Process-based indicators for timely identification of apricot frost disaster on the warm temperate zone, China

Frequent occurrences of late spring frost disaster create severe agricultural/forest damage, even given the background of global warming. In the warm temperate zone of China, which is the largest planting area for fresh apricot, late spring frost disaster has become one of the major meteorological hazards during flowering. To prevent cold weather–induced apricot frost disaster and reduce potential losses in related fruit economic value, it is vital to establish a meteorological indicator for timely and accurate identification of cold weather process–based apricot frost disaster, to provide support for timely apricot frost monitoring and warning in late spring. In this study, daily minimax temperature (Tmin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{\mathrm{min}}$$\end{document}) and apricot frost disaster data during flowering were combined to establish meteorological identification indicators of apricot frost based on cold weather processes. A process-based apricot frost model f(D,Tcum)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(D,{T}_{\mathrm{cum}})$$\end{document} was firstly constructed, and characteristics of Tcum\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{\mathrm{cum}}$$\end{document} (accumulated harmful temperature) were explored under different D (duration days) based on the representation of historical apricot frost processes. Thresholds for the Tcum\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{\mathrm{cum}}$$\end{document} for apricot frost in 1, 2, 3, 4, and more than 5 days of apricot frost process were determined as − 1.5, − 2.9, − 4.4, − 5.8, and − 7.3 °C, respectively. Validation results by reserved independent disaster samples were generally consistent with the historical records of apricot frost disasters, with 89.00% accuracy for indicator-based identification results. Typical process tracking of the proposed identification indicator to an apricot frost event that occurred in North Hebei during April 3–9, 2018, revealed that the indicator-based identification result basically coincides with the historical disaster record and can reflect more detailed information about the apricot frost process.


Introduction
Frost, as a temperature-related meteorological disaster, has been recognized as a major threat to the normal growth and production of agriculture and forest productivity because of cold weather conditions (Vitasse et al. 2018). As the public generally believe that climate change is a scientific conclusion, a decrease in the frequency and intensity of frost events has been speculated with global warming. Paradoxically, as temperatures increase in early spring, perennial crops such as fruit trees gradually become increasingly vulnerable to cold temperatures, because of climate fluctuation that has led to considerable phenological shifts, such as advancing the date of flowering, and accelerating vegetative development, leading to the advancement of frost-sensitive stages due to a warmer winter and spring (Shimono 2011;Saeidi et al. 2012;Chen et al. 2014;Wang et al. 2018). Frost disaster is widely reported in North American ecosystems (Gu et al. 2008;Augspurger 2013) and forest and fruit trees in Europe (Yann et al. 2018). The annual economic loss caused by meteorological disasters exceeds 300 billion dollars in China, with average frost damage caused economic losses of 25 billion dollars from 2011 to 2018 approximately (data was acquired from the "Year book of Meteorological Disasters in China" from 2011 to 2018). Timely monitoring of the occurrence of frost disasters is of considerable importance for adopting targeted measurements to reduce economic and production losses.
Cereals (e.g., maize and wheat) (Wang et al. 2013(Wang et al. , 2019 and economic plantings (e.g., beech, oak, apple, coffee, and tea) (Susan et al. 2018;Antonio et al. 2019;László et al. 2019) have been shown to be vulnerable to frost, with different performance characteristics and mechanisms in response to frost. For example, the reduction of spike number was 8-15% and the yield loss was 5-14% with a 5-day spring frost episode at the elongation stage . Spring frost during or immediately after leaf unfolding of beech causes a strong negative response in radial growth (Dittmar et al. 2006). Additionally, cold tolerance varies between species. When the cold weather continues and frost stress increase, plants are injured irreparably, with destruction in fruit yield and quality, that is, a fruit tree frost. To trigger a fruit tree frost, two fundamental factors must be linked: the first is the cold weather conditions, including the starting/ending time, duration, and degrees of cold. The second is the plants' characteristics against cold, which always differs based upon the target fruit tree species and its phenological phase (Yang et al. 2016(Yang et al. , 2020. To explore the cold weather conditions that cause a frost disaster, studies on the frost effect on fruit trees have been primarily focused on environment-controlled experiments (Hatice et al. 2019), designing specific experiments by controlling cold environment to examine the mechanisms of frost effect and the relationships with yield and quality factors. For example, flower bud frost tolerance of several Italian apricot genotypes was assessed with an artificial frost treatment (− 4 °C) at dormant and swelling phenological stages (Viti et al. 2010). Young leaves of temperate trees have been shown to resist to temperatures below between − 8 and − 3 °C in the laboratory (Lenz et al. 2013;Vitasse et al. 2014). Through controlled experiments, the impacts of low temperatures on crop or fruit trees can be directly observed and detected, following the strategy of changing one element while others are stably maintained.
Largely, attention been focused on the spatiotemporal patterns of meteorological frost and their influences on crops (Xiao et al. 2018) in regional assessment. Long series of historical meteorological and yield data have generally been combined to explore the negative impact of frost on crop and fruit trees, affiliating its spatiotemporal distribution and risk changes. For example, Xiao et al. (2018) analyzed frost risk from a meteorological perspective based on the relationship between meteorological factors and yield changes. Relationships between crop or fruit productivity and low temperature indices can provide basic information on the frequency and magnitude of extremes cold weather conditions and their influence on agricultural or forests, but for timely monitoring or warning, pre-and during frost damage are of more importance because the occurrence of agro-/forest frost disasters is always attributed to certain extreme cold processes. So, evaluating the extent of the process event damage is of more importance.
Apricot (Prunus armeniaca L.), the naturally dominant fresh apricot tree species of the warm temperate zone in China, has suffered late spring frost on the flowers and unripe fruits in orchards, which can dramatically impact apricot production (Ozkan et al. 2018), since it negatively affects the biomass, reproduction, fruit growth, and ultimately the yield and fruit quality. In this paper, a processbased apricot frost model is proposed that could provide timely, accurate monitoring and warning of an apricot frost disaster based on meteorological method throughout the flowering season (from mid-March to late-April), when cold weather occurs in late spring. The main goal is to statistically characterize apricot frost disasters in the China warm temperate zone and to develop a process-based indicator for timely monitoring and warning of apricot frost disaster, with the ultimate goal of facilitating better orchard management to mitigate the effect of disaster weather.

