A Quantitative Method for Long-term Water Erosion Impacts on Productivity with a Lack of Field Experiments: a Case Study in Huaihe Watershed, China

Water erosion causes reduced farmland productivity, and with a longer period of cultivation, agricultural productivity becomes increasingly vulnerable. The vulnerability of farmland productivity needs assessment due to long-term water erosion. The key to quantitative assessment is to propose a quantitative method with water loss scenarios to calculate productivity losses due to long-term water erosion. This study uses the agricultural policy environmental extender (APEX) model and the global hydrological watershed unit and selects the Huaihe River watershed as a case study to describe the methodology. An erosion-variable control method considering soil and water conservation measure scenarios was used to study the relationship between long-term erosion and productivity losses and to fit with 3D surface (to come up with three elements, which are time, the cumulative amount of water erosion and productivity losses) to measure long-term water erosion. Results showed that: (1) the 3D surfaces fit significantly well; fitting by the 3D surface can more accurately reflect the impact of long-term water erosion on productivity than fitting by the 2D curve (to come up with two elements, which are water erosion and productivity losses); (2) the cumulative loss surface can reflect differences in productivity loss caused by long-term water erosion.


Introduction
The soil controls the hydrological, erosional, biological, ecological and geochemical cycles and is also important for humankind as the source of goods, resources and services [1][2][3][4].Approximately 80% of the world's agricultural land suffers moderate to severe land degradation due to long-term soil erosion [5], such as Africa [6], Europe [7][8][9], South America [10] and Asia [11].Cropped land is particularly vulnerable to erosion due to the exposure of bare soil for lengthy periods and the disturbance of the soil structure during farming operations, such as tillage [12].Long-term water erosion results in the loss of organic matter and, as a consequence, the loss of production and the elimination of fertile land [13][14][15].As a result of long-term water erosion, over the last 40 years, about 30% of the world's cropland has become unproductive, and much of it has been abandoned [5,16].Each year, an estimated 10 million ha of cropland worldwide are abandoned due to the loss of productivity caused by water erosion [17][18][19].This in turn threatens food security and poverty, as well [20,21], and affects the sustainable development of agriculture.A quantitative assessment of farmland yield loss caused by long-term water erosion is thus important.
The methods used to assess farmland yield loss are mainly long-term field experiments and model simulations under soil loss scenarios.In long-term field experiments, the relationship between water erosion (amount of water erosion or erosion depth) and crop yield is evaluated with a dimensional curve [22].Gao et al. [23] conducted a long-term field experiment and established the relationship between erosion depth and the reduction of yield and biomass of the past 10 years by a dimensional curve.Larney et al. [24] set erosion depths (0, 5, 10, 15 and 20 cm) of surface soil, which were removed to represent different erosion severity for the past 16 years, and used multiple dimensional curves to find the relationship between water erosion and crop yield.The long-term field experiment method can more accurately simulate the dynamic change and the reduction of yield due to water erosion at present.However, it needs long-term measured data in the field, and there are difficulties to characterize the relationship of yield loss with long-term water erosion by limited field experiments [25].
The model simulation method was used to simulate soil erosion and productivity under soil loss scenarios based on the natural, process-based simulation model [26].Ye and Van Ranst [26] used land degradation scenarios (the no degradation scenario, the current scenario and the double degradation scenario) to quantify the long-term water erosion impact on food crops based on the web-based land evaluation system (WLES) model.However, the yield loss they simulate considered little the cumulative amount of water erosion.Other studies [22,23] have found that with a longer planting time, there would be a cumulative increase in the amount of water erosion and an accelerated reduction in crop yield.Therefore, the combination of the WLES model and land degradation scenarios is not suitable for expressing the relationship between the cumulative amount of water erosion and the loss in productivity.
In order to assess the effect of long-term water erosion on a large or global scale, an effective method with the 3D (three-dimensional) surface [27] considering accumulated soil loss scenarios is needed.The method comes up with three elements, which are time, the cumulative amount of water erosion and losses in agricultural productivity.The 3D surface is able to quantify the relationship by three elements, while the 2D curve can only express two elements, which are soil erosion and productivity losses [24,28].
This paper proposes a quantitative method with the 3D surface to calculate productivity loss due to long-term water erosion at a large scale with a lack of field experiments.This paper selected the agricultural policy environmental extender (APEX) model with localization parameters to estimate soil erosion and its effect on crop.After that, an erosion variable-controlled method was used to consider the effect of the cumulative amount of water erosion and the developed 3D surface to (come up with three elements, which are time, the cumulative amount of water erosion and productivity losses) based on the logistic curve model [28] to study the relationship between long-term erosion and productivity loss.The 3D surface is important for understanding the effects of long-term erosion on productivity.

