This document discusses analyzing seasonal patterns in the United States to test Hopkins' Law of Phenology. The study area spans 29°N to 46°N latitude along 84°W longitude, covering five states. Normalized difference vegetation index (NDVI) data from 2011 were analyzed using ENVI software to determine the first day of spring at 11 locations. Dates ranged from day 87 to 140, generally following Hopkins' Law of 4 days later per degree of latitude northward. While results sometimes deviated from the law, the relationship between date and latitude closely matched an expected curve.

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HOPKIN’S LAW MAKES SENSE: AN APPROACH TO STUDY THE NATURAL LAWS AND
PHENOLOGICAL CALENDARS IN THE UNITED STATES OF AMERICA, THROUGHOUT
THE LONGITUDE 84ºW
Dinesh Shrestha
Sujan Parajuli
Department of Geography,
South Dakota State University.
Abstract. – I am interested in verifying the Hopkin’s Law, that the time of occurrence of a given periodical
event in life activity in temperate North America is at the general average rate of 4 days for each 1 degree of latitude
5 latitude, degrees of longitude, 400 feet of altitude, later northward, eastward and upward in the spring and early
summer, and the reverse in late summer and autumn. The study area lies in the central USA, the latitude ranging from
29°N to 46°N along the longitude 84°W, and covers five states, running south to north: Georgia, Tennessee,
Kentucky, Ohio, and Michigan. The study focuses on seasonal patterns from south to north of the United State,
comparing these patterns with the Hopkins Law of Phenology. This comparison, and the study’s results were not as
predicted: the seasonal patterns often did not follow the law but tended to follow the law at the point of latitude and
longitude aforementioned. The NDVI imagery obtained from WELD Distribution System was analyzed in ENVI 4.8
software to determine the First Day of Spring (FSD). The FSD was then plotted against the latitude which gave a
curve that best fits Hopkin’s Law. The obtained result shows the FSD at the southern-most latitude (29°N) on the 97th
day of the year, and the northern-most latitude (46°N) is on the 129th
day of the year which is close to Hopkin’s Law
(85%).
Keywords: NDVI, Day of Year, Hopkin’s Law of Phenology, Weld Distribution System, First Day of Spring, curve
fitting, longitude and latitude.
1. PHENOLOGICAL CALENDARS IN THE UNITED STATES OFAMERICA
A season is a division of the year, marked by changes in weather, ecology and hours of
daylight (Bennet, 2012). Seasons result from the yearly orbit of the Earth around the Sun and the
tilt of the Earth's rotational axis relative to the plane of the orbit. During May, June, and July, the
northern hemisphere is exposed to more direct sunlight because the hemisphere faces the sun. The
same is true of the southern hemisphere in November, December, and January. It is the tilt of the
Earth that causes the Sun to be higher in the sky during the summer months which increases the
solar flux. However, due to seasonal lag, June, July, and August are the hottest months in the
northern hemisphere and December, January, and February is the hottest months in the southern
hemisphere. Moreover, different human activities and natural disturbances have effects in the
dynamics of ecosystems causing the alteration in the phenological process. For instance, the
modification of vegetation cover, with a predominant clearing of natural vegetation, may have a
long-term impact on sustainable food production, freshwater and forest resources, the climate and
human welfare (Foley et al., 2007).
1.1 Phenology
Phenology is a branch of science dealing with the relations between climate and the timing
of biological phenomena, such as bud burst, leaf out and plant flowering. It is the study of the
timing of recurring biological phases, the causes of their timing with regard to biotic and abiotic
forces, and the interrelation of phases of the same or different species.” Many phenological

