The method proposed herein is based on electronic devices with a built-in clock which maintains a relatively accurate account of time and date. Digital clocks are built into most electronic hardware and can run for many years on one single battery with limited drift. It is therefore assumed that at any time the object can enquire the current time t and date d. It is assumed that the time is set to Universal Time (UTC) which makes the calculations presented herein simple and is represented in decimal form in the range from 0 to 24. Issues such as daylight saving time are thus avoided. Next, the date d is represented as the day of the year, where 1 January is day 1, etc.

Next, the strategy requires that the device has some form of light sensor that is capable of measuring the lighting condition e. The simplest possible device is a light intensity sensor which gives a voltage directly proportional to the incident light intensity. This type of sensors consume no electrical energy since they actually change the light energy into electrical energy without any power source. Some other devices could be used, for instance a simple and low cost exposure value meter such as those built into most low-cost digital cameras [17–19] or it could be a low cost camera (CCD-chip). In the cases of exposure values (EV) then e is a real value between 0 to approximately 20 [20,21]. If a camera is used a simple representation of exposure can be obtained simply using: (1) e = 1 XY ∑ x = 1 X ∑ y = 1 Y I x , ywhere I_x,y is the pixel intensity for the pixel located at (x, y) in the image and X and Y are the numbers of image pixels in columns and rows correspondingly. To reduce computation a small subset of these points can be sampled throughout the image using some two dimensional sampling pattern.

The system proposed in this paper is designed to be as simple as possible. Therefore, apart from the light sensor, a microprocessor is required in order to analyze the data given by the sensor. This element consumes most of the energy in the system.

A feasible option for implementing the proposed system is to use mote technology [22]. These motes are small, low-cost electronic devices which includes sensor support, small microprocessors, wireless and serial communications. This kind of technology is being used during the last years in sensor networks. Furthermore, some of these motes are designed to be energy-efficient and their energy consumption allows to use these devices with single batteries for a long time.

The SGPS operation is summarized in the flowchart presented in Figure 1. Also, an Arduino-based implementation is suggested in Figure 2. The system can be designed in a more complex way, including other data measuring devices or using a more powerful microprocessor, but the system proposed herein is focused in a simple measuring strategy, with low-cost, energy-efficient components.

This paper presents a new algorithm to localize outdoor objects using only sunlight intensity data for a given location. With this information, it is possible to obtain the sunrise and sunset times for that location. Only with these two values, the celestial model determines the coordinates (both longitude and latitude). Thus, the coordinates can be expressed as a function of the sunrise and sunset times as shown in Figure 3.

There is an accurate, well-known algorithm based on Julian Day which allows to find out the sunrise and sunset times for a specific place by only knowing the coordinates of that place and also the date [23]. However, it is not possible to obtain the inverse of the algorithm since some formulas of this algorithm depend on both latitude and longitude, and when they are tried to be separated the result is that the latitude depends on the longitude and vice-versa.

Then, the method described herein can be reduced to the problem of identifying the sunset and sunrise times for a given day. Given an accurate measurement of the sunrise time t_sunrise and sunset time t_sunset, the solar noon t_midday is simply: (2) t midday = t sunrise + t sunset 2

The UTC representation of time assumes that sunrise occurs before sunset. If the sunset occurs before the sunrise within the 24 h UTC time window then it means that it is a fractional day, since the sunrise happens the day before (in UTC format) that sunset. This has to be taken into account when applying this equation, since if the equation is applied directly with a sunset time earlier than the sunrise time, the noon time will not be correct (in fact, the result would be the midnight time). Section 3.1 details how to overcome this issue when applying Equation (2).

As stated in Section 2.1 the times have to be expressed in UTC and in decimal format (from 0 to 24). For the next formulas all the times will be expressed in this format and the angles in radians. The angular sunset can be computed as follows: (3) a sunset = π 12 | t sunset − t midday |and the declination of the sun δ can be approximated by the following Fourier series [24]: (4) δ = 0.006918 − 0.399912   cos ( β ) + 0.070257   sin ( β ) − 0.006758   cos ( 2 β ) + 0.000907   sin ( 2 β ) − 0.002697   cos ( 3 β ) + 0.00148   sin ( 3 β )where β is the fractional year expressed in radians given by: (5) β = 2 π 365   dand d is the day from 1 to 365. In the case of a leap-year, where d can be 366 the value for β is close to 2π, which gives the same result in the equation for δ as when d is 1 and β is equal to 2π/365.