Study Area
This paper chose the middle and upper reaches of the Huaihe River watershed as the study area, as shown in Figure 1.The Huaihe River watershed is one of the seven major river watersheds in China.It flows from the west to the east, bordering the Huanghe River (Yellow River) in the north and the Yangtze River in the south.Starting in the Tongbai Mountains in Henan province, its waterway is about 1000 km long, and its catchment area is about 1.9 ˆ10 5 km 2 [29].The area of its cropland is 120,542 km 2 , and the cropland is characterized by extreme slopes covering 10,413 km 2 of the overall cropland area.Water erosion rates as high as 35 Mg¨ha ´1¨year ´1 occur across nearly 22% of the watershed area.Inappropriate land use practices, such as extensive cultivation, cultivating on steep slopes (over 27%) and deforestation, in combination with the lack of conservation practices and mismanaged construction projects, aggravate water-induced erosion [30].Average annual precipitation is 728 mm; the yearly average maximum daily temperature is 20.6 ˝C; and the yearly average minimum daily temperature is 10.0 ˝C.The main crops are maize and rice.Based on HydroSHEDS (WWF, Washington, DC, USA), the study area can be divided into 21 small watershed units and 68 subarea units.
Sustainability 2016, 8, x 3 of 17 mismanaged construction projects, aggravate water-induced erosion [30].Average annual precipitation is 728 mm; the yearly average maximum daily temperature is 20.6 °C; and the yearly average minimum daily temperature is 10.0 °C.The main crops are maize and rice.Based on HydroSHEDS (WWF, Washington, DC, USA), the study area can be divided into 21 small watershed units and 68 subarea units.The APEX model was developed to extend the Environmental Policy Integrated Climate (EPIC) model [31] capabilities to whole farms and small watersheds, based on watershed and subarea units [32].According to the HydroSHEDS basin [33], the world can be divided into a subarea global scale to fit the APEX model precision requirement.The most important data for each subarea, such as weather and environment, were used as input data to simulate the characteristics of each watershed.

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Numerical simulation of the natural geography process: Natural, process-based simulation models can help to interpret complex natural geography process, environmental evolution and policy [34,35].The agricultural policy environmental extender (APEX) model was developed to estimate soil productivity as affected by erosion, and it simulates approximately eighty crops with a crop growth model using unique parameter values for each crop [34,36].The APEX model was constructed to evaluate various land management strategies considering sustainability, erosion (wind, sheet and channel), economics, water supply and quality, soil quality, plant competition, weather and pests [34].Based on localization parameters and localization data, the APEX model can effectively simulate the natural environment process of the study area.Therefore, this paper selected the APEX model with localization parameters to estimate soil erosion and its effect on crops.

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Relationship between erosion and productivity: Currently, field experiments find that there is a cumulative effect on productivity with long-term water erosion [22,23].However, the models with soil loss scenarios consider little of the cumulative amount of water erosion.Therefore, this paper considered all water erosion scenarios from the most severe erosion scenario to the moderate to no erosion scenarios and used the yield under the no erosion scenario as the optimal yield, as well as the other yield under erosion scenarios as the erosion yield.The residual between the erosion yield and optimal yield is the yield loss by The APEX model was developed to extend the Environmental Policy Integrated Climate (EPIC) model [31] capabilities to whole farms and small watersheds, based on watershed and subarea units [32].According to the HydroSHEDS basin [33], the world can be divided into a subarea global scale to fit the APEX model precision requirement.The most important data for each subarea, such as weather and environment, were used as input data to simulate the characteristics of each watershed.

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Numerical simulation of the natural geography process: Natural, process-based simulation models can help to interpret complex natural geography process, environmental evolution and policy [34,35].The agricultural policy environmental extender (APEX) model was developed to estimate soil productivity as affected by erosion, and it simulates approximately eighty crops with a crop growth model using unique parameter values for each crop [34,36].The APEX model was constructed to evaluate various land management strategies considering sustainability, erosion (wind, sheet and channel), economics, water supply and quality, soil quality, plant competition, weather and pests [34].Based on localization parameters and localization data, the APEX model can effectively simulate the natural environment process of the study area.Therefore, this paper selected the APEX model with localization parameters to estimate soil erosion and its effect on crops.

