Introduction

Water and soil resources are indispensable and important for social and economic development and the ecological environment of a country and a region. These resources are the key strategic resources for the sustainable development of the regional social economy [24]. With the rapid changes in the population and economy, the contradiction between the supply and demand of water and soil resources is becoming increasingly acute. This has caused a series of ecological and environmental problems, such as soil erosion, water pollution and water level decrease, which have led to the degradation of regional agricultural water and soil resources. The average annual precipitation in the Jiansanjiang administration is 383.5–886.1 mm [15, 29]. Because a large quantity of groundwater is pumped for rice irrigation, the groundwater level has decreased annually; this problem is exacerbated by the excessive usage of chemical fertilizers and pesticides because irrigation water infiltrates into the soil and carries pollutants and pollutes the groundwater. This process not only harms the drinking water safety of humans and animals but also destroys the ecosystem security [11]. Therefore, it is urgent to study the resilience of the water and soil resource system of the Jiansanjiang administration. Further research on the resilience of regional agricultural water and soil resource systems is important for the scientific management of water and soil resources.

Research on resilience locally and abroad [10] mainly focuses on the application of ecological systems [18, 21], disaster systems [12, 13], sustainable development and utilization of land and economic fields; research on agricultural water and soil resources is rare [30]. Research on the resilience of water and soil is still at the conceptual level and involves case analysis. Research on resilience diagnosis is still weak, and research on the mechanism and influencing factors of resilience is scarce. The ability of resilience diagnosis results to guide practice should be strengthened. Resilience research includes mathematical statistics [3], modeling [19] and experimental methods [2]. However, traditional methods have some defects and are prone to discretization of evaluation grades, unreasonable results and loss of subjective information. Previous studies have optimized the projection index function using real-coded accelerating genetic algorithms (RAGA) [8], particle swarm optimization (PSO) [5], the immune evolutionary algorithm (IEA) [34] and the simulated genetic annealing algorithm (SGAA) [4, 16]. However, these algorithms are uncertain, easily fall into the local optimum and appear to undergo premature convergence, which leads to unwarranted and unreliable results. Therefore, this paper proposes the firefly algorithm (FA) to optimize the projection pursuit model and overcome the defect of being caught in partial optima, which would make finding the global optimal solution easy. Simultaneously, a projection pursuit model based on the genetic algorithm (GA) is established to provide a systematic and reasonable research model for the study of water and soil resource system resilience.

The objectives of this paper are to apply the FA-PP to evaluate the resilience of the agricultural water and soil resource system of the Jiansanjiang administration in Heilongjiang Province, China, to reveal the local water and soil resource system resilience, to analyze the main driving factor influencing the water and soil resource resilience and to propose suggestions for improvement.

Materials and Methods

Study Area

The Jiansanjiang administration is located in northeastern Heilongjiang Province, China, between the Heilongjiang River to the north and the Ussuri River to the south (latitude 46°50′–48°13′ and longitude 132°32′–134°33′) [14]. The topography from southwest to northeast slopes up and down, is 152.9 km long from east to west and 152.8 km wide from north to south and has an overall altitude between 101.1 and 625.6 m. The total area is 1.24 km2. There are 15 farms in the Jiansanjiang administration; the location and administrative divisions are shown in Fig. 1. There are well-developed river systems and abundant water resources in this area. [32].

Fig. 1
figure 1

Location of the Jiansanjiang administration in Heilongjiang Province, China

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Data Indexing

According to the scientific and objective principles of index screening, this study used the driving force-pressure-state-impact-response (DPSIR) model to construct the evaluation index system of regional agricultural water and soil resource system resilience. The DPSIR model represents a natural system that is divided into five elements: driving force, pressure, state, influence and response. Each element is separated into several indexes [1]. The “driving force” is the probable cause of the change of the system. The “stress” is concentrated on the direct pressure of human behavior on the natural environmental resources. The “state” is a condition under various stress factors. The “influence” refers to the impact of the state of the system on human health and socioeconomic structure. The “response” process indicates the measures that human beings take to promote sustainable development, such as improving resource use efficiency, reducing pollution and increasing investment.

