Mission Specific Solar Radiation Environment Model (MSSREM): Peak Flux Model

Abstract

Coronal mass ejections and solar flares can accelerate high fluxes of energetic particles. Depending on where this solar activity occurs on the sun, these outward moving particles can reach the Earth and enter the Earth's magnetosphere. They can also strike manmade objects in space. If the electronics in space are not protected from these energetic particles, they can cause the spacecraft to reboot, go into “safe mode,” have other anomalies, or cause catastrophic damage and loss of the mission. To protect the mission, the user can employ one or more mitigation strategies. The user may choose to add shielding, choose parts less prone to radiation effects, and/or mitigate by design. Implementing any of these strategies adds cost to the mission, so it is important to frame the design for the purpose of survival in a reference environment, which is severe enough to provide the desired confidence of mission success, but not more. For this reason, models have been developed that construct a design reference environment tailored to a specific mission. In this paper, the Mission Specific Solar Radiation Environment Model (MSSREM) peak flux model will be discussed. MSSREM uses probabilistic modeling techniques to build a design reference environment that can be tailored to a user specified mission start date, mission duration, and confidence level. The model can be run for any space mission outside the Earth's magnetic field and 1 AU from the sun during the years 1953–2055.

Key Points

• Probabilistic modeling of solar energetic particle episodes
• Design reference environments that are tailored to a specific space mission
• Modeling peak fluxes for missions shorter than one half of a year

1 Introduction

One of the first models to predict the solar energetic particle environment was the King model (King, 1974). The King model gave a proton fluence level that would not be exceeded during Solar Cycle 20 at a given confidence level. In 1990, the Jet Propulsion Laboratory (JPL) model was developed to look at event-integrated fluences above two different energy thresholds (Feynman et al., 1990). In 2002, Feynman et al. verified that the JPL model was still valid and in agreement with the newer models (Feynman et al., 2002). The Cosmic Ray Effects on Micro-Electronics, Version 1996 (CREME96) used the October 1989 solar energetic particle event to produce a worst-case week average, worst-case day average, and peak instantaneous flux environments (Tylka et al., 1997). In 1999, Xapsos et al. developed a peak proton flux model using extreme value theory, called the Emission of Solar Protons (ESP) model (Xapsos et al., 1999). This model predicted the peak proton flux for a mission by using data from 1974 to 1994. The ESP model used a truncated power law to model the extreme fluxes. Nymmik (1999) developed a model that produces a probability for ≥10 MeV/nucleon fluences and peak fluxes. This model was able to provide probabilities for elements Z = 1–28. The Solar Accumulated and Peak Proton and Heavy Ion Radiation Environment (SAPPHIRE) model was developed to look at quite a few different things including peak flux (Jiggens, Heynderickx, et al., 2018; Jiggens, Varotsou, et al., 2018). The peak flux model used the Lévy distribution instead of the Poisson distribution (Jiggens et al., 2012). The virtual enhancements−solar proton event radiation (VESPER) model produces virtual time series of proton differential fluxes to create reference environments for the user's mission. In this sense, the VESPER model is fundamentally different than the other solar proton event (SPE) models (Aminalragia-Giamini et al., 2018). Both the SAPPHIRE and VESPER models use an exponential cut-off power law to model the extreme fluxes.

While all these models are useful by themselves, a lot of them are also integrated into toolkits for space environment and effects calculations. Toolkits like CREME96, SPace ENVironment Information System (SPENVIS), and Outil de Modélisation de l'Environnement Radiatif Externe (OMERE) allow users to calculate a space environment and use the resulting environment to calculate the dose on electronics, spacecraft charging, etc. (Heynderickx et al., 2000; Peyrard et al., 2004; Tylka et al., 1997). Recently, Fifth Gait Technologies, Inc., has been working on an updated version of CREME96 called the Space Ionizing Radiation Environment and Effects (SIRE2) toolkit (Adams et al., 2020). One of the main areas that needed to be improved from the CREME96 toolkit was the solar energetic particle (SEP) model. Instead of relying solely on the previous worst-case fluxes offered in CREME96, the Fifth Gait team wanted to provide the user the ability to select their desired confidence level when constructing the SEP environment. While existing models were evaluated for inclusion into the SIRE2 toolkit, the Fifth Gait team decided to develop the Mission Specific Solar Radiation Environment Model (MSSREM) peak flux model.

The MSSREM peak flux model is a new model that produces design reference environments that can be tailored to the user's mission. The user specifies the mission start date, the mission duration, and the desired confidence level, and MSSREM will produce design reference environments including all the elements from hydrogen to uranium. The energy range for each spectrum is from 1.0 MeV to 100 GeV (using the same energy channels as used in CREME96). MSSREM can be run for missions outside the Earth's magnetic field and 1 AU from the sun during the years 1953–2055. Included in the MSSREM peak flux model is a new data driven approach for short missions that uses the sunspot number over phase of the solar cycle to create unique reference environments for missions as short as 5 min.

Used in conjunction with a geomagnetic cutoff transmission function model, it can be applied to any mission within the Earth's magnetosphere and above the atmosphere. This allows users to create design reference environments of the solar radiation component of the flux for their potential mission that would provide the user a baseline for designing their spacecraft. MSSREM focuses on episodes of solar activity that may consist of multiple particle events that overlap in time. Since these episodes do not usually come from the same active regions on the sun, they tend to be uncorrelated. Therefore, their temporal distribution follows Poisson statistics.