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Relationship between erosion and productivity: Currently, field experiments find that there is a cumulative effect on productivity with long-term water erosion [22,23].However, the models with soil loss scenarios consider little of the cumulative amount of water erosion.Therefore, this paper considered all water erosion scenarios from the most severe erosion scenario to the moderate to no erosion scenarios and used the yield under the no erosion scenario as the optimal yield, as well as the other yield under erosion scenarios as the erosion yield.The residual between the erosion yield and optimal yield is the yield loss by erosion.In each year, the yield loss by erosion was affected by the cumulative soil erosion.This study is concerned with the productivity losses in particular years compared to the productivity in a scenario of no water erosion.Therefore, the length of time that the farmland had been planted was not considered.

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Logistic fitness model: The logistic curve model could quantify the relationship of hazard and loss [28].However, curves will lose the information when there are three elements, such as time, the cumulative amount of soil erosion and losses in agricultural productivity.The 3D surface was able to quantify the relationship by three elements [37], while the 2D logistic curve could only express two [24,28].Therefore, this paper selected the 3D surface to quantify the long-term water erosion impact on food crops.
The study consists of three basic steps: (1) build the APEX model with the localization parameters and the localization data of the study area; (2) based on the erosion variable-controlled method, the long-term relationship between erosion and loss of productivity is constructed; (3) the cumulative years, the cumulative amount of water erosion and the loss of productivity are used to build the 3D surfaces expressed as water erosion intensity-cumulative years-yield loss.The research framework is described in Figure 2.
Sustainability 2016, 8, x 4 of 17 erosion.In each year, the yield loss by erosion was affected by the cumulative soil erosion.This study is concerned with the productivity losses in particular years compared to the productivity in a scenario of no water erosion.Therefore, the length of time that the farmland had been planted was not considered.
• Logistic fitness model: The logistic curve model could quantify the relationship of hazard and loss [28].However, curves will lose the information when there are three elements, such as time, the cumulative amount of soil erosion and losses in agricultural productivity.The 3D surface was able to quantify the relationship by three elements [37], while the 2D logistic curve could only express two [24,28].Therefore, this paper selected the 3D surface to quantify the long-term water erosion impact on food crops.
The study consists of three basic steps: (1) build the APEX model with the localization parameters and the localization data of the study area; (2) based on the erosion variable-controlled method, the long-term relationship between erosion and loss of productivity is constructed; (3) the cumulative years, the cumulative amount of water erosion and the loss of productivity are used to build the 3D surfaces expressed as water erosion intensity-cumulative years-yield loss.The research framework is described in Figure 2. We selected the middle and upper reaches of the Huaihe River watershed as a case and used the agricultural policy environmental extender (APEX) model [38], which is an emerging tool for landscape and watershed environmental analyses, as a tool to simulate erosion and productivity, and we used the IPCCAR5 scenario 2P6 data [39,40] as the long-term future climate data in the APEX We selected the middle and upper reaches of the Huaihe River watershed as a case and used the agricultural policy environmental extender (APEX) model [38], which is an emerging tool for landscape and watershed environmental analyses, as a tool to simulate erosion and productivity, and we used the IPCCAR5 scenario 2P6 data [39,40] as the long-term future climate data in the APEX model.Then, we built accumulated soil loss scenarios, with different soil and water conservation measure factors to represent accumulated erosion severity, and we used 3D surfaces to express the relationship between the cumulative amount of water erosion, the loss of productivity and cumulative years; thus, we quantitatively assessed the loss of productivity due to long-term water erosion.