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relations exist in folklore. Jackson et al. (2001) cite the aphorism, “seed should be sown when oak
leaves are as big as sow’s ears”, for example.
1.2 Hopkins’s Law of Phenology.
Andrew Delmar Hopkins (1857-1948) was a prominent scholar that has had a lasting
effect on phenology for his contributions, mainly Hopkins Law. Hopkins Law states that, “… the
time of occurrence of a given periodical event in life activity in temperate North America is at the
general average rate of 4 days for each 1 degree of latitude 5 latitude, degrees of longitude, 400
feet of altitude, later northward, eastward and upward in the spring and early summer, and the
reverse in late summer and autumn”. This means that the further west from the Atlantic Coast, the
further north, or higher in the elevation a site, the later spring arrives.
Phenological metrics describe the phenology of vegetation growth as observed by satellite
imagery. Some standard metrics derived are Onset of greening, Onset of senescence, Timing of
Maximum of the Growing Season, Growing season length, and so forth. In Figure 1, Greenup
Onset is the beginning of measurable photosynthesis. The start of spring is the period when
photosynthesis rate is rapid. The plants are growing green and getting more green leaves. The
Greenup Phase is the duration of photosynthetic activity, or say the time frame from the start of
the season until the maturity onset. During maturity phase, the plants are fully grown and have
maximum leaves and maximum photosynthesis process takes place. The phase, when the plants
begin to shade off the leaves and the photosynthesis rate, is seized is Senescence Onset and
Senescent Phase. During dormancy onset and dormant phase the photosynthesis rate are minimum
and plants exhibit minimum greenness. This is because of shading of the leaves.
Figure 1: The schematic diagram of the Greenup Onset, Greenup Phase, Maturity Onset and Maturity Phase,
Senescence Onset and Senescent Phase, Dormancy Onset and Dormant Phase.
1.3 Data Collection
Select the region in US Consortium and order the NDVI, Day of Year and Cloud Data
from WELD website 2011 CONSUS Spring image for 11 points along the longitude 84ºW. Each
line of the 11 locations order the monthly WELD 2011 data for ~1X1º area. Then, the data were
downloaded. The downloaded data should be organized into separate folders. Data were renamed

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per the name of the months, if necessary. Data were saved in a separate folder. For each ordered
~1X1º area, a single pixel 30m vegetation/soil pixel that for the months January, February through
July is predominantly NOT water, cloud, snow, urban, or missing was located. Then, the WELD
NDVI and Day of Year pixel values for each month (Jan to July) were collected. The above 2
steps were repeated for all 11 areas. The prime objective is to define the first day of spring from
monthly NDVI and Day of Years values. The study area lies in the central USA, the latitude
ranging from 29°N to 46°N along the longitude 84°W, and covers five states, running south to
north: Georgia, Tennessee, Kentucky, Ohio, and Michigan (Figure 2). The study focuses on
seasonal patterns from south to north, comparing these patterns with the Hopkins Law of
Phenology.
Figure 2: Study Area: the latitude ranging from 29°N to 46°N along the longitude 84°W
2. METHODOLOGY.
The entire project involves three major steps as shown in figure 3: (1) Image interpretation
and data analysis using ENVI software, (2) Analysis of the output and comparison with the
Hopkin’s Law, and, (3) Presentation of the output and preparing the map in Arc GIS. All three
steps require the skilled manpower with adequate knowledge in ENVI software, Google Earth,
Arc GIS, Excel, and Mathematics (Regression and RMSE). The project would also demand the
adequate knowledge in Phenology and Hopkin’s Law.
Study Area
Longitude 84°W

