1 Introduction

Total Solar Irradiance (TSI) study is necessary to understand different contributions to global climate changes^[1-5]. Increasing number of Total Solar Irradiance Radiometers (TSIRs) has conducted long-term high-precision measurement on space platforms since 1970, including the Differential Absolute Radiometers (DIARADs) of Royal Meteorological Institute of Belgium (RMIB)^[6], the Active Cavity Radiometer Irradiance Monitors (ACRIMs) of the National Aeronautics and Space Administration (NASA)^[7], the POM-6 type radiometer of World Radiometer Center (WRC)^[8], etc. The Total Solar Irradiance Monitors (TSIMs), developed by Changchun Institute of Optics, Fine Mechanics and Physics (CIOMP) for China Meteorological Administration (CMA), have been operated on the Feng Yun-3A satellites (FY-3A) for nearly seven years since 2008^[9-11]. The TSIM/FY-3B and TSIM/FY-3C were separately launched on 2010 and 2014^[12-14].

In order to calibrate different space-born TSIRs, the ground-based absolute radiometers are applied to transfer World Radiance Reference (WRR). WRR is established by World Meteorological Organization (WMO)^[15]. WMO conducts International Pyrheliometers Comparison (IPC) at WRC every five years. The transfer radiometers are calibrated by the comparison measurement of solar irradiance. The TSIRs' traceability to WRR can be obtained by the transfer radiometers.

Solar Irradiance Absolute Radiometer (SIAR) is the transfer radiometer for calibrating TSIMs, and has attended IPC for three times^[16]. SIAR adopts electrical substitution principle^[17]. The solar irradiance is absorbed by the detector of SIAR, resulting in a temperature rise. When the temperature rise reappeared with an equivalent electrical power, the solar irradiance can be obtained by precisely measuring the equivalent electrical power. The relationship between temperature response and heating power is named as sensitivity. The sensitivity is an important parameter for estimating equivalent electrical power. And the measurement result of SIAR should be corrected by the sensitivity, because of the non-equivalence between optical and electrical heating^[18].

In the practical comparison experiment, the sun is the optical source. The solar irradiance is depended on the atmospheric condition. For higher radiant power, SIAR can obtain a measure precision superior to 0.08%^[19]. But the measure precision rapidly decreases with the reduction of the radiant power. This problem is investigated and solved in this paper.

The sensitivity is assumed as linear within the whole measurement interval in the traditional method^[19]. But the sensitivity has nonlinearity. The nonlinearity comes from complex factors, including the radiant power, the ambient temperature drift, and the convection environment^[20]. Thus this assumption leads to a system error, which decreases the measure precision. The traditional method needs an optimization to compensate the system error^[21].

In this paper, the nonlinearity of sensitivity is experimentally investigated and corrected. Based on the interpolation approximation principle, an optimization method is proposed to improve the measure precision^[22-23]. The optimization method is adopted by the fourth generation of SIAR. The comparison experiment results illustrate that the system error of sensitivity nonlinearity is compensated by the real-time interpolation approximation. Due to the improvement of measure precision, the optimization method significantly improves the calibration precision of ground-based comparison experiment. The investigation of sensitivity nonlinearity also provides the theoretical and experimental basis for optimizing the on-orbit calibration precision for TSIM.

2 SIAR Fundamentals 2.1 Detector Design

The diagrammatic sketch of SIAR is shown in Fig. 1. The primary cavity and the reference cavity constitute the detector of SIAR. They are weakly linked to the heat sink. The cavities and heat sink constitute a heat conduction structure. When loading an invariable power on the primary cavity, the heat conduction structure obtains a thermal equilibrium state. Normally, the establishment time of thermal equilibrium state is between 7τ and 10τ. The τ is the thermal time constant of primary cavity.

2.2 Equilibrium Temperature

In practical engineering application, the output voltage of thermocouple represents the temperature variation of primary cavity. Therefore the output voltage of thermocouple is named as temperature voltage. Meanwhile, the temperature voltage at thermal equilibrium state is named as the equilibrium temperature. Each thermal equilibrium state corresponds to a unique equilibrium temperature.