Data
Table 1 shows the contents of the database used for the global-scale water erosion evaluation.According to the Paris Agreement [50], the expected key result was an agreement to set a goal of limiting global warming to less than 2 degrees Celsius ( ˝C), and the IPCCAR5 scenario 2P6 [38,39] was similar to the Paris Agreement.Therefore, the IPCCAR5 scenario 2P6 was adopted as the meteorological data, and productivity losses for 2000-2099 were simulated based on the year 2000.
Based on the APEX model requirements, the most important environmental data (such as slope, elevation) and field management data were used to simulate each watershed and subarea as Table 2.The basin number adopted a HydroSHEDS number.The soil number was adopted from the ID provided by FAO and ISRIC.The field management data were adopted from the average agro-meteorological station data for the study area; the sowing time is 8 June; the harvest time is 15 September; and the farming system is sole cropping.Meteorological data were adopted from the IPCCAR5 scenario 2P6 in the study area.The daily maximum temperature, daily minimum temperature, daily relative humidity, daily solar radiation and daily wind speed from 2000-2099 were input into the APEX model.
The study area includes four dominant soil types [43], Mollic Gleysols (4339), Eutric Gleysols (4329, 4326), Calcaric Gleysols (4319), lithosols with Chromic Cambisols (3085) [43].The soil property stratification data are shown in Table 3.The APEX model can be subdivided into nine separate components defined as weather, hydrology, soil erosion, nutrients, soil temperature, plant growth, tillage, plant environment control and economics [32,51].The APEX component for water-induced erosion simulates erosion caused by rainfall and runoff and by irrigation (sprinkler and furrow).To simulate rainfall/runoff erosion, APEX contains seven equations: the USLE [52], the Onstad-Foster modification of the USLE [53], RUSLE [54], the MUSLE [32], two recently-developed variations of MUSLE and an MUSLE structure that accepts input coefficients.The USLE depends strictly on rainfall as an indicator of erosive energy.However, it provides only annual estimates.The MUSLE and its variations use only runoff variables to simulate erosion and sediment yield.Runoff variables increase the prediction accuracy, and the MUSLE is calculated by ( 1) and (2).
X " 1.586 ˆpQ ˆqp q 0.56 ˆWSA 0.12 (2 where Y is the sediment yield in t¨ha ´1, EK is the soil erodibility factor, CVF is the crop management factor, PE is the erosion control practice factor, SL is the slope length and steepness factor, ROKF is the coarse fragment factor, Q is the runoff volume in mm, q p is the peak runoff rate in mm¨h ´1 and WSA is the watershed area in ha. The parameters and sensitivity parameters were used for the study area.The parameters and the value of water erosion are the curve number for moisture Condition 2 (CN2), which is 65, the curve number index coefficient (CNIC), which is 1.5, conservation practice factor (PEC), which is 0.72, and the peak runoff rate rainfall energy adjustment factor (APM), which is 0.1 [30].

Soil and Water Conservation Measure Scenarios
We used an erosion control variable method to study the relationship between long-term erosion and productivity loss.Only the erosion was changed, and the remaining factors (such as weather, field management practices, fertilizer) were controlled and unchanged.The conservation measure factor was a number between 0 and 1.We set 12 conservation measures factors (0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1) as soil and water conservation measure scenarios to represent different soil and water conservation measures.The best soil and water conservation measure scenario represents the minimum soil erosion where the soil and water conservation measures factor is 0.01; the worst conditions and water conservation measures represent the maximum amount of potential soil erosion, where the soil and water conservation measures factor is 1.

Soil Erosion and Loss of Productivity
Based on the soil and water conservation scenarios, we used cumulative years, erosion intensity and loss of productivity to construct the surfaces.

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Productivity simulated by the APEX model: A single model is used in APEX for simulating all of the crops considered (about 100).Each crop has unique values for the model parameters.APEX is capable of simulating growth for both annual and perennial crops.Annual crops grow from planting date to harvest date or until the accumulated heat units equal the potential heat units for the crop [55].
Soil erosion has affected the soil supply of nitrogen (N), phosphorus (P), potassium (K) and the depth of the soil layer [34,55,56].Crop yield may be reduced through nutrient stress and temperature stress because of soil erosion [57,58].

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Loss of yield index calculation: The yield loss index is defined here as the percentage of yield loss compared to the yield under the best conservation measure scenarios, and it is calculated using Equation (3).
where Lr i is the yield loss index for crops due to soil loss under i conservation measure scenario, y i is the yield under the i conservation measure scenario and maxpyq is the yield under the best conservation measure scenario.

‚ Water erosion intensity calculation:
Water erosion is the cumulative soil erosion over cultivation years, and the water erosion intensity is the proportion of water erosion under the worst soil conservation measure scenario.It was calculated using Equation (4).
where Se i is water erosion under i conservation measure scenario, Se i is the water erosion after cumulative j year, Se max is the water erosion under the worst conservation measure scenario and WE i is the water erosion intensity under the i conservation measure scenario.