Finally, the coordinates can be obtained. For the longitude λ (in radians), where positive values represent east and negative values represent west, the next equation is employed: (6) λ = 2 π   12 − t midday 24and the latitude φ (in radians) of the objects location can be found by numerically solving for latitude using the following equation, where positive values represent north and negative values represent south: (7) cos ( a sunset ) = sin ( − 0.0145 ) − sin ( δ )   sin ( φ ) cos ( δ )   cos ( φ )The equations provided in this sections allows to find out the coordinates for outdoor objects by means of the times of the sunrise and the sunset for a given day.

The system is designed to work autonomously, being able to localize itself within the first 24 h of being switched on. This implies that the system can determine its coordinates without any previous information besides the time.

Given a sufficient supply of electric power, for instance if the object has a steady power supply, the initialization can be simply performed using brute force by continuously sampling the lighting condition. If the light sensor is sampled at r samples per second, then this is the same as M samples for each 24 h cycle, given by: M = 24 × 60 × 60 × r

The accuracy of the measurements will therefore be in the range of: a = 360 M

It will take maximum 24 h to identify the location and the effort involved is defined by E = M × p, where p is the energy consumed to perform each sample and E is the total energy. This is the optimal solution in terms of speed and accuracy. However, for a power constrained device the strategy is unrealistic as a very simple device could run out of power after just a few hours or sooner, although the latest technologies released are improving the energy consumption and some devices can run for months with standard off-the-shelf AA batteries.

Several theoretical enhancements are possible. Sunrises can be predicted if early measurements are taken. This is because the light intensity increases gradually over some time interval before one passes the threshold. If a light intensity measurement is taken that is above the night baseline value but yet below the threshold then this is a sign that a sunrise is approaching soon and the sample rate can be dynamically increased. This is illustrated by Figure 4(a) which shows an authentic intensity plot obtained using a webcam. Clearly the intensity rises for about 20 min before the sun breaks.

However, it may be more difficult to get a pre-warning of a sunset in this way as this will suddenly drop below the threshold value. One way to overcome this is to observe additional features. If the light sensor is capable of capturing color spectrum information then additional information can be exploited to better predict sunrises and sunsets. For instance, the CCD sensors in digital cameras are capable of detecting color as well as intensity. This is because sunsets and sunrises often are characterized by large changes in hue which affect entire scenes. Such changes in hue occur prior to sunsets and thus a detection in hue change can be used to predict an upcoming sunset. This is illustrated in Figure 4(a) which presents a 24 h plot obtained using a webcam. The hue h(r, g, b) of each pixel is computed from the r, g and b (red, green and blue) components as follows: (8) h ( r , g , b ) = atan 2   ( 2 r − g − b ,   3 ( g − b ) )

The steady line shows the overall image intensity and the other line illustrates overall image hue. Clearly, the hue is changing dramatically just before the sunrise and sunset. In fact, these can be seen as the two peaks in the hue plot. However, the exploitation of such features is a topic of future research.

Section 2.2 mentioned that fractional days require special treatment for the algorithm to work. We define as fractional days those days where the sunset occurs before the sunset within the 24 h UTC time window.

For example, imagine that there is a day in which the sunrise occurs at 22 h and the sunset occurs at 4 h into the following day. Then, applying the algorithm midday is reported to occur at 13 h. This make no sense as midday occur at night. To avoid complex modifications of the algorithm the following strategy is employed: Midday of a fractional day within 24 h window is equal to the midday time calculated by the standard t_midday Equation (1) using the sunrise of the previous day adding 12 h and obtaining the module of this number by 24. Or, more precisely: (11) t md ′ ( d ) = mod ( t md , frac ( d ) + 12 ,   24 )where (12) t md , frac ( d ) = SR ( d − 1 ) + SS ( d ) 2

Applying this strategy to the previous example, in which the standard t_midday equation gives a result of 13 h UTC, the new result yields a midday at 25 h UTC (13+12) or 1 h UTC of the same day (since 25 h UTC refers to d − 1). The correct midday time is thus obtained. Figure 7 depicts the effect of this modification.