Study area
The warm temperate zone of China is a major fresh apricot production area located in Northern China between 24° ~ 42°N and 125° ~ 104°E that includes Hebei, Henan, Shandong, Shanxi, Shaanxi, Beijing, and Tianjin provinces, as well as the areas of east of Lanzhou (Gansu province), south of Shenyang (Liaoning province), South of Ningxia, and northern part of Anhui and Jiangsu provinces (Fig. 1). Fresh apricots account for more than 50% of the country's total output in this region (Zhang and Zhang 2003). The unique geographical location makes it vulnerable to the extreme cold surges from eastern Siberia or eastern Mongolia. The cold air goes south via North China, resulting in frost damage in winter and early spring (Ding and Krishnamurti 1987;Wang 2018;Ding et al. 2021). Severe cold weather processes have been witnessed frequently in this region, resulting in severe damage to the public and the economy in recent decades. For example, during the early spring period in 2018, an extreme cold weather process assaulted the north Hebei province (located in the north part of the warm temperate zone), with a temperature decrease of more than 14 °C, dropping from 8.1 to − 5.9 °C. This resulted in huge areas of crops and fruit trees, including apricot, suffering from late spring frost, and it caused an economic loss of more than 2 billion dollars (Beijing Climate Center 2018).

Meteorology and disaster records
Datasets including meteorological data and disaster records were used to construct apricot frost samples, facilitating the identification of indicators of apricot frost in the flowering phase in this study. Meteorological data from 150 weather stations were obtained from the National Meteorological Information Centre, China Meteorological Administration (NMIC, CMA), including daily minimum temperature datasets from 1981 to 2018. Apricot frost disaster records can be derived in the China Meteorological Disasters Book (Hebei, Shandong, Henan, Shaanxi, Shanxi, Gansu, Ningxia, Beijing, Tianjin, Jiangsu, Anhui), the Yearbook of Meteorological Disasters in China, forest and fruit disasters surveys, county-based disaster records for the fruit industry, and relevant media reports. Apricot frost disaster records include the time, location, and the destruction of apricot flowers covering the period from 1981 to 2018, clearly recorded with the late spring frost injury time and disaster occurrence areas.