In this paper, the particle databases used in the MSSREM peak flux model are explained. Then, in-depth explanations of the probabilistic modeling approaches used in MSSREM are given. This includes the long, short, and intermediate mission length approaches used for six elements, and the scaling approach used to calculate the reference environmental spectra for the other elements. Finally, some results from the MSSREM peak flux model are shown and compared to other current models.

2 Particle Databases

The probabilistic modeling techniques used in MSSREM require a large database of particle fluxes for each element. These databases were constructed by fusing data from different satellites. The proton database spans the years 1986–2016 and comes from the Geostationary Operational Environmental Satellite (GOES) series. The data were cleaned and the multiple satellites were normalized to one another in the same manor to the process described in Robinson et al. (2018). The authors decided to use the GOES data for the years 1986–2001 instead of the Goddard Medium Energy (GME) experiment (as was used in Robinson et al., 2018) because GOES is still operational and future years could be more easily added to the databases. It is important to note here that GOES-6 has slightly different energy channels than the later satellites. However, the differences are small and insignificant for this study. The GOES energy channels used in this work are the same ones presented in Table 2 of Robinson et al. (2018). There has been recent work by Sandberg et al. (2014) to redefine the GOES channels due to the broadness of the energy channels and the difficulty to resolve the mean effective energies. The authors decided to use the energy channels specified by the instrument team. The process used to identify the new mean energies for each channel in the MSSREM peak flux model will be addressed in the next section.

The use of the GOES data instead of the GME data originally published by Xapsos and Stauffer (2005) for the years 1986–2001 did have some minor unintended side effects. Since the GME data identified episodes using a threshold level in two different energy channels, the start times of the episodes are slightly different between the two data sets. In addition, GME had a much lower background across all energy channel. Finally, the number of episodes in each year differed slightly between the two instruments, with GOES usually having slightly more. This can be attributed to the differences in the background and the episode identification method.

Episodes in the proton data set were identified by eye as first done in Robinson et al. (2018). The data were examined for periods of time when the particle flux rose above the residual background. If just one or two measurements rose above background or returned to background, this did not signal a start or end to an episode. While automating the episode identification process would result in episodes that could be reproduced more easily, the authors decided against the automation to avoid missing smaller episodes. To see how the smaller episodes are missed, the Robinson et al. (2018) work used the same episodes as in this work for the years 2002–2015. Comparing the number of episodes during this period from Robinson et al. (2018) to the National Oceanic and Atmospheric Administration (NOAA) (https://umbra.nascom.nasa.gov/SEP/) and Solar Energetic Particle Environment Model (SEPEM) (http://www.sepem.eu/help/event_ref.html) event lists found online, there are approximately double the number of episodes in Robinson et al. compared to NOAA's or SEPEM's lists. The additional episodes included in Robinson et al. (2018) are smaller than the others identified but help to complete the lower end of the distributions.

The lowest usable energy channel in the GOES proton data set is P2. It was used to define the temporal extent of an episode. In the higher energy channels, if flux returned to background and rose again, these additional rises were considered to be part of the same episode if the flux in channel P2 remained above background. The authors decided to visually identify the episode rather than use a threshold level so that the cumulative distributions contain all the flux measurements above background rather than the flux above some threshold. In this work, as mentioned above, episodes are assumed to be statistically independent so that time-independent Poisson statistics were used to describe them.

An example of an episode is shown in Figure 1. In this episode, the flux can be seen rising above background around day 187 in channel P2 (left plot). The episode ends around day 221. While there was one measurement that returned to background around day 210, it was taken as a fluctuation in the data and not a signal of the end of the episode. In channel P4 (right plot), the episode returned to background twice, once around day 194 and again between 197 and 199. While the data appear to just reach background on day 194, they return to background again for several successive measurements near day 200 before raising above background another time.

Figure 1

The 4 July 2012 episode is shown above. The time periods highlighted by the red line show where the flux is above background in that channel. On the left, the data are plotted for protons in channel P2. The plot on the right shows the proton channel P4 data. The flux returns to background in channel P4 multiple times, but all the data between day 187 and 221 are considered to be part of the 4 July 2012 episode. This figure first appeared in Robinson et al. (2018).

The helium data come from two different sources. MSSREM uses data from the SEPEM reference data set, version 2.0 and the Solar Isotope Spectrometer (SIS) onboard the Advanced Composition Explorer (ACE). The SEPEM data comes from the GOES satellites along with their precursors, the Synchronous Meteorological Satellite spacecrafts. These satellites enable this data set to span the years 1974–2015. The 5-min data from all these satellites were normalized and rebinned into new energy channels by Crosby et al. (2015). The process of identifying episodes in the SEPEM data was the same as was used for the protons.

The second set of helium data comes from the SIS on board the ACE. The ACE data were taken from the Level 2 data available from http://www.srl.caltech.edu/ACE/ASC/level2/new/intro.html website. The ACE data span the years 1997–2016 in 1-hr average measurements. All elements available from SIS (He, C, N, O, Ne, Na, Mg, Al, Si, S, Ar, Ca, Fe, and Ni) were used in this work and were processed in the following manner. The ACE energy channels can be found at http://www.srl.caltech.edu/ACE/ASC/level2/sis_energy_levels/SIS_ebands.txt website. Since only a few heavy ions were detected during each measurement due to the rarity of some of these elements, Poisson statistics come into play. This means that the background cannot be found as was done for the protons. Instead, the flux measurements were converted into counts. The counts for all the time outside of a proton episode were considered to be at background and averaged together for each year. The average count rate was then subtracted from each measurement in the year to produce background subtracted count rates. The counts were then converted back into a flux. There were a few instances were there average background counts for a year was 0. In these cases, the Badhwar-O'Neill Galactic Cosmic Ray Flux Model was used to find the background cosmic ray flux for the year (O'Neill et al., 2015).