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Cumulative loss 3D surface fitting: From the basic idea section, we know that the 3D surface with the logistic model is better than the 2D logistic curve [28].This paper established a cumulative loss 3D surface model by adding the cumulative years as the time dimension variable, calculated using Equation (5).
LR " pa{ p1 `b ˆexp pc ˆWE i qq ´a{ p1 `bqq pa{ p1 `b ˆexp pcqq ´a{ p1 `bqq ˆpd ˆpi ´eq 2 `f q (5) where a, b, c, d, e, f are the parameters of the surface model and LR is the loss yield index of crop due to soil loss under all kinds of conservation measures.i is the simulation of the cultivation i year, and WE i is the accumulated soil loss after cultivation i year.As can be seen, in any year, the cultivation relationship is in the logistic curve shape.

Statistical Analysis
To confirm the significance of the results obtained by the cumulative loss 3D surface and the loss curve, we performed a paired t-test for all small watershed units [59][60][61][62].The test statistic for the paired samples t-test, denoted t, follows the same formula as the one-sample t-test, calculated using Equations ( 6) and (7).

t "
x di f f ´0 s x (6) where: where x di f f is the sample mean of the differences; n is the sample size (i.e., the number of all small watershed units); s di f f is the sample standard deviation of the differences; s x is the estimated standard error of the mean (s/sqrt(n)).
The calculated t-value is then compared to the critical t value with df = n ´1 from the t distribution table for a chosen confidence level.If the calculated t value is greater than the critical t value, then we conclude that the means are significantly different [62].

Productivity Simulation and Verification
Summer maize was selected to evaluate its productivity in different water erosion scenarios and to provide information on the status of regional productivity.The simulated summer maize yield of Watershed 434298 (Lushan) and the observed yield of the Huaihe River Agricultural Meteorological Station were compared to calibrate the simulated yield as in Figure 3.
where d iff x is the sample mean of the differences; n is the sample size (i.e., the number of all small watershed units); diff s is the sample standard deviation of the differences; x s is the estimated standard error of the mean (s/sqrt(n)).
The calculated t-value is then compared to the critical t value with df = n − 1 from the t distribution table for a chosen confidence level.If the calculated t value is greater than the critical t value, then we conclude that the means are significantly different [63].

Productivity Simulation and Verification
Summer maize was selected to evaluate its productivity in different water erosion scenarios and to provide information on the status of regional productivity.The simulated summer maize yield of Watershed 434298 (Lushan) and the observed yield of the Huaihe River Agricultural Meteorological Station were compared to calibrate the simulated yield as in Figure 3.

Comparison of Cumulative Loss 3D Surfaces and 2D Curves
The key to comparing the 2D curve with the cumulative loss 3D surface is whether yield loss is affected by the cultivation year during long-term water erosion.For all watersheds, paired t-tests were used for testing the R-square of the 3D surface and 2D curve (Tables 4 and 5).  Figure 3 shows the region's productivity.In this region, the measured and simulated average yields were 6.58 t/ha and 6.87 t/ha, respectively, in 2007-2012.In 2007-2012 and 2008-2012, the RMSE was 1.53 and 0.78, respectively.Because the IPCCAR5 meteorological data had observed values for 1971-2004 and simulated values for 2005-2099, the simulated results contain an error, but the results still reflect the productivity level in the region.

Comparison of Cumulative Loss 3D Surfaces and 2D Curves
The key to comparing the 2D curve with the cumulative loss 3D surface is whether yield loss is affected by the cultivation year during long-term water erosion.For all watersheds, paired t-tests were used for testing the R-square of the 3D surface and 2D curve (Tables 4 and 5).From Tables 4 and 5, the mean R-square of the curve method is significantly lower than that of the surface method (paired samples t-test, t = 13.813,df = 20, 2-tail Significance (p) < 0.05) (Table 5).Thus, the mean R-square of the curve method and that of the surface method are significant.Therefore, fitting by the surface has a much higher R-square than fitting by the curve for all watersheds.The loss curve is fragmented, and the loss surface is continuous.The loss curve can be understood as a certain point in time on the surface.The loss surface has a more accurate meaning and expression.The surface fitting can thus more accurately reflect the impact of long-term water erosion on productivity.