This strategy allows the coordinates having the sunset of one day and the sunrise of the next day to be found on hardware platforms where the UTC days are treated independently. A small error is introduced, since the times of two different days are used. This error is practically insignificant since the difference of a sunrise (or sunset) times of two consecutive days is only a few minutes and by itself lower than the error of the light sensors used to detect sunrise or sunset times.

The problem of accurately identifying the sunrise and sunset times is difficult using only light intensity data. Comparing the measured data with the times given by the NOAA celestial model, it can be assumed that when the data is close to 0 W/m² a transition occurs, that is, a transition from day to night, or vice versa. Using such a threshold to identify the sunrise and sunset times is a very simple solution, and its accuracy depends on the atmospheric conditions for that day since a cloudy day the light intensity value will be lower at the same hour than for a sunny day. However, due to the measurement system offset, setting a threshold of 0 gives accurate sunrise and sunset measurements. Of course, this threshold depends on the hardware employed and should be calibrated.

However, due to signal noise, using a simple zero crossing condition is not enough, as it may result in many transitions per day. For days with more than two transitions, the mean time of the two transitions closest to those of neighbouring days are used (see Figure 8).

The strategy for handling fractional days relies on information about the previous day. When this information is not available the sunrise time for the current day is used instead.

The main steps of the approach can be summarised as follows:

The celestial model used is described in Section 2.2.

Figure 9 shows the error histograms for latitude and longitude using the equation of time correction. They show the absolute error in percentage in the X axis and in the Y axis the number of measures in which the error is given by the X axis. The error percentage for the latitude is: e latitude % = e latitude 180   100Similarly, for the longitude: e longitude % = e longitude 360   100

The interpretation of these histograms is that most days only have a small error. For instance, more than 10.000 days (almost 35% of the total number of tested days) had less than 1% error in longitude. The red line shows the accumulated percentage. The errors for longitude is less than those for latitude.

Figure 10 shows the same histograms as in Figure 9 but with kilometers as unit converted using 1°= 111.12 kilometers in latitude and 1°= 111.12 cos φ kilometers in longitude. Moreover, Figure 11 shows a dispersion plot with the error obtained for each day. The results obtained prove that the algorithm works well using only sunlight intensity data. However the error in kilometers is high for most common applications. The dispersion plot (Figure 11) shows that most points are close to the origin. The longitude accuracy is high while the latitude accuracy is somewhat worse, though acceptable. Thus, the dispersion plot is concentrated in a vertical line due to the equinoxes.

Next, Figure 12 shows results from days where the sunlight signal is noisy and where there were more than two transitions per day. Here, the points are concentrated around zero error and in a vertical line. However, in this case the accuracy is not as high as in the previous examples.

Figure 13 also reveals how the error percentage for a given value (manually chosen) varies among different stations as a function of the elevation over the sea level. This image also includes a linear regression plot. Although there too few stations to reliably assess the tendency, it is possible to assert that the higher the station the lower the error. If a station is higher, it will see the sunrise earlier (or the sunset later). This can be translated in terms of decreasing the threshold values (Section 4.1) in function of the height of the sensor over the sea level. This has the same effect as the height: anticipating the sunrise or delaying the sunset. But this parameters depends on the light sensor employed and it also requires a height sensor.

Finally, Figure 14 shows the error of the algorithm for 2010 data for one of the stations. It is interesting to compare this graph with Figure 6. It is possible to see how the equinoxes increases the error enormously in the latitude. On the other hand, the longitude error points out that the error of the system is higher than the error introduced because of the Spencer EqT. Thus, the critical part of the algorithm is to identify accurately the sunrise and sunset times.