis of length 1) and arrayFlatten (puts every array element into its own band).

responseName = ee.String('ndvi')
predictorNames = ee.List(['constant', 'time', 'cos', 'sin'])

trend = trainingCollection.select(predictorNames.add(responseName)) \
            .reduce(ee.Reducer.linearRegression(predictorNames.length(), 1))

coefficients = trend.select('coefficients') \
                    .arrayProject([0]) \
                    .arrayFlatten([predictorNames])

Visualizing the coefficients reveals different patterns in seasonality and trend (time) for different land cover types. The harmonic model compresse the NDVI time series into a few parameters that still represent the spectral-temporal dynamics. NOTE: the visualization of the map can take a while. The pixelwise trend fitting is processing intenstive. Keep the zoom level steady. The process restarts every time you zoom in or out.

Map.addLayer(coefficients.select(['sin']), {"min": -0.23, "max":0.1}, 'sin')
Map.addLayer(coefficients.select(['cos']), {"min": -0.23, "max":0.1}, 'cos')
Map.addLayer(coefficients.select(['time']), {"min": -0.03, "max":0.03, "palette": ['red', 'white', 'green']}, 'time')

Fitted values

We can obtain the fitted NDVI values to better understand how our models fits the data. To do this, we need to write a funciton that takes a training image (containg NDVI, \(t\) and the harmonic terms) and applies the estimated coefficients. We then map that function to the training collection to obtain a collection of fitted NDVI images.

def linearFit(image):
    fit = image.select(predictorNames) \
               .multiply(coefficients) \
               .reduce('sum') \
               .rename('fitted')
    return image.addBands(fit)

fitted = trainingCollection.map(linearFit)

Pixel observations

Let’s evaluate the model fit for individual pixel observations. To visualize these pixels, we first extract the NDVI time series for a single pixel. Second, we extract the coefficients. And third, we extract the fitted values.

Extract NDVI time series

pt_dbf = ee.Geometry.Point([13.18496707443565,53.06189771226909])
pt_ag1 = ee.Geometry.Point([13.134970698398053, 53.06835763719942])
pt_ag2 = ee.Geometry.Point([13.047938380282819, 53.04421500010952])
pt_ag3 = ee.Geometry.Point([13.0798673963961, 53.010351264908145])

ee_vals = ndviS2.getRegion(geometry=pt_dbf, scale=scale, crs=crs)

# use data converter to return values as python list
vals = ee.data.computeValue(ee_vals)
vals[0:3]

[['id', 'longitude', 'latitude', 'time', 'ndvi'],
 ['20180721T102021_20180721T102024_T32UQD',
  13.184850310654017,
  53.06195679914768,
  1532168424461,
  0.8832853025936599],
 ['20180721T102021_20180721T102024_T33UUU',
  13.184850310654017,
  53.06195679914768,
  1532168424461,
  0.8791130185979972]]

# convert returned list to pandas data.frame
# the first element (vals[0]) contains the column names
# the second element contains the first observation and so on 
df = pd.DataFrame(vals[1:], columns=vals[0]).dropna()

# extract date string and convert it to date object
df['date'] = pd.to_datetime(df.id.str.slice(0, 8))

 # copy date to row index
df.index  = df.date
df.head()

	id	longitude	latitude	time	ndvi	date
date
2018-07-21	20180721T102021_20180721T102024_T32UQD	13.18485	53.061957	1532168424461	0.883285	2018-07-21
2018-07-21	20180721T102021_20180721T102024_T33UUU	13.18485	53.061957	1532168424461	0.879113	2018-07-21
2018-07-23	20180723T101019_20180723T101036_T32UQD	13.18485	53.061957	1532340636891	0.903789	2018-07-23
2018-07-23	20180723T101019_20180723T101036_T33UUU	13.18485	53.061957	1532340636891	0.909469	2018-07-23
2018-07-26	20180726T102019_20180726T102150_T32UQD	13.18485	53.061957	1532600510652	0.886746	2018-07-26

The dataset may contain duplicate values due to the overlapping Sentinel-2 tiles. That is, there should only be one observation for a given day. So, let’s drop duplicate values.