Episode identification in the ACE data had to be handled differently than for the protons and SEPEM data. Since the count rate per measurement is small in the heavy ion data, Poisson Statistics once again come into play. Poisson statistics allow for the background to be higher than the average background that was determined for each year, so measurements above background could just be fluctuations in the counting statistics for that energy channel. To avoid instances where data fluctuations in the background lead to the identification of an active episode, multiple criteria were applied to each channel of data. The combined criteria produce a data set where there was less than one measurement above background that could be statistically attributed to background fluctuations in each channel. All channels had to have a measurement 2.5σ above background to be considered part of an active episode. In channel 1 for each element, the data were searched during all protons episodes as defined by GOES channel P2. In addition, in channel 1, there had to be at least three consecutive measurements above background for an episode to be identified while channels 2–8 had to have at least two consecutive measurements above background. Finally, episodes were identified in channels 2–4 only when there was an episode in channel 1 during the same time while channels 5–8 had to have an active episode in channel 1 and at least one of channels 2–4 above background.

This work uses both the SEPEM and ACE helium data. ACE SIS has a priority system in place that will cause the instrument to focus on the Z > 2 elements over the helium data during large solar storms. This priority system causes there to be very little helium data collected during large solar storms. For this reason, the ACE data is less reliable during large solar storms. Meanwhile, the SEPEM database uses the GOES satellites. These satellites do not have an anticoincidence shield and therefore do not saturate during large solar episodes. However, since the background in the instruments on the GOES satellites is much higher than on the ACE instrument, the smaller episodes are hidden underneath the high background. There are less than half as many SEPEM helium episodes as there are ACE helium episodes. Figure 2 shows the residual background that remains in both instruments after the data has been background subtracted. While the two instruments have different energy channels, there is overlap between the two instruments in ACE channels 2–8 and SEPEM channels 1–6.

Figure 2

The 4 July 2012 episode is plotted above for the helium data sets. The top plot shows the ACE data while the bottom plot shows the SEPEM data. As can be clearly seen, the SEPEM data has a much higher residual background compared to the ACE data.

To normalize the two helium data sets to one another, a similar process as described in Robinson et al. (2018) for protons was used. Periods of time were examined where both sets of data were well above background and the flux was believed to be isotropic. The flux measurements during such periods of time were converted into fluence. The SEPEM fluence spectra were then fit with the Weibull function (Xapsos et al., 2000). The Weibull function fit was then used to find the fluence at the mean energy of each ACE channel. These fluences were then used to create SEPEM to ACE ratio for each ACE channel. These ratios were used to scale the ACE data to the SEPEM data. Figure 2 compares the SEPEM data with the scaled ACE data for the 4 July 2012 episode. The selection criteria for choosing which helium data set to use will be discussed in section 3.

While the ACE SIS data are used for the heavy ion data in this work, there are some drawbacks to using this data set. First, the statistics from heavy ions are low. This can prevent the rarer elements from being observed during an episode. The ACE SIS instrument also has anticoincidence shielding. This means that if the particle storm is large enough, there can be dead times in the observation period around these large episodes. Finally, the SIS instrument only detects particles in a limited energy range. Particles that are outside the energy range detected by the instrument are not measured.

3 Probabilistic Modeling Approaches

For the peak flux model, MSSREM calculates the design reference environments for six elements. These elements are hydrogen (protons), helium, carbon, oxygen, silicon, and iron. Both sets of helium data are probabilistically modeled before MSSREM selects the data set to use going forward. The selection criteria are discussed later in this section. For the remaining 86 elements, MSSREM scales the calculated design reference environments using elemental ratios and heavy ion enhancement factors as described later in this paper. The reference environments for protons will be for 30-min-averaged fluxes, the environments for helium will be for either 1-hr-averaged or 5-min fluxes, while all the environments for the heavier elements will be for 1-hr-averaged fluxes.

For the six elements that are probabilistically modeled, MSSREM uses different approaches for long and short mission lengths in the peak flux model, with both being used for intermediate mission lengths. The long mission approach is used for missions longer than 0.5 years. The short mission approach is only used for missions lasting between a few minutes to a few hours or days, depending on the channel. For missions that do not fit into the short or long mission approach, the intermediate approach is used. Depending on the mission length specified by the user, MSSREM will automatically select the appropriate approach for each channel.

The MSSREM peak flux model can be used for missions between the years 1953 and 2055. The following probabilistic modeling approaches use the sunspot numbers to determine the amount of solar activity that will be encountered in the mission. MSSREM uses the smoothed monthly sunspot number from 1953 to 2018. For the years 2019–2022, MSSREM uses the NOAA sunspot predictions available from https://www.swpc.noaa.gov/products/predicted-sunspot-number-and-radio-flux website. For the years 2023–2055, MSSREM uses an average sunspot cycle from the previous four cycles. MSSREM does not use the recently released prediction for sunspot cycle 25 since sunspot predictions are not that accurate until the solar cycle has begun. Future versions of MSSREM will allow the user to input their own sunspot predictions rather than only being able to use the predictions described in this paper.