2.3 Electrical Substitution Principle

SIAR applies the principle of photoelectric equivalence. When the shutter is opened, the primary cavity absorbs solar irradiance. Then the equilibrium temperature is obtained. This process is called irradiance observation. When the shutter is closed, the primary cavity is only heated by electrical power. The electrical power is adjusted in order to get the same equilibrium temperature with irradiance observation. This process is called electrical calibration. The adjusted electrical power is named as the equivalent electrical power. The solar irradiance is obtained by precisely measuring the equivalent electrical power.

3 Optimization Method Principle

In order to compensate the nonlinearity error of sensitivity nonlinearity, an optimization method is proposed. Based on the interpolation approximation principle, the real-time correction sensitivity is obtained by two point interpolation approximation.

3.1 Self-test Process

Before the irradiance measurement, the shutter is closed. Two calibration electrical powers (P_L and P_H) are respectively loaded on the primary cavity, resulting in the temperature variations as shown in Fig. 2(a). Then the equilibrium temperatures (V_L and V_H) are obtained. The sensitivity (S) of SIAR is calculated by

$ S = \left( {{P_H} - {P_L}} \right)/\left( {{V_H} - {V_L}} \right) $

(1)

where S is used to approximately represent the sensitivity of the wide power interval (P_L-P_H) as shown in Fig. 2(b). This process is called Self-test. When an equilibrium temperature (V) is obtained, the corresponding electrical power (P_E) can be calculated by

$ {P_E} = S \cdot V - b $

(2)

where b is the intercept of sensitivity curve.

3.2 Measurement Procedure

The optimization method has one irradiance observation and two electrical calibrations as shown in Fig. 3. Before the irradiance observation, the primary cavity is warmed by P_E1, resulting in an equilibrium temperature of V₁.

The shutter is opened before the irradiance observation. The primary cavity absorbs P_O. Meanwhile a compensation electrical power (P_E2) is calculated and loaded on the primary cavity in order to maintain the equilibrium temperature of V₁^[17]. Due to the non-equivalence between optical and electrical heating, the equilibrium temperature of irradiance observation is V₂. Shutter is closed before the electrical calibrations. According to V₂, the primary cavity is only heated by the electrical power (P_E3) calculated by Eq. (2). The equilibrium temperature of the first electrical calibration is V₃.

In the second electrical calibration, the primary cavity is heated by the electrical power (P_E4), which is expressed as

$ {P_{E4}} = {P_{E3}} + S \cdot \left( {{V_2} - {V_3}} \right) $

(3)

The equilibrium temperature of the second electrical calibration is V₄. The real-time correction sensitivity (S₁) is obtained by two point interpolation approximation. Then S₁ is calculated by

$ {S_1} = \left( {{P_{E4}} - {P_{E3}}} \right)/\left( {{V_4} - {V_3}} \right) $

(4)

Then P_O is derived by

$ {P_O} = {P_{E3}} - {P_{E2}} + {S_1} \cdot \left( {{V_2} - {V_3}} \right) $

(5)

The area (A) of primary aperture is determined at high accuracy. Meanwhile the absorptivity (ρ) of primary cavity is precisely measured. The calibration coefficient (ε) is obtained by the calibration experiment. Thus solar irradiance (E) is calculated by

$ E = \frac{{\varepsilon {P_O}}}{{\rho A}} $

(6)

However, the traditional method only has one electrical calibration, and uses S to calculate P_O. Thus P_O is given by

$ {P_O} = {P_{E3}} - {P_{E2}} + S \cdot \left( {{V_2} - {V_3}} \right) $

(7)

where S₁ is obtained in the solar irradiance measurement process, and represents the sensitivity of the little power interval (P_E3-P_E4). Compared with the traditional method, the system error of sensitivity is compensated by real-time correction. Thus the optimization method can improve measure precision.

4 Sensitivity Nonlinearity Investigation

S and S₁ are experimentally measured, so as to investigate the nonlinearity effects to the measure precision of SIAR.