df = df.drop_duplicates(subset=['date'])
df.head()

	id	longitude	latitude	time	ndvi	date
date
2018-07-21	20180721T102021_20180721T102024_T32UQD	13.18485	53.061957	1532168424461	0.883285	2018-07-21
2018-07-23	20180723T101019_20180723T101036_T32UQD	13.18485	53.061957	1532340636891	0.903789	2018-07-23
2018-07-26	20180726T102019_20180726T102150_T32UQD	13.18485	53.061957	1532600510652	0.886746	2018-07-26
2018-07-31	20180731T102021_20180731T102024_T32UQD	13.18485	53.061957	1533032424462	0.914293	2018-07-31
2018-08-07	20180807T101021_20180807T101024_T32UQD	13.18485	53.061957	1533636624457	0.902853	2018-08-07

Now we can plot column ndvi against the dataframe index, which is date.

df.ndvi.plot(style='.', figsize=(5,3));

Extract coefficients

We can extract the coefficients of the harmonic model for a single pixel using a reducer. Coefficients is an ee.Image, which has the sampleRegions reducer. You may get an exception “User memory limit exceeded”. To avoid this error, you can limit the processed area size or save intermediate results to your assets.

# returns feature collection
ee_coef = coefficients.sampleRegions(collection=pt_dbf, scale=scale, 
                                     projection=crs)

# extracts feature collection (server-side) to pandas dataFrame (client-side)
pt_coef = ee.data.computeFeatures({'expression': ee_coef, 'fileFormat': 'PANDAS_DATAFRAME'})
pt_coef.drop(columns="geo", inplace=True)
pt_coef

	constant	cos	sin	time
0	0.723115	-0.162878	-0.213892	-0.002603

Extract fitted values

Let’s extract the fitted NDVI values for a single pixel. Note, fitted is an ee.ImageCollection, which we can reduce with getRegion(). The output is a list that we can turn into a Pandas DataFrame for further analysis.

ee_fit = fitted.getRegion(geometry=pt_dbf, scale=scale, crs=crs)
pt_fit = ee.data.computeValue(ee_fit)

dfit = pd.DataFrame(pt_fit[1:], columns=pt_fit[0]).dropna() # create DataFrame and drop NA
dfit["date"] = pd.to_datetime(dfit.id.str.slice(0, 8)) # extract date from tile-id
dfit.index = dfit.date # put date into index to facilitate plotting
dfit = dfit.drop_duplicates(subset=['date']) # EE fits to duplicate values
dfit = dfit.iloc[:, np.delete(np.arange(dfit.shape[1]), 4)] # drop first 'time' column
dfit.iloc[:,4:].head(3)

	cos	time	constant	ndvi	fitted	date
date
2018-07-21	-0.946117	48.552486	1	0.883285	0.820100	2018-07-21
2018-07-23	-0.934459	48.557941	1	0.903789	0.825084	2018-07-23
2018-07-26	-0.914793	48.566177	1	0.886746	0.832095	2018-07-26

Visualize extracted values

dfit[["ndvi", "fitted"]].plot(style=['.', 'x'], figsize=(6,4));

Make predictions

def fit_harmonic(t, constant, time, sin, cos, period=1):
    
    result = constant + time*t + \
                sin*np.sin(2.0 * np.pi * (t / period)) + \
                cos*np.cos(2.0 * np.pi * (t / period))
    return result

def fit_trend(t, constant, time, sin, cos ):
    result = constant + time*t
    return result
    

# date difference in days
t = pd.date_range('2018-01-01', '2021-12-31', freq='D') 

# date difference in yeas
td = (t - pd.to_datetime('1970-01-01')) / np.timedelta64(1, 'D') / 365.25

# convert params stored in data frame to dictionary
params = pt_coef.to_dict(orient='records')[0]

# the result is a pandas.core.indexes.numeric.Float64Index
# we can extract the values of t with t.values
fitted_harm = fit_harmonic(td.values, **params)

fitted_trend = fit_trend(td.values, **params)

dfm = pd.DataFrame({'ndvi_harm':fitted_harm, 
                    'ndvi_trend': fitted_trend}, index=t)

ax = dfit[["ndvi", "fitted"]].plot(style=['.', 'x'], figsize=(6,4))
dfm.plot(ax=ax);

Mann-Kendall

https://developers.google.com/earth-engine/tutorials/community/nonparametric-trends

Sen’s slope

https://developers.google.com/earth-engine/tutorials/community/nonparametric-trends#sens_slope

Further reading

• Time Series Modeling (EE Community Tutorial)

• Time Series Analysis in Earth Engine (Lecture: Nick Clinton): Video

• Time Series Analysis in Earth Engine (Lecture: Nick Clinton): Slides