Derivation of apricot frost disaster-causing factors
Frost is mainly induced by the cold weather process, which can be derived from (1) radiation frost, that is, surface energy continuously radiates to the atmosphere, during calm winds, clear skies, and a temperature inversion; (2) advection frost, that is, cold air advection to a region during moderate-to-high winds, overcast skies, without inversion; and (3) mixed type of both conditions (Wang et al. 2019).
The daily minimum temperature, low temperature duration, diurnal temperature range, and accumulated harmful temperature should always be used as disaster-causing factors for the analysis of the three types of frost. The intensity and duration of disaster weather determines whether it can trigger plant injury as was discovered by previous studies (Yang et al. 2016). To demonstrate a frost process on apricot, the frost duration days (D) and the accumulated harmful temperature ( T cum ) were chosen to demonstrate a process-based apricot frost (as showed in Fig. 2), which can be defined as f (D, T cum ).

Derivation of T cum and D
Relevant frost trigger thresholds, optimum thermal environment, and the resistance abilities of fruit trees vary with the development phases of the species (Álvaro et al. 2020). Flowering or leaf-out of temperate trees can resist temperatures between − 8 to − 3 °C in experimental work or laboratory studies (László et al. 2019;Hatice et al. 2019). Low temperatures (− 4 °C) were applied at full flowering to select optimum apricot genotypes (Prunus armeniaca L.) with high resistance to cold in the Cappadocia region (Hatice et al. 2019). However, during clear and windless nights, temperatures measured at 2-m height under the sheltered conditions of a Stevenson screen were established to be 4 to 8 °C higher than those in plant tissues (Ducrey 1998). Temperatures below 0 °C, as recorded in a standard weather station, can potentially be used in frost risk analysis on leaves, flowers, and young fruits, for the mismatch of temperatures between plant tissues and weather stations (Yann et al. 2018). Commonly, a measured air temperature The theoretical diagram of the apricot frost disastercausing factors for a processbased apricot frost disaster that is near 0 °C is considered a frost trigger threshold for cultivated trees in spring (Ducrey 1998;Perraudin and Fellay 1975). So, 0 °C was chosen as the potential temperature threshold demonstrated as T thr , and T cum was used as the ultimate determination to identify an apricot frost. T cum in an apricot frost was calculated as follows: Here, T cum is the accumulated harmful temperature in the apricot frost; T min is the daily minimum temperature below T thr . T thr is a threshold temperature, that is, 0 °C. D is the consecutive days that T min < T thr .

Historical apricot frost process representation
To represent historical apricot frost processes in this study, times and locations of historical apricot frost disaster records, as well as the daily minimum temperature data, were firstly integrated, and afterwards, D and T cum in the apricot frost process were rechecked and calculated according to Formula (1). Extreme cold weather processes causing apricot frosts were determined after considering the D in a catchment. We took an apricot frost disaster in early April 1993 in Changzhi county (Shanxi province) as an example. Records describe this apricot frost disaster as, "At the beginning of April 1993, cold weather condition caused apricot, peach and pear petals fell off in Changzhi, Shanxi province." Daily T min data were extracted from the Changzhi Meteorological Station, with T min < 0 °C from April 7 to April 11, with T min which were − 1.8, − 3.6, − 2.2, − 5.7, and − 2.9 °C. A historical apricot frost process dataset was built, including D and T cum , that is, D (5 days) − T cum (− 16.2 °C). According to this method, 202 samples were represented. We randomly chose 10% of disaster samples for frost identification indicator validation, while 90% of the disaster samples were used for indicator construction. Detailed information of disaster samples for apricot frost are shown in Table 1.

Identification of indicators for an apricot frost
Disaster sample sequence distribution fitting, interval estimation, and other methods to acquire the disaster population characteristics and construct agricultural meteorology disaster indicators have been applied and confirmed in the construction of disaster indicators such as agricultural floods (Yang et al. 2016), waterlogging, drought (Wu et al. 2018), and heat damage (Yang et al. 2020). For apricot frost disasters of different duration (in days), the threshold defines, for "D-T cum " combinations, the amount of T cum likely to be triggered for the same D for apricot frost processes. In this study, characteristics of disaster population can be attached through the probability density functions or cumulative probabilities of T cum amount in each of the duration datasets.
Afterwards, sample coverage rates (SCRs) were calculated for different D to identify the most suitable indicators, considering the distribution of disaster samples in the disaster population probabilities.