The long mission approach follows the approach described in Xapsos et al. (1998). This approach uses extreme value theory to derive an equation that will give the peak flux that can be expected during the mission at a given confidence level. This upper bound flux can be found by solving

(1)

where F_T(M) is the user-specified confidence level, μ is number of episodes per year, T is the mission length in years, and M = log(ϕ), where ϕ is the peak flux during an episode. P(M) is defined as the probability that the next solar energetic particle episode will have a peak flux <ϕ. This equation can be used whenever the mission length is much greater than the length of an episode. For this reason, this approach can only be used for missions that are longer than 0.5 years. Since the entire derivation makes no reference to the element or the energy of the particle, it can be used for any element and energy range. A more detailed derivation of Equation 1 can be found in either Xapsos et al. (1998) or Robinson (2015).

The peak fluxes from the episodes found in our databases are used to build a cumulative distribution for each elemental energy channel. An example of the peak flux distribution used for the long mission approach is given in Figure 3. The peak flux distribution is fit by three different functions. Since the distribution is well represented in the upper and middle portions of the distribution, all that is required is a function that matches the shape of the distribution. Power law and logarithmic quadratic functions are used here to describe the upper and middle portions of the peak flux distribution. Since the lower region of the distribution has less data in it, a theoretical model is needed to fit the data. A Fréchet distribution is used to fit this region of the distribution (Xapsos et al., 1998). The Fréchet distribution is given by

(2)

where y is the cumulative distribution value, f is the peak flux of an episode, and a, b, c, and d are fitted parameters of the Fréchet distribution. Looking at the equation, y will equal 0 when f is equal to d. This means that d can be considered the maximum peak flux that can be observed during an episode. Following Xapsos et al. (1998), the maximum peak flux used in this work will be twice the largest flux measured in each channel. Since this follows the work of Xapsos et al., (1998), the extreme fluxes in this distribution are also modeled by a truncated power law.

Figure 3

This figure shows the channel 1 peak flux distribution for carbon. The distribution is fit by three functions. The low flux regime is fit by a power law function, the middle of the distribution is fit with a logarithmic quadratic, and the high flux regime is fit with the Fréchet function.

Using the three different functions to describe the cumulative distribution is a unique approach to representing the distribution. The main concern of this approach is overfitting the data. Since the functions are piecewise continuous, the way to verify that each model does not overfit the data is to verify that the function does not overfit the portion of the distribution that it covers. Both the Fréchet and Power Law functions have been shown to fit the cumulative distributions in the previous work (Gabriel & Feynman, 1996; Xapsos et al., 1998). Since these functions can fit the peak flux distribution, they can also be used to fit portions of the distributions. For the logarithmic quadratic function, it only has three free parameters. The Fréchet function has four free parameters to fit the distributions. Since an equation with more parameters does not overfit the distribution, the logarithmic quadratic function does not overfit the distribution either.

The number of episodes per year used in the long mission approach can be found in a few different ways. For the years covered in the database, the actual number of episodes observed in each year can be used. For the historical years before our database, the sunspot number can be used to determine the number of episodes each year. This method uses the smoothed monthly sunspot number and the number of episodes found in each year of our database to build a model to predict the number of episodes using the sunspot number. The smoothed monthly sunspot number was obtained from the World Data Center Sunspot Index and Long-term Solar Observations (SILSO), Royal Observatory of Belgium, Brussels (SILSO, 2019). The smoothed monthly sunspot number is summed over 12 months to produce an annual sunspot number, with a year starting on 1 October and ending on 30 September of the following year. This sunspot proxy method uses the same equation commonly used for dead time corrections in charged particle detectors. This equation is

(3)

where N is the number of episodes in a year, n is the number of sunspots in a year, and a, b, and q are fitted parameters. The sunspot proxy is shown in Figure 4. The sunspot predicts can be found on NOAA's website (https://www.swpc.noaa.gov/products/predicted-sunspot-number-and-radio-flux). With these data, the sunspot proxy can be used to calculate the number of episodes to the end of the current solar cycle or in a year where there is no flux measurement.

Figure 4

The sunspot proxy is shown above. The values for the parameters for the fit are a = 0.0297, b =  − 0.9894, and q = 0.0144. For years with a low number of sunspots, Equation 3 is almost linear because most episodes consist of single isolated events. The number of events is proportional to the sunspot number in this case. As more and more sunspots occur each year, the number of episodes in a year begins to level off before decreasing at the highest annual sunspot rates. This makes sense because as the Sun is more active, more events occur. However, when there are more events per year, they start to merge into episodes. At the highest sunspot rates, episodes merge into longer episodes, thus reducing the number of episodes seen in a year.

With the sunspot proxy, the number of solar energetic particle episodes in a year can be estimated by knowing the annual sunspot number. For years with a low number of sunspots, Equation 3 is almost linear because most episodes consist of single isolated events. The number of events is proportional to the sunspot number in this case. As more and more sunspots occur each year, the number of episodes in a year begins to level off before decreasing at the highest annual sunspot rates. This makes sense because as the Sun is more active, more events occur. However, when there are more events per year, they start to merge into episodes. At the highest sunspot rates, episodes merge into longer episodes, thus reducing the number of episodes seen in a year.