This study constructed an evaluation index system of water and soil resource resilience using three levels. The first layer of the index system is the target layer, which is the level of the water and soil resource resilience of the Jiansanjiang administration. The second layer is the influencing factor layer, which includes the five factors of driving force, pressure, state, impact and response. The third layer is the indicator, which contains 13 index factors for the five influencing factors, as shown in Table 1.

Table 1 Definition of the water and soil resources system resilience evaluation index
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Data Sources

In addition to the X10 (groundwater quality F value) evaluation index, 12 evaluation indexes were collected from the “Jiansanjiang Farm Reclamation Statistical Yearbook” and the “Jiansanjiang Water Resources Annual Report” in 2014.

Evaluation index X10 (groundwater quality F value) was determined in the field by harvesting water samples from the irrigation water of 15 representative farms. At each sampling point, three water samples were collected, and from each water sample, the pH, the CODMn(chemical oxygen demand), and the NH3-N, NO3−-N, Cl−, F−, SO42−, Fe and Mn contents were determined. The pH was measured using a PHS-3C pH meter, and the other parameters were determined using a Hach DR2800 portable spectrophotometer.

Projection Pursuit Model

The projection pursuit model is not limited to problem size and data structure, can achieve high-dimensional data dimension reduction and has low-dimensional space for data on a unified evaluation [7]. For the processing of complex multi-dimensional data and analysis of high-dimensional observation data, especially the distribution of non-normal high-dimensional data with complex morphology, the projection pursuit model can provide a variety of analyses with strong anti-interference ability, good robustness and high hit rate, among other features [17].

  1. 1.

    Normalize the evaluation index

The normalization processing of the sample evaluation index set. Supposing that sample sets are \( \left\{ {x^{*} \left( {i, j} \right)|i = 1,2, \ldots ,n;j = 1,2, \ldots ,p} \right\} \), where \( x^{*} \left( {i,j} \right) \) is the parameter value \( j \) of sample \( i \), and \( n \) and \( p \) are the capacities of sample and quantity of index, respectively. Aiming to eliminate dimension and unify range of variation, this equation can be normalized as follows:

$$ x\left( {i,j} \right) = \frac{{x^{*} \left( {i,j} \right) - x_{\hbox{min} } \left( j \right)}}{{x_{\hbox{max} } \left( j \right) - x_{\hbox{min} } \left( j \right)}} $$
(1)
$$ x\left( {i,j} \right) = \frac{{x_{\hbox{max} \left( j \right)} -x^{*} \left( {i,j} \right)}}{{x_{\hbox{max} \left( j \right)} -x_{\hbox{min} \left( j \right)} }} $$
(2)

where \( x_{\hbox{min} } \left( j \right) \) and \( x_{\hbox{max} } \left( j \right) \) are the maximum and the minimum values of \( j \), respectively, and \( x\left( {i,j} \right) \) is the normalized sequence.

  1. 2.

    Construct a projection function

Construct the projection index function \( Q\left( a \right) \). The main idea is to compress the \( p d \) data into the one-dimensional projection value \( a = \left( {a\left( 1 \right),a\left( 2 \right), \ldots ,a\left( p \right)} \right) \) in the direction \( z\left( i \right) \).

$$ z\left( i \right) = \mathop \sum \limits_{j = 1}^{p} a\left( j \right)x\left( {i,j} \right) $$
(3)

where \( a \) is the unit length of the vector. The characteristics of \( z\left( i \right) \) are that local projection points should gather as much as possible. The projection index function could be expressed as follows:

$$ Q\left( a \right) = S_{z} D_{z} $$
(4)

where \( S_{z} \) is the standard deviation of projection value \( z\left( i \right) \), and \( D_{z} \) is the local density of the projection value \( z\left( i \right) \).

$$ S_{z} = \sqrt {\mathop \sum \limits_{i = 1}^{n} \frac{{\left( {z\left( i \right) - E_{z} } \right)^{2} }}{n - 1}} $$
(5)
$$ D_{Z} = \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = 1}^{n} \left( {R - r_{ij} } \right)\left( t \right) $$
(6)
  1. 3.