Distribution fitting test of T cum sets
Normal, exponential, and uniform distributions are commonly adopted in the expression of disaster characteristics, because of the succinct parameters and simple algorithm. So, we chose the three discrete probability distributions of T cum amount in different D datasets as candidate distributions, representing the historical apricot frost processes. Kolmogorov-Smirnov (K-S) was applied for the goodness of fit testing of T cum . Kolmogorov-Smirnov (K-S) is a test method for comparing a frequency distribution and a theoretical distribution, or the value distributions of two observations. The null hypotheses of T cum with a theoretical distribution function in each D were tested with statistical analyses of maximum difference between the empirical and theoretical distribution functions (that is, normal, exponential, and uniform). The null hypothesis is rejected at a given significance level if the test statistic exceeds a critical value (Yang et al. 2016).
Here, D n is a random variable with distribution dependent on n; D n,a is a critical value at the level of significance.

Identification of the trigger value of T cum for different D
The ideal threshold can express most of the disaster population to ensure the accuracy of identification, while reflecting the concentration of independent samples to avoid misjudgement in non-disaster populations. The most suitable trigger value or disaster indicator can be derived from SCRs. Firstly, inverse function values of the best fitting of T cum sets were calculated with 5% step size. Secondly, SCRs were calculated under the inverse function values, and we compared and chose the upper limit with the biggest SCR slope.
Here, SCR i is the coverage of disaster samples at the Dth duration days; n i is the number of historical disaster samples whose T cum reaches the cumulative probability of the overall disaster population with 5% step size; N i is the total historical disaster sample at the Dth duration days.

Reserve independent sample test
The rationality of the threshold for apricot frost was verified by comparing the disaster occurrence consistency between the indicator-based results and historical documentation, using the reserved independent apricot frost samples, as showed in part 2.4 and Table 1.

Typical process tracking
An extreme snowfall process and strong wind cooling weather occurred in Northern Hebei in 2018, as a result of which there was frost damage to apricot trees in the northern area. The frost process of apricot in flowering is tracked in this research, through the occurring points and intensity demonstrated in D and T cum , based on the indicators constructed above.

f (D, T cum )characteristics in the historical apricot frost
The percentage of duration days for apricot frost process (D) was calculated based on the disaster samples. As shown in Fig. 3, the minimum value of D is 1, while the maximum is 6 days in the warm temperate zone of apricot in China. A 3-day process was detected as having the highest possibility of apricot frost, followed by a 4-day process, with 23.60% and 23.03% of the frost process continued for 3 days and 4 days, respectively. 7.61% and 6.52% apricot frost samples lasted for 5 and 6 days, respectively. Considering the cold weather condition and the phenological characteristics that the flowering period of apricot trees generally lasts 7-20 days, 5-and 6-day frost processes will be discussed in a unified manner for the convenience of application. Thus, the characteristics of (D, T cum ) were represented as in Fig. 4, and the information of T cum in 1, 2, 3, 4, and ≥ 5 days for apricot frost samples are shown in Table 2. The average T cum s were − 2.6, − 5.5, − 11.9, − 17.0, and − 22.8 °C for 1, 2, 3, 4, and ≥ 5 days processes, respectively.