There has been some thought recently that larger episodes have a greater chance of producing larger peak fluxes due to the overlapping of the events. Raukunen et al. (2018) addressed this by splitting episodes and then randomly sampled the number and size of events in an episode to address this effect. The author agrees that more events piling up in episodes can cause the peak flux in the episode to be higher. However, the degree to which the peak flux is changed due to overlapping event is minimal to the peak flux of the episode. The biggest event in the episode is usually marginally affected by riding on the tail or front of another smaller event within the episode.

To determine the number of episodes per year for the far future, an 11-year solar cycle fit was used. The 11-year cycle was fit to the number of episodes identified during each year in our database and used to predict the number of episodes per year through the year 2055. The 11-year solar cycle prediction is shown in Figure 5. As one can see, this fit does a reasonable job, giving a reduced chi squared value of 2.47 for the last four solar cycles. This fit is only used to extend MSSREM three solar cycles into the future. While the solar cycle averages about 11 years, each solar cycle can be a few years longer or shorter. If this fit is used to extend the number of episodes in a year farther into the future, there is a greater chance that MSSREM will not be synchronized with the actual solar cycle due to the varying length of future solar cycles.

Figure 5

This above fit demonstrates the 11-year solar cycle fit that is used in MSSREM to estimate the number of episodes in a year through 2053. Since episodes can be described by time-independent Poisson statistics, the error on the number of episodes in a year is approximately the square root of the number of episodes per year.

For the short mission approach, a data-driven approach was developed to find the upper bounding flux for the user's mission at a specified confidence level. The short mission approach uses a custom cumulative distribution for each energy channel that is specific to the user's mission together with the active time ratio for that mission. The custom cumulative distribution is created from the entire database of flux measurements above background while the active time ratio is created from the percentage of time a hypothetical mission experiences flux above background in each sunspot grouping. The sunspot groupings are used to determine how active the sun will be during the user's mission.

The custom cumulative distribution is created by taking the chronological list of all flux measurements for a given element and energy channel and then removing all measurements that are not part of an episode. This creates a list that only contains fluxes that are above background. Next, the data are grouped into bins, each having the length of one mission, creating a data set containing as many bins as will fit in the list. For example, the proton data base is comprised of 30-min-averaged measurements. If the user's desired mission length is 5 hr, each bin will hold 10 measurements. The maximum flux in each bin is found and used to build the custom cumulative distribution. This distribution is then constructed such that the Y axis is the fraction of all maximum fluxes ≥ the flux value on the X axis. At 0 on the X axis, the Y axis will be 1 and the Y axis will have its lowest value at the largest flux value on the X axis.

Next, the active time ratio (ATR) was found, which is a ratio of the number of hypothetical missions that include at least part of an episode divided by the total number of hypothetical missions. To find the ATR, a sequential list (with one entry for each flux measurement) was created. This list is used to record whether or not each measurement from our data set was above background. Such a list was created for each of the four different groupings of the monthly sunspot number. Each measurement in these lists is then used as a start date for a hypothetical mission with the same mission length as the user's mission. The number of hypothetical missions that have at least one flux measurement above background is tallied for each list. The ATRs are then found for each sunspot grouping. Finally, using the mission start date and duration, the percentage of the mission in each of the sunspot groupings is found. The ATR for the entire mission is calculated from each sunspot grouping ATR, weighted by the part of the mission in each grouping.

With the custom cumulative distribution and the ATR found, the flux for the elemental energy channel can be found. The user's confidence level, K, is then corrected for the ATR to obtain H = (1 − K)/ATR, where H is the Y axis value in the custom cumulative distribution. If H ≥ 1, the design reference environment is the galactic cosmic ray background and the flux is reported as 0. If H < 1, then there is flux above background in this channel. The value of H is used to find a flux value for the design reference environment from the cumulative distribution. Depending on the number of flux measurements in the custom cumulative distribution, one of two approaches is used to find the flux corresponding H. If there are more than 1,000 flux measurements in the distribution, linear interpolation between consecutive points in the distribution can be used to find the flux value. If the custom cumulative distribution has less than 1,000 flux measurements, it needs to be fitted with smoothing functions. The power law, logarithmic quadratic, and Fréchet functions are used to fit the distribution. These three functions are used to build a distribution with 1,000 evenly spaced data points. If the value of H is less than the smallest value in the distribution, a Fréchet function fit is used to find the corresponding flux by extrapolation. Otherwise, linear interpolation is used between the nearest two points in this distribution to find the correspond flux for the reference environments.

The short mission approach is limited by the number of data points that are above background in the cumulative distribution. This is to ensure that the short mission approach does not give an upper bounding flux higher than the long mission approach for any confidence level. As the mission length increases, fewer hypothetical missions can be constructed. Also, because they are longer the peak fluxes in each tend to be larger. The bins used to create the custom cumulative distributions also increase in size, causing there to be fewer bins. If the mission length is increased enough, the custom cumulative distribution will be a more severe environment than the peak flux distribution used in the long mission approach. To prevent this from happening, the short mission approach needs twice as many flux measurements in the custom cumulative distribution as the number of episodes in an elemental energy channel. This limits the short mission approach to being used for missions no longer than a few days, depending on the energy channel.