Cumulative Loss 3D Surface from Long-Term Accumulated Water Erosion
We used the accumulated loss data of all of the hydrological response units in the Huaihe River watershed to construct a representation of the cumulative loss surface.Figure 4a shows a long-term loss rate of less than 100%, and Figure 4b shows a long-term loss rate of up to 100%.From Tables 4 and 5, the mean R-square of the curve method is significantly lower than that of the surface method (paired samples t-test, t = 13.813,df = 20, 2-tail Significance (p) < 0.05) (Table 5).Thus, the mean R-square of the curve method and that of the surface method are significant.Therefore, fitting by the surface has a much higher R-square than fitting by the curve for all watersheds.The loss curve is fragmented, and the loss surface is continuous.The loss curve can be understood as a certain point in time on the surface.The loss surface has a more accurate meaning and expression.The surface fitting can thus more accurately reflect the impact of long-term water erosion on productivity.

Cumulative Loss 3D Surface from Long-Term Accumulated Water Erosion
We used the accumulated loss data of all of the hydrological response units in the Huaihe River watershed to construct a representation of the cumulative loss surface.Figure 4a shows a long-term loss rate of less than 100%, and Figure 4b shows a long-term loss rate of up to 100%.The 3D surfaces "water erosion intensity-cumulative years-loss rate" fit well.The R-square is 0.54 in Figure 4a and 0.78 in Figure 4b, both of which reflect a strong relationship between accumulated water and soil loss, tillage years and crop yield loss. Figure 4a shows that the maximum potential loss rate for one year is 9%.The maximum potential loss rate for 100 years is 35%. Figure 4b shows that the maximum potential loss rate for one year is 40%, and the maximum potential loss rate for 100 years is 100%.Both figures indicate that long-term water and soil loss have an accumulated The 3D surfaces "water erosion intensity-cumulative years-loss rate" fit well.The R-square is 0.54 in Figure 4a and 0.78 in Figure 4b, both of which reflect a strong relationship between accumulated water and soil loss, tillage years and crop yield loss. Figure 4a shows that the maximum potential loss rate for one year is 9%.The maximum potential loss rate for 100 years is 35%. Figure 4b shows that the maximum potential loss rate for one year is 40%, and the maximum potential loss rate for 100 years is 100%.Both figures indicate that long-term water and soil loss have an accumulated effect on crop yield loss. Losses in crop yield increase with increasing tillage years, and the loss is irreversible.
The minimum loss for one year is almost zero, and the minimum accumulated productivity loss is 0.2%, indicating that farmland productivity can be significantly maintained with soil and water conservation measures.Inappropriate tillage practices may make the region's productivity loss rise to 100% in the future 100 years.However, the loss can be reduced with proper soil and water conservation measures, making the case for the importance of soil and water conservation practices on agricultural productivity.

Spatial Differences in Long-Term Water Erosion Impacts on Productivity
In order to understand the spatial differences long-term loss, the 3D surfaces "water erosion intensity-cumulative years-loss rate" were obtained for all of the watersheds in the study area.The basins located in the study area are as shown in Figure 5. to 100% in the future 100 years.However, the loss can be reduced with proper soil and water conservation measures, making the case for the importance of soil and water conservation practices on agricultural productivity.

Spatial Differences in Long-Term Water Erosion Impacts on Productivity
In order to understand the spatial differences in long-term loss, the 3D surfaces "water erosion intensity-cumulative years-loss rate" were obtained for all of the watersheds in the study area.The basins located in the study area are as shown in Figure 5. Figure 5 shows a pattern difference in potential productivity losses for all basins located in the study area.The southern portion of the study area exhibits higher loss, especially Basins 434284 and 434282, where the largest potential productivity loss rate can reach 100%.The northern portion of the study area, which is relatively more suitable for crops, exhibits less loss.This indicates that long-term water and soil loss have a significant impact on productivity loss, particularly in the southern region of the study area.The cumulative loss 3D surface can in this way reflect the differences in potential productivity loss.Figure 5 shows a pattern difference in potential productivity losses for all basins located in the study area.The southern portion of the study area exhibits higher loss, especially Basins 434284 and 434282, where the largest potential productivity loss rate can reach 100%.The northern portion of the study area, which is relatively more suitable for crops, exhibits less loss.This indicates that long-term water and soil loss have a significant impact on productivity loss, particularly in the southern region of the study area.The cumulative loss 3D surface can in this way reflect the differences in potential productivity loss.