Identification of an apricot frost process
Results of K-S tests showed that four datasets of T cum series, that is, Ds 1, 2, 4, and ≥ 5, followed normal distribution with K-S Sig: 0.189, 0.438, 0.147, and 0.398, respectively (Table 3). One dataset, that is, T cum series in 2 days, passed the uniform distribution significance test, with K-S Sig. 0.058, while none passed exponential distribution significance. Comparing the 3-candidate distribution fitting, normal distribution showed better performance than uniform or exponential distributions. Mathematical transformation was taken in T cum in 3-day frost processes, and the final K-S Sig. was 0.078, which passed the significance fitting test. Normal distribution of T cum s at different duration days of apricot frost is shown in Fig. 5. The inverse values of normal distribution fitting functions accumulative probability from 70 to 95% are shown as Table 4. The threshold for the apricot frost at each growth stage was identified according to the SCR at 5% cumulative probability step (Fig. 6). Comparing the distribution of SCR in 70 to 90%, the slope of SCR was highest from 75 to 80% for 1-to 3-day apricot frost processes, while SCR changed gently after 80%, meaning that the disaster samples for 1-to 3-day frost processes have a high degree of aggregation in range between 75 and 80% disaster population cumulative probability. Considering most independent disaster samples are detected in the inverse value range between 75 and 80% for 1-to 3-day frost processes, and the context of cumulative probability, an inverse value of 80% cumulative probability is more suitable as a potential threshold, for 80% of the disaster population can be detected in such T cum for 1-to 3-day frost processes. Inverse values of T cum at 80% normal cumulative probability were firstly calculated as potential thresholds for 1-day, 2-day, and 3-day frost processes, respectively. For instance, we took the threshold identification of 1-day frost: the inverse values of T cum were − 1.9, − 1.7, − 1.5, − 1.3, − 1.0, and − 0.6 °C for normal distribution fitting function at 0.7 to 0.95 cumulative probability at 5% step, meaning that the inverse values can identify 70%, 75%, 80%, 85%, 90%, and 95% of the disaster population for 1-day apricot frost processes. Considering the independent sample distribution and deserter population characteristics, the inverse value of 80% cumulative probability, that is, − 1.5 °C, was identified as the threshold for 1-day frost processes.
Change of SCR for 4 days and ≥ 5 days of apricot frost processes showed that more independent disaster samples can be detected with inverse values of higher normal cumulative probability (or lower T cum ). Looking back at apricot frost processes, we found that frosts can be detected in the preceding 3 days, meaning that the disaster processes can be timely recognized from the 1-3 days as soon as possible, although the process can persist long, and the continuous accumulation of T cum aggravates the degree of frost damage. For 4 day and ≥ 5 days of frost processes, the identification thresholds were firstly constructed by the inverse value of 80% cumulative probability, and then modified according to the results of thresholds of 1-to 3-day processes. For instance, we took the threshold identification of 4-day frost: the inverse value of 80% cumulative probability of T cum in 4 days of apricot frost processes was − 7.2 °C, which is lower than 4 times of 1-day frost threshold (− 1.5 °C), 2 times of 2-day frost threshold (− 2.9 °C), and the sum of thresholds for 1-day and 3-day processes (− 1.5 and − 4.4 °C). The minimum value of the above was taken as the threshold for 4 days of frost processes, that is, − 5.8 °C.
On that basis, thresholds for the T cum for apricot frost in 1, 2, 3, 4, and ≥ 5 days of apricot frost processes were determined, and indicators of apricot frost processes and their activity in disaster sample identification are shown in Table 5. The totally consistent ratio was more than 80% for disaster samples identification.

Validation of reserved independent samples
Eighteen random historical apricot frost samples were independently selected to validate the applicability of the proposed apricot frost indicators. Table 6 shows the validation results of the reserved independent samples according to the  apricot frost indicator constructed previously. All samples of 1-day, 2-day, and ≥ 5-day frost processes can be identified by the indicators, and the coincidence rate of the frost indicator-based is 100%. One sample in 3 days and 4 days of apricot frost processes failed in the indicator-based identification, with frost indicator accuracies of 75% and 80% for 3-day and 4-day apricot frost processes, respectively. Overall, the results calculated by the frost indicators were generally consistent with disaster records in historical documents, with 89.00% of indicator-based results completely consistent with historical records. Take the reserved independent samples in 2006 for instance. Apricot frosts were historically represented in Shanxi and Hebei provinces, with samples including Nangong (Hebei province), Taiyuan (Shanxi province), Yushe (Shanxi province), and Chengde (Hebei province) randomly selected for validation of the constructed frost indicators. T cum s that are − 3.0, − 6.5, − 14.5, and − 18.7 °C were detected for 1-day, 2-day, 3-day, and ≥ 5-day processes, lower than the temperature thresholds (that are − 1.5, − 2.9, − 4.4, and − 7.3 °C) respectively. Identification results were well consistent with the historical records in such reserved independent samples.  -45 -40 -35 -30 -25 -20 -15 -10 -5 0 Cumulative probability (°C) Normal type (-11.91, 8.93) Table 6), indicating that the apricot frost indicators can reasonably reflect actual apricot frosts in regional scales.