The intermediate mission approach consists of running the short and long mission approaches twice for different mission lengths and then fitting the data with the Sigmoid function. The long mission approach lengths used are 0.5 and 5 years. The short mission lengths used are different for each elemental energy channel. The longest short mission length possible for each channel and a mission length that is 2 orders of magnitude shorter are used. With these four points, the flux can be plotted as a function of mission length and the data can be fit with a sigmoid function. The sigmoid fit is then used to find the corresponding flux for missions whose length fall between the regions of applicability of the short and long mission approaches. An example of the sigmoid fit is shown in Figure 6.

Figure 6

This figure shows the Sigmoid fit for protons in channel P6 for the intermediate mission approach. The mission started on 2001.5 and was run at the 95% confidence level. The chi-square value for this fit was 0.1488. This fit would be used to find the flux for any mission length between 0.004 and 0.5 years for this channel.

For the helium data, both ACE and SEPEM data were probabilistically modeled. The ACE data were examined to see if a predetermined threshold level for each channel was exceeded. The threshold levels used in MSSREM are shown in Table 1. These threshold levels were chosen to be at least 2 orders of magnitude below the maximum flux measured in each ACE channel. If none of the channels exceeded the threshold level, the ACE data were used for the helium data. However, if one of the ACE channels exceeds the threshold level, the SEPEM data were used instead. Since the ACE data have been normalized to the SEPEM data, the data are interchangeable. Using the ACE data for the smaller fluxes (less intense spectra) and switching to the SEPEM data for the larger fluxes (more intense spectra) reduces the deficiencies of both data sets and allows for a more accurate design reference environment to be built.

Table 1. ACE Helium Threshold Levels

ACE helium channel	Threshold level (cm^2*sr*sec*(MeV/nuc))^−1
1	1.0
2	1.0
3	1.0
4	0.5
5	0.1
6	0.05
7	0.008
8	0.005

With the flux found in all channels for the six elements, the resulting upper bounding spectra for each element are fitted with the Band function (Band et al., 1993). The Band function enables smooth energy spectra to be estimated using the available data. In addition, the Band function fit can be used to distribute the spectra into smaller energy bins. The Band function was used in MSSREM since it has been found to characterize solar proton energy spectra well (Band et al., 1993; Mewaldt et al., 2005; Robinson, 2015; Tylka et al., 2005). Before the spectra could be fitted with the Band function, the mean energies for each channel of data had to be calculated. The mean energy is a value such that half the flux in the channel fell below and half above this energy value. The mean energy can be found by

(4)

where is the mean energy for a channel and E₀ and E₁ are the lower and upper bounds of an energy channel (Robinson, 2015). To find the mean energy of a channel, two power laws are fit between the channel and the proceeding and following channels. The slope from each power law is then averaged together. Using the average slope, the range of the energy channel is broken into 25 equal segments. The flux from all 25 segments are averaged and plugged into the power law equation to find the mean energy of the channel. For the first and last channel in each spectrum, only one power law fit was used to fit the mean energy.

With the mean energy for each channel now calculated, the Band function could be used to fit the spectra for each element. Once the Band function fit was found, the spectra are divided into 1,002 energy channels logarithmically spaced between 0.1 MeV and 100 GeV. The spectra are always reported as having zero flux for the energy channels between 0.1 and 1 MeV. The energy channels below 1 MeV being reported as having zero flux in them follows the approach used in CREME96. This ensures that there is a seamless transition between MSSREM and the old CREME96 flux model in SIRE2. While there is a significant amount of flux in the 0.1–1 MeV energy range, MSSREM does not include these particles since they can be stopped with very little shielding.

For the remaining 86 elements, the design reference environments can be created by scaling the reference environments of the six probabilistically modeled spectra. The scaling of the reference environments also needs to include the heavy ion enhancement that can in different episodes. However, an average episode and the coronal abundances are needed to assist in this scaling.

The average episode was created from the 14 elements available in the ACE SIS data. Except for data during the periods of time when caution is recommended in the ACE SIS release notes (http://www.srl.caltech.edu/ACE/ASC/DATA/level2/sis/sis_release_notes), all the flux measurements above background were converted into a fluence and summed up for each elemental energy channel. The spectra for each element were fit with a power law and the power law was used to distribute the fluence into the 1,002 energy channels described earlier. The average episode spectra for the 14 elements are shown in Figure 7.

Figure 7

The ACE average episode is shown above. This average episode was used to create the design reference environments for nine elements that were not calculated probabilistically.

During a solar particle episode, particles that are sent outward from the Sun originate from in the Sun's corona. To accurately model the particle composition of an episode at Earth, the coronal abundances need to be used. The coronal abundances used in this work were from Schmelz et al. (2012). Since Schmelz et al. only included 20 coronal abundances, the remaining abundances were calculated from the photospheric or meteorites abundances. Using the coronal abundances from Schmelz et al. (2012), a coronal to photospheric abundance ratio could be found. This ratio, and the first ionization potential (FIP) energy, can be used to find the coronal abundance for the remaining elements. For elements with a FIP energy less than 10 eV, the coronal to photospheric abundance ratio gives a ratio greater than 1. Elements with a FIP energy greater than 10 eV will have a ratio less than one (Schmelz et al., 2012). To calculate the abundance ratio in between the reported ratios in Schmelz et al. (2012), a power law was used to extrapolate the abundances ratios to the other elemental FIP energies. This allows for the abundance ratios to be found for other elements. The coronal abundances can then be found if the photospheric abundances are known. The photospheric abundances used in this work are found in Scott, Grevesse, et al. (2015), Scott, Asplund, et al. (2015), and Grevesse et al. (2015). For a select few elements, the meteorite abundances (or cosmic abundances) from Asplund et al. (2009) were used since there is no photospheric abundance data. In these cases, the meteorite abundances were treated as the photospheric abundances.