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Figure 3: Flow diagram for the Image interpretation and data analysis using ENVI software.
2.1 Image interpretation and data analysis in ENVI software.
The first step was to locate the working area and download the data, and then work with
ENVI software to process the image. Image interpretation and data analysis using ENVI software
are categorized into four steps. The first step is to order and download the data. For this, I selected
the region in US Consortium and ordered the NDVI, Day of Year and Cloud Data from WELD
website 2011 CONSUS Spring image for 11 points along the longitude 84ºW. After I downloaded
the data, the second step was Pre-processing the data I organized and renamed the data into
separate folders, and saved them in a separate folder. The third step was processing, where I
extracted the WELD NDVI and Day of Year pixel values for each month (Jan to July) for all 11
areas. The fourth step was Post-processing where the data was plotted in the graph (X-axis
showing the Day of year and Y-axis showing the NDVI). Table 1 and Figure 4 give the graph
showing the FSD for the point P8 (Clermont, Georgia). The FSD is DAY 105 i.e. April 15th, in
this location. The method adopted is NDVIratio = (NDVI-NDVImin) / (NDVImax-NDVImin.
The First Day of Spring (FSD) was then determined and a final graph showing, X- axis as the
derived First Day of Spring (FSD) and Y- Axis as the latitude of the area was plotted (Table 2 and
Figure 5).
Table 1: Determining the FSD for the point P8 (Clermont, Georgia). The FSD is determined to be DAY 105 i.e. April 15th,
in this location.
Point ID Latitude Longitude
Land
cover
/use
Location
description DOY NDVI
NDVI
ratio
First
Day of
Spring
P8 38.9 -84.1 Forest
Clermont,
Georgia
30 0.10 0.068
105
46 0.10 0.143
85 0.32 0.328
117 0.44 0.479
149 0.74 0.630
181 0.77 0.781
190 0.77 0.824

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Figure 4: The graph showing the FSD for the point P8 (Clermont, Georgia). The FSD is determined to be DAY 105 i.e.
April 15th, in this location. The method adopted is NDVIratio = (NDVI-NDVImin) / (NDVImax-NDVImin. The
polynomial curve fitting gives R² = 0.9
Table 2: Summary of the Day of Year obtained from NDVI ratio for all 11 locations along with its land cover type, in the
central USA, the latitude ranging from 29°N to 46°N along the longitude 84°W.
Point ID Latitude Longitude
Land
cover/use Location description
DOY by (NDVI
Ratio)
P1 30.54 -84.43 Forest Seminole, Georgia 87
P2 31.55 84.44 Forest Stewart, Georgia 62
P3 32.80 -84.76 Forest Harris, Georgia 110
P4 33.20 -84.14 Forest Lamar, Georgia 78
P5 34.40 -84.15 Forest Fulton, Georgia 89
P6 35.90 -84.69 Forest Cumberland, Tennessee 100
P7 36.88 -84.65 Forest Wayne, Kentucky 80
P8 38.88 -84.14 Forest Clermont, Georgia 105
P9 41.55 -84.14 Forest Preble, Ohio 140
P10 42.35 -84.79 Forest Mercer, Ohio 148
P11 43.89 -84.54 Forest Jackson, Michigan 125
y = -6E-06x2 + 0.0083x - 0.3061
R² = 0.9666
0
0.2
0.4
0.6
0.8
1
1.2
0 20 40 60 80 100 120 140 160 180 200
NDVIratio
Day of Year
Determining the FSD at Clermont, Ohio
FSD

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Figure 5: The graph showing the FSD for the point P8 (Clermont, Georgia). The FSD is determined to be DAY 105 i.e.
April 15th, in this location. The method adopted is NDVIratio = (NDVI-NDVImin) / (NDVImax-NDVImin. The
polynomial curve fitting gives R² = 0.9
2.2 Data Analysis
There are different methods adopted to study Phenology. A diversity of satellite measures
and methods has been developed. These methods can be divided into four main categories: (1)
NDVI Thresholding (absolute or relative values), (2) Derivatives (temporal derivatives i.e. slopes),
(3) Smoothing Algorithms, (4) Model fit (NDVI curve fitting- regression, splines). I adopted first
two methods.
2.2.1 NDVI Thresholding (absolute or relative values)
The NDVI is an index used to identify vegetation and its health through the Levels of
chlorophyll detected in the leaves. NDVI is calculated from the visible and near-infrared light
reflected by vegetation. Healthy vegetation absorbs most of the incoming visible light, and
reflects a large portion (about 25%) of the near infra-red (NIR) light, but a low portion in the red
band (RED). Unhealthy or sparse vegetation reflects more visible light and less NIR light. To
apply the NDVI, the following formula is used: NDVI = (NIR – RED) / (NIR + RED).
The NDVI is a common and widely used transformation for the enhancement of
vegetation information. It is used to measure vegetation cover characteristics and incorporated
into many forest assessment studies. It can be used for an accurate description of land cover,
vegetation classification and vegetation phenology. In some cases, multi-resolution imagery and
integrated analysis method were included along with NDVI for land cover classification.
Temporal dynamics of the NDVI or adding an NDVI image with the multispectral image is also
useful in differentiating the vegetation. The classification accomplishing of NDVI using GIS
software depend on remotely sensed satellite data as the primary information source. However,
y = 5.4865x - 94.575
R² = 0.5963
y = 2.8647x - 4.6759
R² = 0.9769
50
70
90
110
130
150
30 32 34 36 38 40 42
Dayoftheyear
Latitude (°N)
Relationship between FSD and the Latitude
DOY obtained from NDVI ratio Hopkin's Law
Linear (DOY obtained from NDVI ratio) Linear (Hopkin's Law)