4.1 Sensitivity Measurement 4.1.1 Equipment set up

Laser radiant power (λ=532 nm) is the optical source. The laser radiant power is modulated by power stabilizer in order to improve the stability. The Semi-reflecting mirror separates the laser radiant power into a single optical path and a reference optical path as shown in Fig. 4. The single optical path is measured by SIAR. The reference optical path is used for monitoring the stability of laser radiant power. The reference optical path enters the integrating sphere, resulting in a response voltage of photodiode (Hamamatsu S1406). The response voltage is measured by a digital multimeter (Keithley 2 700). The different power level of single optical path is obtained by regulating the attenuator. Then S₁ is repeatedly measured at different radiant power level.

4.1.2 Experiment result

S is measured according to the method of self-test. S₁ is repeatedly measured at different radiant power level. The average relative standard deviation (σ_r1) of S₁ is calculated by

$ {\sigma _{r1}} = \frac{1}{N}\sum\limits_{j = 1}^N {\left[ {\frac{1}{{{\mu _j}}}\sqrt {\frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{\left( {{S_{1ji}} - {\mu _j}} \right)}^2}} } } \right]} $

(8)

where N is the irradiance power levels; n is the measurement times; S_1ji is the measurement results; μ_j is the average value of S_1j.

The sensitivity nonlinearity is represented by the average relative nonlinearity error (σ_r2), which is calculated by

$ {\sigma _{r2}} = \frac{1}{N}\sum\limits_{j = 1}^N {\left[ {\frac{1}{{{\mu _j}}}\sqrt {\frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{\left( {{S_{1ji}} - S} \right)}^2}} } } \right]} $

(9)

At different radiant power level, the sensitivity of SIAR is obviously nonlinear as shown in Fig. 5.

When the sensitivity of SIAR is assumed as linear, the relative system error (σ_r3) of S is expressed as

$ {\sigma _{r3}} = \sqrt {\sigma _{r1}^2 + \sigma _{r2}^2} $

(10)

The measurement result illustrates that σ_r3 is 2.2%, and S is 0.023 70 mW/mV. Thus the systemic error (σ₃) of S is 0.000 521 mW/mV.

4.2 Precision Analysis

According to Eq. (4) and Eq. (5), the relative precision (δ_r) of SIAR is expressed by

$ {\delta _r} = \sqrt {2\delta _{r1}^2 + 2\delta _{r2}^2 + \delta _{r3}^2} $

(11)

where δ_r1 (≈0.041%) is the relative measure precision of electrical power; δ_r2 (≈0.040%) is the relative measure precision of temperature voltage; δ_r3 is the relative precision of sensitivity. δ_r3 can be calculated by

$ {\delta _{r3}} = \frac{{\left( {{V_2} - {V_3}} \right){\sigma _3}}}{{{P_O}}} = \frac{{\mu {\sigma _3}}}{{{P_O}}} $

(12)

where μ is the deviation between V₂ and V₃.

The μ is random in practical measurement, and changes in the range of 1 mV to 20 mV. δ_r3 and P_O are inversely related as shown in Fig. 6. When P_O is large enough, δ_r3 is smaller than 2.0×10^-4, which is little enough to be neglected compared with δ_r1 and δ_r2. Then the optimal δ_r is 0.08%. In case of μ is 10 mV. When P_O is larger than 30 mW, δ_r3 is lower than 0.02%. Then δ_r3 can be neglected. When P_O is lower than 20 mW, δ_r3 is more than 0.03%. δ_r3 cannot be neglected in this situation. This result explains that the measurement precision of SIAR rapidly decreases with the irradiance reduction.

5 Comparison Experiment

The precisions of optimization method and traditional method are compared by the experiment equipment in Fig. 4. The stability of laser radiant power is better than 0.02% within 80 min as shown in Fig. 7. The laser radiant powers at each power level are respectively measured by traditional method and optimization method.

The optimization method obtains a better measurement precision approximated to 0.1% as shown in Fig. 8. The measure precision of traditional method rapidly decreases with the reduction of irradiant power. The comparison result illustrates that the system error of sensitivity is compensated by real-time correction. Thus the optimization method improves the measure precision of SIAR.

6 Application in External-Field Calibration Experiment

The fourth generation of SIAR is manufactured for the coming IPC-Ⅻ. They are named as SIAR-4A, SIAR-4B and SIAR-4C. Before attending IPC-Ⅻ, SIARs are tested by External-Field calibration experiment. The second generation SIAR, which is named as SIAR-2C, is adopted as the transfer detector.