The scaling of the design reference environments uses a specific method for each element. For the Z < 29 elements, the reference environments can be calculated by

(5)

where f_i(j) is the calculated scaled flux for the ith element in the jth energy channel. The g_i(j) term is the standard flux for the ith element in the jth energy channel. The standard fluxes are either the average episode or the coronal abundances. The standard flux is the same across all energy channels when the coronal abundances are used for an element. The h_i(j) term is the heavy ion enhancement for the ith element in the jth energy channel. The heavy ion enhancement for each element can be calculated by

(6)

where Z_u and Z_l are the atomic numbers for the upper and lower bounding elements, respectively. The upper and lower bounding elements are one of the six probabilistically modeled elements. The f_u(j) and f_l(j) terms are the probabilistically modeled fluxes in the jth energy channel of the upper and lower bounding elements, respectively. The g_u(j) and g_l(j) terms are the standard fluxes in the jth energy channel of the upper and lower bounding elements, respectively.

To calculate the scaled flux for the first energy channel of calcium, the heavy ion enhancement for the first energy channel needs to be found. Silicon is the lower element and iron is the upper element in this example. Calcium's atomic number is 20 while silicon's atomic number is 14 and iron's is 26. The first part of the first term in Equation 6 is equal to 0.5. Then using silicon and iron reference environmental spectra along with the average episode, the fluxes in the first energy channel can be used to find the heavy ion enhancement for the first calcium energy channel. The heavy ion enhancement is then multiplied by the standard flux for calcium in the first energy channel. For cobalt and nickel, there is no upper element since iron is the largest element that was probabilistically modeled. Instead, the upper and lower bounding elements used are silicon and iron.

For elements 29 ≤ Z ≤ 92, a heavy ion enhancement could not be calculated since ACE SIS does not have any data on elements Z ≥ 29. The reference environments for these elements can be calculated by

(7)

The lower element used for calculating all the reference environments for Z ≥ 29 elements is iron. In other words, the remaining reference environments are all scaled from the iron reference environment.

In this way, the design reference environmental spectra for all 92 elements are created. The environments created are 30-min-averaged peak fluxes for protons, 1-hr-averaged peak fluxes for Z = 3–92, and either 5-min- or 1-hr-averaged peak fluxes for helium, depending on the data set used by MSSREM. Since the Band fit is used to interpolate the spectra into the 1,002 energy channels, the spectra are checked to see if the flux in any channel is smaller than 10% of the lowest galactic cosmic ray background found in Badhwar-O'Neill 2014 model. In such cases, the flux for that energy channel is dominated by galactic cosmic rays so the flux from the MSSREM peak flux model is reported as zero.

4 MSSREM Peak Flux Model

In Figure 8, the design reference environments produced by the MSSREM peak flux model for two different missions are shown. The first mission was a hypothetical mission that lasted 168.98 min and launched on 1 January 2001 at 12:00 a.m. The confidence level was selected to be 92% for this mission. The design reference environmental spectra for silicon and iron were both 0 for this mission. This mission utilized the short mission approach in the MSSREM peak flux model. The second mission was a hypothetical 2-year mission that launched on 1 January 2016 at 12:00 a.m. The confidence level selected for this mission was 99%. This mission utilized the long mission approach in the MSSREM peak flux model.

Figure 8

The design reference environment for two different missions are shown above. The first mission was for a mission lasting 168.98 min that launched on 1 January 2001 at 12:00 a.m. The 92% confidence level was used for this mission. The reference environmental spectra for this mission are represented by the solid lines in the figure. The second mission was for a mission starting on 1 January 2016 at 12:00 a.m. and lasts 2 years. The 99% confidence level was chosen for this mission. The design reference environmental spectra for this mission are the dashed lines on the plot.

To check the accuracy of the MSSREM peak flux model, the short and long mission approaches described in this work need to be verified. For the short mission approach, there is no other model that can produce a reference environment for missions in the tens of minutes time frame. Instead, the short mission approach was tested by comparing the custom cumulative distributions for a mission to distributions created by random sampling. Distributions were computed both ways for each of the sunspot grouping. This testing was done using a 5-hr mission length. The MSSREM peak flux model run four times, once for each sunspot grouping. The custom cumulative distributions for the proton channel P2 from these runs were used for the comparisons.

To create distributions by random sampling, flux lists were created from the data set for each sunspot grouping. In addition, a fifth list was created that contained all the flux measurement for the entire data set. These flux lists included all the flux measurements. Those that were at background were set to 0 flux. Since the proton channel P2 data consists of 30-min-averaged measurements, 10 measurements were needed to span the mission length. Each list was randomly sampled 10 times and then the 10 measurements were binned together to represent the fluxes encountered during a mission. The maximum flux in each bin was found. The random sampling was repeated until the number of samples equaled the number of flux measurements in the list. All the maximum fluxes from the samples were used to create a cumulative distribution.