    Optimize the projection function

The maximal objective function is as follows:

$$ \hbox{max} :Q\left( a \right) = S_{z} D_{z} $$
(7)

The constraint condition is as follows:

$$ {\text{s}}.{\text{t:}}\quad \mathop \sum \limits_{j = 1}^{p} a^{2} \left( j \right) = 1 $$
(8)
  1. 4.

    Evaluate the program

The best projection direction \( a \) in Eq. (3) will be used to obtain the program projection value \( z^{*} \left( i \right) \); for \( z^{*} \left( i \right) \), a greater value is better.

Firefly Algorithm (FA)

The FA is a collection of intelligent algorithms based on natural enlightenment that was proposed by Cambridge University scholar Yang Xinshe [27] in 2008. A random optimization algorithm is built by simulating the population behavior of fireflies. At present, few scholars have applied the FA to evaluate agricultural water and soil resource resilience. In this study, the FA is used to establish the projection pursuit model (FA-PP) to evaluate the regional agricultural water and soil resource system resilience.

The definition of the author is introduced from a mathematical point of view of the FA for an explanatory description [25].

The relative brightness of the fireflies is as follows:

$$ I\left( l \right) = I_{0} \cdot e^{{ - \gamma l_{{ij^{2} }} }} $$
(9)

where \( I_{0} \) is the maximum fluorescence brightness of the firefly, i.e., its own (\( l \) = 0) brightness; \( \gamma \) represents the increase in distance and the degree of attenuation of the absorption of the medium as the intensity absorption coefficient, which is a constant; and \( l_{ij} \) represents all fireflies between the Cartesian points, which is expressed as follows:

$$ l_{ij} = x_{i} - x_{j} = \sqrt {\mathop \sum \limits_{k = 1}^{d} \left( {x_{i,k} - x_{j,k} } \right)^{2} } $$
(10)

where \( x_{i} {\text{and }}x_{j} \) are the spatial positions of fireflies \( i \) and \( j \), respectively; \( d \) is the search space dimension; and \( x_{i,k} {\text{and }}x_{j,k} \) are the kth component of fireflies \( i \) and \( j \) in the \( d \)-dimensional space.

The attractiveness of fireflies is

$$ \beta \left( l \right) = \beta_{0} \cdot e^{{ - \gamma l_{{ij^{2} }} }} $$
(11)

where \( \beta_{0} \) is the attractivity of the light source (\( l \) = 0).

Firefly \( i \) is moved by the larger firefly \( j \) to update its position:

$$ x_{i} = x_{i} + \beta \left( {l_{ij} } \right) \cdot \left( {x_{j} - x_{i} } \right) +\, {\text{alpha}}\left( {{\text{rand}} - 0.5} \right) $$
(12)

where \( {\text{alpha}} \) is the step size factor, which is usually assumed to be constant on [0,1]; and rand obeys a uniform distribution of random factors on [0,1].

The specific optimization algorithm steps are as follows:

  1. 1.

    Initialize the basic parameters of the algorithm; set the number of fireflies \( m \), the light intensity absorption coefficient \( \gamma \), the maximum attraction \( \beta_{0} \), the step factor \( \alpha \), the search accuracy \( \varepsilon \) or the maximum number of iterations \( \hbox{max} N \).

  2. (2)

    Randomly initialize the location of the firefly and calculate the firefly’s objective function values as the respective maximum fluorescent brightness \( I_{0} \).

  3. (3)

    Calculate the relative brightness \( I \) and attractiveness \( \beta \) of the fireflies in the population and determine the direction of movement of the fireflies according to the comparative brightness. Update the space position of the fireflies and randomize the fireflies at the optimal position. According to the location of the updated fireflies, re-calculate the brightness of fireflies.

  4. 4.