Significance of Cumulative Loss 3D Surface
Long-term water erosion has resulted in reduced farmland productivity and the reduction or even disappearance of fertile land, affecting the sustainable development of agriculture.The relationship between long-term water erosion and productivity under different soil conservation measure scenarios can more explicitly guide farming at a macroscopic scale, increasing the importance of soil and water conservation measures.
Using the accumulated loss 3D surface, we can assess the potential loss of productivity under long-term water erosion, the maximum loss of productivity and the time when the maximum loss happens.For a watershed that would reach a 100% loss rate within 100 years, planning structures should be converted to grass or forest.The earlier the 100% loss occurs, the more vulnerable the watershed productivity becomes due to soil erosion and the more urgent are reforestation measures.For a watershed that would not reach 100% loss within 100 years, the lower the loss proportion is, the more suitable the watershed becomes for planting.Higher loss proportions mean that greater investment in soil and water conservation measures is needed to improve soil and water conservation capacity.Using the accumulated loss 3D surface can effectively predict the future productivity due to long-term water erosion, quantitatively assess regional vulnerability and guide regional investment in conservation projects.
Inappropriate conservation measures have the potential of resulting in 100% productivity losses in the mountainous areas of the southern study area over the next 100 years, and this will eventually lead to desertification.This is consistent with the studies of areas in the Mediterranean at the same latitude [63].de la Rosa et al. [63] used an ImpelERO model to study soil vulnerability caused by soil erosion in Western Europe.They found that, the maximum impact according to the long-term productivity reduction (97%) was shown for the Odiaxere-Albufeira site in the Mediterranean region and for the 2100 time horizon [63].According to the EEA [64], the EU Mediterranean countries have severe soil erosion problems, which can reach the highest levels and lead to desertification.However, this loss can be reduced by soil and water conservation measures.

The Validation and Uncertainty of the Cumulative Loss 3D Surface
In order to verify the cumulative loss, we used long-term field experiment data from the published literature.However, some studies did not report on cumulative loss.Zhou et al. [65] conducted a field experiment in 2005-2013 in Heilongjiang.Comparing erosion conditions at an erosion of 20 cm with 30 cm, the yield loss at 30 cm of soil erosion was for the most part greater than what it was at 20 cm of soil erosion.This suggests that the more the erosion, the greater the yield loss. Reductions in maize yield in the first five study years were significant.However, after the first five years, reductions became less consistent; and in this case, no cumulative erosion was found, and yield losses did not increase.Gao et al. [23] studied the relationship between soil erosion and time in black soil and found that with the accumulation of time, erosion continues to accumulate.The experiments of Zhou et al. [65] did not set the cumulative erosion scenarios and resulted in loss without accumulation.For promotion, this study considered the erosion in the second year based on the first year's erosion, which conformed to the real situation better.Some research has suggested that soil management contributes to reducing the soil losses and maintaining productivity, such as reduced tillage, mulching, appropriate cover crops or organic amendments [66][67][68][69][70][71].Some research [22,23] has suggested that with a lengthened cultivation period, productivity would be reduced and erosion would increase, meaning that there is a cumulative effect on productivity.In a study of Cambisol soil, Tenberg et al. [22] found a relationship between yield and the long-term erosion, as shown in Figure 6a.In this study, we compared our results by using the lowest level soil and water conservation measure scenarios to assess the relationship between time and yield, as shown in Figure 6b.For the same soil, different water conservation measure scenarios lead to different levels of soil erosion intensity.The worst soil and water conservation measure scenario is similar to 18% Sombrite, which means an effective vegetable coverage of 30%.The loss of yield in the Cambisol at Itapiranga [22] is found to be similar to that in Watershed 434298 in the study area, both experiencing a significant loss in productivity due to long-term water erosion.Meanwhile, the similar soil and water conservation measures would result in similar losses of productivity.