Typical process tracking
In early spring 2018, a cold weather process spread from north to south across the warm temperate zone, with a rare snow fall process and strong wind cooling weather. Part of the north region had a minimum temperature below 0 °C (Zhao et al. 2020). Extremely serious destruction of apricot trees was recorded in North Hebei province (located in the north part of the warm temperate zone) because of the cold weather-lead frost. T min s of April 2018 were extracted from 8 stations in North Hebei, and the average T min s were below 0 ℃ from April 3 to 9 (Fig. 7). Based on the indicator constructed previously, apricot frosts according to 8 stations were identified daily in North Hebei, as showed in Fig. 8. Two stations, accounting for 25% stations, detected suffered from apricot frost in 3 April. Apricot frost developed in April 4, with 6 stations identifying apricot frost, and this lasted through April 5 and 6. On April 7, all stations were identified to have suffered from apricot frost. The frost process alleviated gradually from 8 onward, with 88% (7 stations) and 38% (3 stations) detecting apricot frost.

Rationality of theory and method
The formation of crop frost is governed by the interacting effects of climatic conditions, topography, soil structure, frost tolerance of crop species, and field/orchard management (Mosedale et al. 2015;Susan et al. 2018;Ma et al. 2015), among which weather conditions are the most important frost trigger factor for causing a disaster (Wang et al. 2019). To investigate regional frost on crop or fruit trees, yield loss and the amount of temperature-related indexes were quantitatively combined, no matter whether the temperature-related indexes are based on meteorological data or remote sensing (Tao et al. 2017;Xiao et al. 2018). Among them, meteorological-based methods are recognized as the most convenient monitoring and are uniquely effective early warning tools (Shi et al. 2020).   Previous indicators of crop frost (such as accumulated frost days or accumulated frost-degree days) (Xiao et al. 2018) could represent frost stress or characteristics in crop planting seasons under climate change, based on various agricultural/meteorology index calculations during the growing stages, whereas the occurrence of a crop/ fruit frost attributing to cold weather conditions could be identified according to the process-based indicators. It is accepted that the occurrence of frost can be triggered by a single threshold, as has been demonstrated by previous studies on frost (Snyder 2000;Simões et al. 2015). For example, 285.5 K was recognized a damage indicator for plants in the tropics (Snyder 2000). However, such single threshold cannot characterize a cold weather process exactly. The unique effects of an agriculture disaster can be predicted based on historical disaster representation and process analysis (Yang et al. 2016;Wang et al. 2019;Yang et al. 2020). As duration and intensity are two basic factors used in the demonstration of disaster weather processes (Yang et al. 2016(Yang et al. , 2020, f (D, T cum ) was adopted to demonstrate process-based apricot frost event in this study, for D is the duration of a frost event and T cum is the intensity of the frost process. The construction process of apricot frost including D and T cum is mainly based on the historical disaster representation and re-analysis in stations. The disaster samples cover the spatial region of the study area and confirmed with long series sequence, which is more vital in the representation of T cum s in Ds. Statistical analysis, such as the optimal distribution fitting test and probability inverse function, are methods for retracing the occurrence and development processes of the apricot frost in North temperate zone, and such methods have been used in the agrometeorological disaster's identification, such as the identification of rice flood (Yang et al. 2016), heat (Yang et al. 2020), and maize chilling (Wang et al. 2019), with both scientific and regionally applicative. The results demonstrate that daily minimum temperature date-based theoretical model f D, T cum is possible in the monitoring apricot frost at flower stage, if real-time and forecast weather data are available.