    If the search accuracy meets the requirements or reaches the maximum number of searches, go to step 5. Otherwise, the number of searches increases by 1; go to step 3 and begin the next search.

  5. 5.

    Output global extreme points and optimal individual values.

The flowchart of the FA is shown in Fig. 2.

Fig. 2
figure 2

Flowchart of the firefly algorithm (FA)

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The FA parameter values were set according to the algorithm documentation and related practical application [20, 26, 28]. The standard genetic algorithm (GA) was used for comparison. Algorithmic descriptions and parameter values are shown in Table 2, and MATLAB 2015B programming was used [6, 9, 23, 31].

Table 2 Algorithm description and parameter settings
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Results

Reliability Analysis of Evaluation System

Reliability analysis is used to evaluate the stability or reliability of the index system as a measurement tool. Cronbach’s α is used to analyze the reliability of the index system. The formula [33] is as follows:

$$ \alpha = \frac{{k\left( {\overline{\text{cov}} /\overline{\text{var}} } \right)}}{{1 + \left( {k - 1} \right)\left( {\overline{\text{cov}} /\overline{\text{var}} } \right)}} $$
(13)

where α is the reliability coefficient. A larger α indicates a greater stability and reliability of the evaluation index system.

The 15 farms’ water and soil resource resilience index data from the 2014 Jiansanjiang administration (see in Table 1) were normalized and then analyzed using SPSS software to perform a reliability analysis; the results are provided in Table 3.

Table 3 Indicator confidence analysis results and Hotelling’s T2 test results
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As shown in Table 3, Cronbach’s α of the unspecialized index in 2014 was only 0.03, and the reliability of the index was poor. However, Cronbach’s α of the standardized index reached 0.954, and the reliability of each index was elevated. For the standardization of the impact of indicators of credibility, objective evaluation plays an important part. The data of the 13 indicators in 2014 were all tested by Hotelling’s T2 test, and the equality of the average score among the indicators was established. Therefore, the evaluation index system of the water and soil resource resilience of the Jiansanjiang administration, after being standardized, had highly credible average scores with good agreement among the projects and can objectively evaluate the water and soil resources resilience of Jiansanjiang.

Evaluation Criteria

Combined with Table 1, the classification criteria were determined according to previous research and assessments and the local situation [22], as shown in Table 4.

Table 4 Level of the evaluation index system of water and soil resource system resilience
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Resilience Level

Depending on the evaluation index system and evaluation criteria (shown in Table 3), each farm has 13 indicators belonging to 13-dimensional data. The boundary value of each grade interval is considered a sample value to form a sample set. The objective function is optimized by the FA and the GA, after which the corresponding classification standard of the restoring force grade is obtained, as shown in Table 5.

Table 5 Computational values of the resilience level
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Spatial Distribution of the Water and Soil System Resilience

The best projection direction of the projection pursuit model based on the FA was found when substituting \( a \) into formula (3) when \( a \) = [0.1047,0.4124,0.0972,0.3807,0.3325,0.1124,0.4386,0.3617,0.2322,0.4118,0.3562,0.0882,0.4105]. These results show that the projection value (score value) of the Jiansanjiang administration’s water and soil resource system resilience is Z = [2.00553,1.07020,2.84385,1.25351,0.88836,0.70353,1.74081,1.22897,0.97056,0.63705,1.58045,0.69064,0.47243,0.91923, 0.83498]. The projection values from largest to smallest can be used to build the 2014 Jiansanjiang administration’s farms water and soil resource resilience order, as shown in Table 6.

Table 6 Simulated values and corresponding grades based on the FA-PP
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Combined with Table 5, the results in Table 6 can be used to draw the Jiansanjiang administration’s water and soil resource resilience spatial distribution map, as shown in Fig. 3.