Conclusions
Long-term water erosion results in reduced farmland productivity and the reduction or even disappearance of fertile land, affecting the sustainable development of agriculture.The relationship between the soil erosion and productivity under different soil conservation measure scenarios can be made more explicit to guide farming on a macroscopic scale.This paper used an erosion variable-controlled method to obtain soil and water conservation measure scenarios to study the relationship between soil erosion, loss of productivity and cumulative years.It was concluded that fitting by the 3D surface can significantly and more accurately reflect the impact on productivity due to long-term water erosion than fitting by a curve.What is more, the accumulative loss 3D surface can reflect regional differences in potential productivity loss.The research can help with understanding the effects of long-term erosion on productivity.
This article seeks to add to the current body of knowledge on the impact of long-term soil erosion on productivity through a vulnerability assessment.The methodology is limited by the local parameter for the APEX model on large scales, so further research can focus on parameter adjustment.Additional research could focus on the maximum productivity loss caused by water erosion for different conservation measure projects.Research could also focus on the point when the maximum loss would occur under a current soil and water conservation measure scenario.For the same soil, different water conservation measure scenarios lead to different levels of soil erosion intensity.The worst soil and water conservation measure scenario is similar to 18% Sombrite, which means an effective vegetable coverage of 30%.The loss of yield in the Cambisol at Itapiranga [22] is found to be similar to that in Watershed 434298 in the study area, both experiencing a significant loss in productivity due to long-term water erosion.Meanwhile, the similar soil and water conservation measures would result in similar losses of productivity.

Conclusions
Long-term water erosion results in reduced farmland productivity and the reduction or even disappearance of fertile land, affecting the sustainable development of agriculture.The relationship between the soil erosion and productivity under different soil conservation measure scenarios can be made more explicit to guide farming on a macroscopic scale.This paper used an erosion variable-controlled method to obtain soil and water conservation measure scenarios to study the relationship between soil erosion, loss of productivity and cumulative years.It was concluded that fitting by the 3D surface can significantly and more accurately reflect the impact on productivity due to long-term water erosion than fitting by a curve.What is more, the accumulative loss 3D surface can reflect regional differences in potential productivity loss.The research can help with understanding the effects of long-term erosion on productivity.
This article seeks to add to the current body of knowledge on the impact of long-term soil erosion on productivity through a vulnerability assessment.The methodology is limited by the local parameter for the APEX model on large scales, so further research can focus on parameter adjustment.Additional research could focus on the maximum productivity loss caused by water erosion for different conservation measure projects.Research could also focus on the point when the maximum loss would occur under a current soil and water conservation measure scenario.

Figure 1 .
Figure 1.Study area (the basin ID was adopted from HydroSHEDS).

Figure 1 .
Figure 1.Study area (the basin ID was adopted from HydroSHEDS).

Figure 2 .
Figure 2. Research framework: (a) build the agricultural policy environmental extender (APEX) model with localization parameters and localization data of the study area in order to simulate long-term water erosion and productivity; (b) use the soil and water conservation measures' scenarios to get a group of samples of long-term water erosion intensity and yield and build the relationship between cumulative years, cumulative soil erosion and the yield loss, which is the residual between the erosion yield and optimal yield; (c) use the 3D surface based on the logistic model to quantify the loss of productivity affected by long-term water erosion.

Figure 2 .
Figure 2. Research framework: (a) build the agricultural policy environmental extender (APEX) model with localization parameters and localization data of the study area in order to simulate long-term water erosion and productivity; (b) use the soil and water conservation measures' scenarios to get a group of samples of long-term water erosion intensity and yield and build the relationship between cumulative years, cumulative soil erosion and the yield loss, which is the residual between the erosion yield and optimal yield; (c) use the 3D surface based on the logistic model to quantify the loss of productivity affected by long-term water erosion.

Figure 3
Figure 3 shows the region's productivity.In this region, the measured and simulated average yields were 6.58 t/ha and 6.87 t/ha, respectively, in 2007-2012.In 2007-2012 and 2008-2012, the RMSE was 1.53 and 0.78, respectively.Because the IPCCAR5 meteorological data had observed values for 1971-2004 and simulated values for 2005-2099, the simulated results contain an error, but the results still reflect the productivity level in the region.

Figure 4 .
Figure 4. Cumulative loss surface for long-term accumulated water erosion in the Huaihe River basin (a) 4342980; (b) 4342840.

Figure 5 .
Figure 5. Spatial differences in the long-term accumulated loss for all basins in the study area.

Figure 5 .
Figure 5. Spatial differences in the long-term accumulated loss for all basins in the study area.

Table 2 .
Major natural geographic features and attributes of the field management of maize.
* Note: The soil ID was adopted from FAO and ISRIC.

Table 3 .
Soil properties used in the simulation for the study site.

Table 4 .
Paired samples statistics of the surface and curve.

Table 5 .
Paired samples Student test.

Table 4 .
Paired samples statistics of the surface and curve.

Table 5 .
Paired samples Student test.