Utilization of apricot frost indicators
The effect of frost on fruit yield and quality is a complex process and is variable, as temperature affects so many biological processes in plants, with different species in different phenological phases each having different responses (Rodrigo 2000; Dittmar et al. 2006;Vanoni et al. 2016;Hatice et al. 2019). Based on worldwide experimental studies, the plant tissue frost damage occurs when temperature falls below temperature 0 °C (Chen et al. 2014;Porter and Gawith 1999;Single 1966Single , 1984. Apricot has been recognized as a species that is robust to cold weather, but frost sensitive when flower buds have fulfilled the endodormancy with the beginning of active growth, and flower destruction occurs as a result of frost disaster (Hatice et al. 2019). The indicators identified in our study demonstrate how cold weather conditions accumulate to frost disaster weather for apricot production in regional scale. The use of the apricot frost indicators can serve as a direct meteorological method to assist in the identification of apricot frosts or processes. Nowadays, operational numerical weather prediction models provide long-term and short-term effective weather forecasts, which help guide the identification of extreme weather phenomena based on space regions and time scales (Susan et al. 2020). The process-based threshold in the identification of apricot frost is operational for its frost forecasting and warning under the given weather forecast productions. More information and measures for apricot frost prevention and mitigation can be implemented according to the indicator-based calculations and results, and these can inform discussions of how to face an apricot frost disaster with the development of weather warning and forecasts.

Uncertainties and limitations
Apricot frost occurrence depends on the climatic conditions, species, and stand ages. For example, the relationship between altitude and frost occurrence has been found in experiments (Laughlin 1982;Laughlin and Kalma 1987;Susan et al. 2018 and 2020) that daily T min was lower at higher altitudes in the absence of absorbed ground heat and thinner air. Apricot trees planted in the northward of a hill are likely to experience more harmful cold accumulation than their counterparts southward (Bao 2011). Eastern slope has been shown to have a higher frequency of frost occurrence than western, for the prolonged cooling hours due to early sunset on the eastern slopes and late sunset on the western slopes lead to heat retention on the ground, which is released at night (Susan et al. 2018). In addition, the degree of cold environment exposure in the context of climate change affects apricots' ability to resist frost. Plants frequently exposed to cold conditions may have evolved a greater capacity to conduct a life in such circumstance or to adapt to frost stress (Hatice et al. 2019). For example, frost stress has increased the frost resistance of apricot in Taihang Mountain (including West Shanxi, North Hebei), for meteorological cold weather in such areas has recurred more frequently than in the warm plain in North China. Therefore, it is necessary to continuously optimize and revise the apricot frost trigger thresholds in specific areas according to factors such as soil, terrain, orchard management, and frost resistance abilities. Overall, the apricot frost indicators constructed in this paper is universal in the main apricot-producing areas in the warm temperate zone in China, and they can provide a basis for targeted apricot frost monitoring and warning.
Owing to the complicated interacting effects of external factors, such as weather conditions, water and fertilizer conditions, and orchard management, on perennial fruit trees, it is very difficult to accurately predict loss in quality and yield that can be attributed solely to frost processes. Additionally, the impact of frost on trees is the result of the interaction between the timing of the event, i.e., full winter vs. active growing season, the low temperature weather processes, and phenological plant status (Marco et al. 2018;Yann et al. 2018;Emilia et al. 2019). Data of apricot phenology, especially the flowering data, is an important factor with regard to frost damage, which is outside of this study's scope. Under global warming, earlier leaf unfolding and flowering for perennial trees has been confirmed by biological observation simulation, as a response to a warmer winter. Obviously the most effective and reliable method to resist late spring frosts in apricots, thus avoiding frost damage, is late flowering (Hatice et al. 2019). Development of cold weather-resistant species in winter and spring is another strategy to prevent frost damage (Hatice et al. 2019).

Conclusion
Cold extremes are unavoidable, but the lessons learned from past experience can be used to reduce the damage they inflict. In this study, a process-based apricot frost model f (D, T cum ) were explored under different D based on the representation of historical apricot frost processes. Thresholds for the T cum for apricot frost in 1, 2, 3, 4, and ≥ 5 days of apricot frost processes were determined as − 1.5, − 2.9, − 4.4, − 5.8, and − 7.3 °C, respectively.
Given the intensification of climate extremes, improving and innovating apricot frost management measures based on such process-based identification indicators is essential to reduce agro-forest losses associated with frost disasters of apricot in the warm temperate zone. Our findings have important implications for government, orchard farmers, and agricultural insurance to take measures for apricot frost prevention and mitigation. With the continuous improvement of yield and quality data and the supplementation of disaster documents, as well as geological, topographical, and orchard managements, apricot frost trigger thresholds will be continuously optimized and revised in specific areas. Additionally, threshold-based classification of disaster evaluation level needs to be elaborated and enhanced, thereby creating a complete assessment of apricot frost risk in the main planting area of the warm temperate zone.