Fig. 3
figure 3

Spatial distribution map of the water and soil resource resilience in the Jiansanjiang administration according to the FA-PP

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The best projection direction of the projection pursuit model based on the GA was found when substituting \( a \) into formula (3) when a = [0.0053,0.3995,0.1574,0.3648,0.3618,0.0952,0.4051,0.2529,0.0706,0.3953,0.3850,0.0964,0.3899]. These results show that the projection value (score value) of the Jiansanjiang administration’s water and soil resource system resilience is Z = [1.91944,0.92890,2.71293,1.16236,0.81761,0.62149,1.57380,1.08199,0.79520,0.61799,1.35940,0.49873,0.26137,0.65279,0.60552]. The projection values from largest to smallest can be used to build the 2014 Jiansanjiang administration’s farm water and soil resource resilience order, as shown in Table 7.

Table 7 Simulated values and corresponding grades based on GA-PP
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Combined with Table 5, the results presented in Table 7 can be used to draw the Jiansanjiang administration’s water and soil resource resilience spatial distribution map, as shown in Fig. 4.

Fig. 4
figure 4

Spatial distribution map of the water and soil resource resilience in the Jiansanjiang administration according to the GA-PP

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Discussion

According to the results presented in Fig. 3 and Fig. 4, the resilience of water and soil resources gradually increases from east to west. When using the GA and FA to calculate the best projection direction value, the main indicators affecting the resilience of water and soil resources are the effective irrigation area (ha) of each farm, each farm’s current GDP (million Yuan) and the groundwater quality F value.

According to the collected data, “Jiansanjiang Farm Reclamation Statistical Yearbook” and the “Jiansanjiang Water Resources Annual Report” in 2014, in addition to Tables 4, 6 and 7, the effective irrigation area of Qixing farm (III level) is 67,365 ha, the current GDP is 2414 million Yuan, and the groundwater quality F value is 2.20; these three values are the best of the 15 farms. The reason for these results is that Qixing farm is the official capital of the location in the Jiansanjiang administration and has the highest degree of agricultural mechanization. Thus, although Qixing farm’s rainfall was 454 mm in 2014 and ranked 14th out of the 15 farms, its amount of natural resources is less than that of the other farms, its groundwater quality F is 1.38, the water and soil erosion control rate is 83%, the groundwater over-exploitation rate is 11%, and the water and soil resilience is the strongest of the 15 farms. The results of the FA-PP model and GA-PP model agree with the actual situation.

In contrast, Yaluhe farm’s (I level) effective irrigation area is 27,087 ha, the current GDP is 681 million Yuan, and the groundwater quality F value is 7.33; these three values are the lowest of the 15 farms. However, Yaluhe farm’s rainfall was 638.9 mm in 2014, which is the highest of the 15 farms, indicating that in the presence of abundant water and soil resources, the Yaluhe farm’s water and soil resource resilience is not high; the results show that Yaluhe farm’s water and soil resilience capacity is the lowest of the 15 farms. This evaluation of resilience is the same as that of the I level farms, Qianshao farm, Nongjiang farm and Honghe farm, which have the same problems; that is, in the presence of abundant water and soil resources, the effective irrigation area and current GDP are ranked low. These farms have not fully exploited natural resources, resulting in water and soil erosion and desolation. Under the premise of undermining the natural environment, that is, having a high groundwater quality F value, the current GDP, GDP per capita, the tertiary industry output value and other economic indicators are still lower than those of other farms.

According to the results, water and soil resource resilience is closely related to economic indicators. If a farm has strong resilience, its economic indicators are among the best, such as Qixing farm, 859 farm and Qianjin farm. However, if a farm has weak resilience, such as Yaluhe farm, Qianshao farm and Honghe farm, its economic indicators may be poor. This suggests that farms with strong resilience not only maintain ecological balance but also gain the greatest economic benefit with the lowest level of damage to water and soil resources.

The FA-PP model and GA-PP model results are basically the same, which confirms the FA-PP model accuracy and validates testing the method. However, compared with the GA-PP model, the FA-PP model has its advantages.