SIAR-2C, SIAR-4A, SIAR-4B and SIAR-4C are installed on the high-precision solar tracking turntables as shown in Fig. 9. The solar irradiance is used as optical source. Based on GPS and servo control system, the high-precision solar tracking turntables can guarantee the same solar irradiance to be absorbed by SIARs. Meanwhile SIARs start and finish measurement at the same time.

SIAR-2C has attended IPC-Ⅹ, obtaining the calibration coefficient (ε) of 0.999 839. The WRR irradiance (E_WRR(i)) can be calculated by

$ {E_{WRR}}\left( i \right) = 0.999839 \times {E_{SIAR - 2C}}\left( i \right),\left( {i = 1,2, \cdots ,N} \right) $

(13)

where E_SIAR-2C(i) is the irradiance measurement result of SIAR-2C. N is the total measurement number. Calibration coefficient is calculated by

$ \varepsilon = \frac{{\sum\limits_{i = 1}^N {{E_{WRR}}\left( i \right)/{E_{SIAR}}\left( i \right)} }}{N},\left( {i = 1,2, \cdots ,N} \right) $

(14)

where E_SIAR is the irradiance measurement result of SIAR. The calibration precision (σ_ε) is expressed as

$ {\sigma _\varepsilon } = \sqrt {\frac{{\sum\limits_{i = 1}^N {\left[ {{E_{WRR}}\left( i \right)/{E_{SIAR}}\left( i \right) - \varepsilon } \right]} }}{{N - 1}}} ,\left( {i = 1,2, \cdots ,N} \right) $

(15)

The traditional method is adopted by SIAR-4A and SIAR-4C. The optimization method is adopted by SIAR-4B. The calibration coefficients are obtained as shown in Table 1. The differences (d) between WRR and the new generation SIARs are respectively 0.001 3, 0.000 5 and 0.001 5. The calibration precisions of new generation SIARs satisfy the IPC standard. Meanwhile, σ_ε-SIAR-4A=0.001 1, σ_ε-SIAR-4B=0.000 7, σ_ε-SIAR-4C=0.001 2. Thus the optimization method improves the calibration precision as shown in Fig. 10.

Measurement results (E) and relative standard deviations (σ) at five irradiance power levels are shown in detail. When the solar irradiance is higher than 700 W/m², the relative standard deviations of the two methods are both approximated to 0.08% as shown in Table 2. With the reduction of solar irradiance, the relative standard deviation of optimization method is stable at 0.1%. But the relative standard deviation of traditional method changes from 0.1% to 0.3%. Thus, due to the improvement of measure precision, the optimization method significantly improves the calibration precision of SIAR.

7 Conclusions

The fourth generation of SIAR is optimized both in measurement method and development technology, in order to attend IPC-Ⅻ. In the past, there is a problem that the measure precision of SIAR rapidly decreases with the reduction of irradiance power. The optimization of calibration method is introduced in this paper.

The sensitivity is assumed as linearity within the whole measurement interval in the traditional method. The sensitivity of SIAR is experimentally investigated and corrected. When the sensitivity is assumed as linear, the relative system error of nonlinearity is 2.2%. The system error cannot be neglected for little solar irradiance measurement. Thus the sensitivity nonlinearity reduces measure precision. In order to compensate the sensitivity nonlinearity, the optimization method is proposed based on the interpolation approximation principle. During the irradiance measurement process, the real-time correction sensitivity is obtained by two point interpolation approximation. The real-time correction sensitivity represents the sensitivity of the little radiant power interval, which is exceedingly close to the solar irradiance power. The comparison experiment result illustrates that the optimization method compensates the system error of sensitivity nonlinearity, and obtains a stable measure precision approximated to 0.1%.

The optimization method is adopted by the fourth generation SIAR. The experimental results illustrate that the optimization method significantly improves the calibration precision of ground-based comparison experiment. Influenced by the measure environment on satellite, the sensitivity of TSIM has also nonlinearity. The investigation of sensitivity nonlinearity also provides the theoretical and experimental basis for optimizing the on-orbit calibration precision for TSIM.