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the WELD gives us the NDVI processed image so that we should not spend much time in NDVI
calculations.
The threshold selection is commonly based on a normal distribution characterized by its
mean and its standard deviation threshold values are scene-dependent; they should be calculated
dynamically based on the image content. However, the thresholds can be determined by three
approaches: (1) interactive, (2) statistical and (3) supervised. In the first approach, thresholds are
interactively determined visual tests. The second approach is based on statistical measures from
the histogram of techniques for selecting appropriate thresholds are based on the modeling of the
signal and noise, which is carried out in this study (Brink, A. D., and N. E. Pendock, 1996). Third,
the supervised approach derives thresholds based on a training set of change and no-change
pixels.
In order to work with the phenology, the NDVI ratio is more accurate and widely practiced
method. In this approach, first, we need to translate NDVI to a ratio based on the annual minimum
and maximum as [NDVIratio = (NDVI -NDVImin) / (NDVImax - NDVImin)] and analyze the
Greenup Onset for the particular pixel value (White et al., 1999). NDVI thresholding is the simplest
method to determine FSD and LSD. The threshold is arbitrarily set at a certain level (e.g. 0.09, 0.17,
0.3, and so forth). However, across the conterminous US, NDVI threshold can vary from 0.08 to
0.40. Thus, it is inconsistent when applied towards large areas. 50% is the most often used threshold.
The increase in greenness is believed to be most rapid at this threshold. Some believe that rapid
growth is more important than first leaf occurrence or bud burst. Lower likelihood of soil –
vegetation confusion than at lower thresholds.
2.2.2 Derivative is calculated based on 3 composites.
We employ the recursive least squares procedure described by Hermance (in press) to
create an average annual phenology by simultaneously fitting non-orthogonal low order
polynomial and harmonic components while minimizing model roughness. The polynomial,
typically 4th order, fits any instrument drift or long-term trends during the observation period,
while the harmonic components, typically a 6th order series, fit the average phenology of the data.
Harmonic analysis (specifically using Fourier series) has been shown to produce an accurate
representation of a single year phenology across a range of land cover (Jakubauskas et al., 2001,
Jakubauskas et al., 2002, Moody &Johnson, 2001). Here, we found that 6 harmonic components
(periods of 1 year, 6 months, 4 months, 3 months, 2.4 months and 2 months) were sufficient to
capture the variety of phenologies tested (e.g., differences in length of growing season and
steepness of onset of greenness). However, the lack of dataset bound me to use the second order
polynomial. NDVI ratio was then calculated: NDVIratio = (NDVI-NDVImin) / (NDVImax-
NDVImin). The obtained curve was fitted with a second-degree polynomial curve: f(x) =
ax2+bx+c. The FDS was determined considering 2 things: a) the point at which the fitted curve
reached 50% NDVI threshold upward, or b) the point where NDVI value abruptly increased.