First, according to the evaluation results, the resilience of Qianshao farm and Honghe farm is I level under the FA-PP and GA-PP models. However, under the FA-PP model, the resilience of Honghe farm was stronger than that of Qianshao farm, and that of Qianshao farm in the GA-PP model was stronger than that of Honghe farm. According to the collected data, “Jiansanjiang Farm Reclamation Statistical Yearbook” and the “Jiansanjiang Water Resources Annual Report” in 2014, in addition to Table 4, the effective irrigation area of Honghe farm (I level) is 37,824 ha, the current GDP is 924 million Yuan, and the rainfall in 2014 was 637.2 mm. The effective irrigation area of Qianshao farm is 36,666 ha, the current GDP is 840.64 million Yuan, and the rainfall in 2014 was 506.2 mm. When comparing Honghe farm and Qianshao farm, the effective irrigation area, current GDP and rainfall of Honghe farm were greater than those of Qianshao farm. However, while the GA-PP indicated that Qianshao farm was more resilient than Honghe farm, the FA-PP model was more suitable for evaluating water and soil resource resilience, and its evaluation results were similar to the actual situation.

Second, according to the evaluation results, the resilience of Qinglongshan farm under the FA-PP model was I level, and the resilience of Nongjiang farm was II level. Under the GA-PP model, the resilience of Qinglongshan farm was II level, and the resilience of Nongjiang farm was I level. According to the collected data, “Jiansanjiang Farm Reclamation Statistical Yearbook” and the “Jiansanjiang Water Resources Annual Report” in 2014, in addition to Table 4, we can see that the effective irrigation area of Nongjiang farm is 3903 ha, the current GDP is 968.56 million Yuan, and the rainfall in 2014 was 538.1 mm. The effective irrigation area of Qinglongshan farm is 34,407 ha, the current GDP is 849.86 million Yuan, and the rainfall in 2014 was 422.9 mm. The effective irrigation area and the current GDP were the greatest contributors to the driving force of resilience, and those of Nongjiang farm were higher than those of Qinglongshan farm; the rainfall in 2014 of Nongjiang farm was far greater than that of Qinglongshan farm. According to Fig. 1, Nongjiang farm had a large amount of river recharge and rich groundwater resources, which are characteristic of the overall terrain for the basin. Therefore, Nongjiang farm’s resilience should be stronger than that of Qinglongshan farm. However, the results of the GA-PP model showed that the resilience of Qinglongshan farm was stronger than that of Nongjiang farm, and the FA-PP model showed no difference in the evaluation of resilience. This demonstrates the superiority of the FA-PP model.

Finally, the convergence iteration of the GA was 142 times, and the convergence iteration of the FA was 38 times. The FA had characteristics of few convergence iterations and rapid speed, which were much better than the characteristics of the GA. The GA was built on the recommended parameters of the relevant literature, which may lead to poor accuracy. The FA has a low dependence on parameter selection, can prevent falling into the local optimum and has high searching efficiency.

Conclusions

  1. 1.

    In this paper, the DPSIR model is established to construct the evaluation index system of water and soil resource resilience, and the FA-PP evaluation model is introduced. The evaluation results show that the resilience of soil and water resources in 15 farms is worrying, and the following suggestions are put forward: increase the level of agricultural mechanization, achieve effective irrigation of farmland and reduce the amount of fertilizer applied.

  2. 2.

    Compared with the GA-PP model, the FA-PP model is more suitable for evaluating the regional agricultural water and soil resource system resilience, and the evaluation results agree better with the actual situation. In the FA-PP model, the FA has a few parameters, the search efficiency is high, and the dependence of the parameter selection is low, but the FA theory and mathematical expression need to be further developed. The specific parameter setting for the different problems also requires further investigation; the FA performance can be improved by cross-combining between different algorithms.

  3. 3.

    This paper only analyzed the characteristics of the water and soil resource resilience of the Jiansanjiang administration in 2014. Therefore, it is impossible to analyze the interannual differences and the evolution of water and soil resource restoration. In addition, this paper only considered the groundwater quality F value in the soil and the water quality index and, due to limited conditions, could not obtain surface water and soil quality monitoring data. The results of this research will have a certain impact. In the future, with an improved farmland environmental monitoring system and increased data sharing, the above problems could be further researched.