To compare the MSSREM custom cumulative distributions and the randomly sampled distributions, one more step was needed. For the short mission approach of the MSSREM peak flux model, the ATR used is specific to the sunspot groupings while the custom cumulative distributions use flux measurements from the entire distribution. This is because the sizes and lengths of episodes are independent of sunspot grouping, so to ensure that the custom cumulative distribution for each sunspot group is not limited by the size of the data set in a sunspot group, the custom cumulative distribution used the flux measurements from the entire database.

For the randomly sampled approach to realistically compare to the MSSREM custom cumulative distributions, the randomly sampling method had to be applied to the entire data set. The data sets for each sunspot grouping were only used to determine the fraction of the mission-length bins that were at background. The fluxes for each sunspot grouping came from the randomly sampled distribution for the entire database of flux measurements. These two components were then combined to create a distribution that can be directly compared to the custom cumulative distribution created in the MSSREM short mission approach. In Figure 9, the cumulative distributions from MSSREM and this random sampling process are compared for each sunspot grouping. The cumulative distributions across the two approaches are very consistent with each other. The only minor differences are seen in the tails of the spectra where the variance from random sampling comes into play.

Figure 9

The plot above shows the testing of the short mission approach. The dashed lines show the cumulative distribution for the custom cumulative distributions used in the short mission approach of MSSREM. The solid lines show the distributions that were built through the random sampling approach. The distributions created in both methods are very similar, with the slight differences in the distribution coming from the random sampling.

To test the accuracy of the long mission approach, the MSSREM peak flux model was compared to SAPPHIRE, CREME96, and ESP models. Six elements were compared to each other (hydrogen, helium, carbon, oxygen, silicon, and iron) and are shown in Figure 10. For this test, MSSREM and SAPPHIRE were both run for 2-year missions that started on 1 January 2016 at the 95% confidence level. The SAPPHIRE model was run through SPENVIS 1 and will interpret this mission as 2 years during solar maximum. The ESP model was also run for a 2-year mission at the 95% confidence level. The ESP model was run through the SEPEM application server and used the SEPEM database. The decision to run ESP using the SEPEM database was to include a comparison of models using the same data for the helium reference environments. This allows for a true validation comparison of the long mission approach. For CREME96, the 5-min peak instantaneous flux from the October 1989 storm was used. The CREME96 model was run through SPENVIS.

Figure 10

The design reference environments produced by MSSREM, SAPPHIRE, ESP, and CREME96 are shown in the plot above. The MSSREM, SAPPHIRE, and ESP models were run for a 2-year mission at the 95% confidence level. The mission start date for MSSREM and SAPPHIRE was 1 January 2016. The 5-min peak instantaneous flux from the October 1989 storm was selected for CREME96.

For the proton design reference, the MSSREM peak flux environment has a nearly similar slope for the upper energy portion of the Band function fit. The lower energy portion seems to have a slightly shallower slope, so MSSREM gives a slightly lower flux in the lower energy portion of the reference environment. For the helium reference environment, MSSREM (using the SEPEM data) provides a slightly less intense environment than the one provided by SAPPHIRE. ESP and MSSREM seem to agree very well throughout all energy ranges where the models overlap. All three environments are higher than the 5-min peak instantaneous flux from the October 1989 in CREME96. For the other four elements, MSSREM provides a less severe environment than SAPPHIRE for the higher energy ranges. The two models provide closer agreement for the lower energy ranges. The reader should be aware that all these models extrapolate the environment above 200 MeV/nuc. The exact energy where the models start to extrapolate each element is dependent on the highest energy channel for the element, which varies by element. For this reason, the extrapolation to higher energies needs to be carefully considered. MSSREM and SAPPHIRE both use the Band function to handle this extrapolation. Further testing and validation of the MSSREM peak flux model is ongoing and will be available in 2020.

5 Conclusions

The probabilistic modeling approaches discussed in this paper allow design reference environments to be built that are tailored to a user's mission. The MSSREM peak flux model is a new tool that uses these probabilistic modeling approaches. MSSREM allows the user to select the mission start date, mission length, and confidence level. Based on these inputs, MSSREM will compute a design reference environment for the user's mission. While multiple models are available that provide the user with these options, MSSREM is unique in the sense that missions shorter than 0.5 years can be run. MSSREM can provide unique reference environments for missions as short as 5 min. In addition, MSSREM uses the sunspot number to determine solar activity rather than the phase of the solar cycle like most models do.

The MSSREM peak flux model agrees with the results from SAPPHIRE, ESP, and CREME96 in the limited testing done so far. A much more thorough V&V of MSSREM is currently underway and will be available in 2020. MSSREM is currently only available in the SIRE2 toolkit. Since SIRE2 is a distribution limited toolkit, MSSREM is currently a distribution limited model. However, the authors hope to be able to provide the community access to a distribution unlimited version of MSSREM in the future.

Distribution Statement

DISTRIBUTION STATEMENT A. Approved for public release; distribution is unlimited. Approved for Public Release 20-MDA-10477 (8 May 2020).

Acknowledgments

MSSREM is only available in SIRE2. SIRE2 is currently only available to U.S. Persons (U.S. citizens or green-card holders). To obtain a copy of SIRE2, send a request to SIRE2's email address ([email protected]). The authors are seeking approval for a version of SIRE2 that can be released to the public in 2020. There is no new data from this paper. European Space Agency Space Environment Information System (n.d.) [Online]. Available: http://www.spenvis.oma.be/spenvis.