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3. RESULTS
The graph partly confirms Hopkin’s Law of Phenology. The Hopkin’s law states that there
is a 1-day delay for every 15 minutes of latitude upwards. Figure 7 shows the map showing the
locations of all 11 points along with its land cover type, in the central USA, the latitude ranging
from 29°N to 46°N along the longitude 84°W. There was a delay but not necessarily by 1 day in
every 15 minutes. The number of days the FSD is being delayed is not consistent and does not
always fit the law. There is a delay in the starting of spring as a function of latitude. The fitted
line exhibits a correlation of is .596 i.e. 59.6% which can be considered acceptable. Some
discrepancies observed are an underestimation at the first 36°N and at 39°N.
3.1 Discrepancies
3.1.1 Possible natural source of discrepancies include:
The local starting of the agricultural season which may be linked to the local
microclimate, such as crop types are unknown, the forest types were different (figure 6), and most
of the land in the northern part was covered by snow for Jan, Feb and March months. This
affected the NDVI. Some other factors may include local disturbance such as fire.
Figure 6: Figure showing one of the natural discrepancies at Point 5, Fulton, Georgia. The land cover is forest,
but seems to be a sparse forest with bushes and small trees.
3.1.1 Possible technical sources of discrepancies:
The possible technical sources of discrepancies are as follows: the latitude 37N did not have data
for July month, cloud cover that may impact the image processing, inconsistencies in pixel
sampling (i.e. not evenly spaced northward or different distances from the longitude line to avoid
missing data, and the polynomial curve fitted does not always work perfectly.

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Figure 7: The Map showing the locations of all 11 points along with its land cover type, in the central USA, the latitude
ranging from 29°N to 46°N along the longitude 84°W.

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4. CONCLUSION
Hopkin’s law is considered to be the prime law to define the season pattern in the North
America (White et al., 1999). Since the terrain, elevation and the land cover is not uniform
throughout the continent, Hopkin’s Law does not always work. For vegetation mapping, the
NDVI approach has been widely utilized in various sensors. However, the different characteristic
of sensors would affect the classification results. Furthermore, the threshold selection is difficult
for extracting vegetation information from various scenes. In this work, I have proposed an
approach with a fixed-threshold scheme (NDVIratio) and a correlation approach, which can be
suitable for WELD imagery and is illustrated with more accurate results than NDVI. My project
results with WELD images partly confirms the theoretical inference of the proposed approach. In
future work, I plan to extend the proposed approach to other high-resolution satellite images, e.g.
QuickBird images, and to find a solid theoretical support for the fixed threshold scheme.
5. REFERENCES
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Brink, A. D., and N. E. Pendock. Minimum cross-entropy threshold selection. Pattern
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Foley, J. A., Ramankutty, N., Leff, B., and Gibbs, H. K. (2007) Global land use changes. In M. D.
King, C. L. Parkinson, K. C. Partington, & R. G. Williams (eds.), Our changing planet: The
view from space. New York: Cambridge University Press, pp. 262-265.
Jackson, R.B., Lechowicz, M.J., Li, X. and Mooney, H.A. 2001. Phenology, growth and allocation
in global terrestrial productivity. In: Terrestrial Global Productivity. Eds Roy et al.
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White, M.A., S.W. Running and P.E. Thorton. 1999. The impact of growing season length
variability on carbon assimilation and evapotranspiration over 88 years in the eastern US
deciduous forest. International Journal of Biometeorology. 42: 139-145.
Hopkins AD (1918) Periodical events and natural laws as guides to agricultural research and
practice. US Dept Agric. Monthly Weather Review, Supplement 9
Jakubauskas et al., 2001 M.E. Jakubauskas, D.R. Legates, J.H. Kastens Harmonic analysis of
time-series AVHRR NDVI data Photogrammetric Engineering and Remote Sensing, 67
(2001), pp. 461–470
Jakubauskas et al., 2002 M.E. Jakubauskas, D.R. Legates, J.H. Kastens Crop identification using
harmonic analysis of time-series AVHRR NDVI data Computers and Electronics in
Agriculture, 37 (2002), pp. 127–139 Moody and Strahler, 1994 A.
Moody, A.H. Strahler Characteristics of composited AVHRR data and problems in their
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Disclosure: “The author(s) declare(s) that there is no conflict of interest regarding the publication of
this manuscript.”